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!-modality | https://ncatlab.org/nlab/source/%21-modality | +-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### Type theory
+-- {: .hide}
[[!include type theory - contents]]
=--
#### Modalities, Closure and Reflection
+-- {: .hide}
[[!include modalities - contents]]
=--
=--
=--
#Contents#
* table of contents
{:toc}
## Idea
{#Idea}
In full [[linear logic]] [[Girard 1987](#Girard1987)] and more generally in [[linear type theory]] there is assumed a (comonadic) [[modality]] traditionally denoted "!" whose role is to model the "underlying" classical [[intuitionistic type theory|(intuitionistic) type]] of a given [[linear type]].
This is alternatively called the "exponential modality" (for good reasons discussed [below](#TheExponentialCondition)), or "storage modality" (as it allows to duplicate and hence "store" otherwise linear data) and sometimes pronounced "of course" (intended alongside "necessarily" and "possibly" as used in [[modal logic]]) or even "bang" (just in reference to the symbol "!").
In classical linear logic (meaning with involutive [[de Morgan duality]]), the de Morgan dual of "!" is denoted "?" and then sometimes pronounced "why not".
Today it is understood (cf. [Melliès 2009](#Melliès09), [p. 36](/nlab/files/Mellies-CategoricalSemanticsLinear.pdf#page=36))
that the exponential modality is best and generally to be thought as as, first of all a *[[comonad]]* on [[linear types]] ([Seely 1989 §2](#Seely89), [dePaiva 1989](#dePaiva89), [Benton, Bierman, de Paiva and Hyland 1992](#BBPH92)) which, secondly, is [induced](monad#RelationBetweenAdjunctionsAndMonads) by an adjunction to classical (meaning here: non-linear [[intuitionistic type theory|intuitionistic]]) types with special [[monoidal functor|monoidal]] properties ([Seely 1989 §2](#Seely89), [Bierman 1994 pp. 157](#Bierman94), [Benton 1995](#Benton95)).
In a most elementary and illuminating example (which may not have received the attention it deserves, cf. the history at *[[quantum logic]]*):
* classical intuitionistic types form (are interpreted in) the category of [[Sets]]
$ClaTypes \,\coloneqq\, Set$
* linear types form (are interpreted in) the category of [[vector spaces]] (cf. [Murfet 2014](linear+logic#Murfet14))
$LinTypes \,\coloneqq\, Vect_{\mathbb{K}}$
and the [[adjoint functors]] in question are
* forming the [[linear span]] of a set
$$
\array{
Set &\overset{\mathrm{Q}}{\longrightarrow}& Vect
\\
W &\mapsto& \oplus_W \mathbb{1}
}
$$
* forming the [[underlying set]] of a [[vector space]]:
$$
\array{
Vect &\overset{\mathrm{C}}{\longrightarrow}& Set
\\
\mathscr{V} &\mapsto& Hom(\mathbb{1}, \, \mathscr{V})
}
$$
in which case the exponential modality acts by sending a vector space to the linear span of its underlying set
\[
\array{
Vect &\overset{\;\; ! \;\;}{\longrightarrow}& Vect
\\
\mathscr{V} &\mapsto& \underset{\mathscr{V}}{\bigoplus} \mathbb{1}
\,.
}
\]
One sees that this construction takes ([[direct sums|direct]]) sums to ([[tensor product of vector spaces|tensor]]) products
$$
!(\mathscr{V} \oplus \mathscr{V}')
\;\simeq\;
(! \mathscr{V}) \otimes (! \mathscr{V}')
$$
and zero ([[zero object|objects]]) to unit ([[unit object|objects]])
$$
! 0
\;\simeq\;
\mathbb{1}
$$
as expected of any [[exponential map]], whence the name "exponential modality".
This example is the 0-sector of the more famous *[[stabilization]]* ([[(infinity,1)-adjoint functor|$\infty$-]])adjunction $(\Sigma_+^\infty \dashv \Omega_+^\infty)$ between (the [[homotopy theory]] of) [[topological spaces]] and (the [[stable homotopy theory]] of) [[module spectra|module]] [[spectra]] equipped with the [[symmetric smash product of spectra]] (given by forming [[suspension spectra]] $X \mapsto \Sigma_+^\infty X$), which points to a deeper origin of the "exponential modality", see [below](#RealizationInLinearHomotopyTypeTheory) and at *[[linear homotopy type theory]]* for more on this.
<center>
<img src="https://ncatlab.org/nlab/files/QuantClassExponential-230821.jpg" width="600"/>
</center>
## Categorical semantics
Following [Seely 1989](#Seely89), [dePaiva 1989](#dePaiva89), [Benton, Bierman, de Paiva and Hyland 1992](#BBPH92) it is common convention that "!" should be a [[comonad]] on the type system (and ? should be a [[monad]], see also at *[[monads in computer science]]*), but there is some room in which further axioms to impose.
The goal is to capture the syntactic rules allowing assumptions of the form $!A$ to be duplicated and discarded.
### Intuitionistic case
The definition from [Seely](#Seely89), adapted to the intuitionistic case and modernized, is:
+-- {: .num_defn}
###### Definition
Let $LinTypes$ be
* a [[cartesian monoidal category]], with [[Cartesian product]] "$\times$" and [[tensor unit]] the [[terminal object]] $\ast$.
which *in addition* carries the [[structure]] of
* a [[symmetric monoidal category]] with respect to a [[tensor product]] $\otimes$ with [[tensor unit]] $\mathbb{1}$.
A **Seely comonad** on $LinTypes$ is a [[comonad]] that is a [[strong monoidal functor|strong monoidal]] as a functor from cartesian monoidal structure $\times$ to the other monoidal structure $\otimes$:
$$
(LinTypes, \times)
\xrightarrow{ \; ! \; }
(LinTypes, \otimes)
$$
meaning that there are [[natural transformations]] of the form
\[
\label{StrongMonoidalPropertyOfExponential}
A,\, B \colon LinTypes
\;\;\;\;\;\;\;
\vdash
\;\;\;\;\;\;\;
!(A \times B)
\;\cong\;
(!A) \otimes (!B)
\]
and
\[
! \ast \;\;\simeq\;\; \mathbb{1}
\,.
\]
=--
(There is also an additional [[coherence]] [[axiom]] that should be imposed; see [Melliès 2009, section 7.3](#Melliès09).)
\begin{remark}\label{TheExponentialCondition}
**(the exponential condition)**
\linebreak
In [[linear logic]], the cartesian monoidal structure on linear types is often denoted "$\&$" ("[[additive conjunction]]"), in which case the condition (eq:StrongMonoidalPropertyOfExponential) reads
\[
!(A \& B)
\;\cong\;
(!A) \otimes (!B)
\,.
\]
But in key examples of categories of (the multiplicative fragment of) linear logic (such as [[Vect]], cf. [Murfet 2014](linear+logic#Murfet14)), the [[cartesian product]] is actually a [[biproduct]], hence a [[direct sum]], in which case the condition on the exponential modality is that
*it takes (direct) sums to (tensor) products*
\[
!(A \oplus B)
\;\cong\;
(!A) \otimes (!B)
\]
*it takes zero (objects) to unit (objects)*
\[
! 0
\;\cong\;
\mathbb{1}
\]
as befits any [[exponential map]] and may explain the choice of terminology here.
\end{remark}
This condition implies that the [[Kleisli category of a comonad|Kleisli category]] of $!$ (i.e. the category of cofree !-coalgebras) is cartesian monoidal. If $LinTypes$ is *[[closed monoidal category|closed]]* symmetric monoidal then the Kleisli category of a [[cartesian closed category]], which is a categorical version of the translation of intuitionistic logic into linear logic.
Of course, the above definition depends on the existence of the cartesian product. A different definition that doesn't require the existence of $\times$ was given by [Benton, Bierman, de Paiva, and Hyland](#BBPH92):
+-- {: .num_defn}
###### Definition
Let $LinTypes$ be a [[symmetric monoidal category]]; a **linear exponential comonad** on $LinTypes$ is a [[lax monoidal functor|lax monoidal]] comonad such that every cofree !-coalgebra naturally carries the structure of a [[comonoid object]] in the category of coalgebras (i.e. the cofree-coalgebra functor lifts to the category of comonoids in the category of coalgebras).
=--
It follows automatically that all !-coalgebras are comonoids, and therefore that the category of all !-coalgebras (not just the cofree ones) is cartesian monoidal. Note that for a comonad on a [[poset]], every coalgebra is free; thus the world of pure propositional "logic" doesn't tell us whether to consider the Kleisli category or the Eilenberg-Moore category for the translation.
A more even-handed approach is the following (see [Benton (1995)](#Benton95) and [Melliès (2009)](#Melliès09)), based on the observation that both Kleisli and Eilenberg-Moore categories are instances of adjunctions.
\begin{definition}
\label{LinearNonlinearAdjunction}
A **linear-nonlinear adjunction** is a [[monoidal adjunction]] $F \colon ClaTypes \rightleftarrows LinTypes \colon G$ in which $LinTypes$ is symmetric monoidal and $ClaTypes$ is cartesian monoidal. The induced !-modality is the induced [[comonad]] $F G$ on $LinTypes$.
\end{definition}
This includes both of the previous definitions where $M$ is taken respectively to be the Kleisli category or the Eilenberg-Moore category of !. Conversely, in any linear-nonlinear adjunction the induced comonad $F G$ can be shown to be a linear exponential comonad. Moreover, if $!$ is a linear exponential comonad on a symmetric monoidal category $LinTypes$ with finite products, then the cofree !-coalgebra functor is a right adjoint and hence preserves cartesian products; but the cartesian products of coalgebras are the tensor products in $C$, so we have $!(A\times B) \cong !A \otimes !B$, the Seely condition.
### Classical case
For "classical" linear logic, we want $LinTypes$ to be not just (closed) symmetric monoidal but $\ast$-[[star-autonomous category|autonomous]]. If an $\ast$-autonomous category has a linear exponential comonad $!$ one can derive a ? from the ! by [[de Morgan duality]], $?A = \big(!(A^*)\big)^*$. The resulting relationship between ! and ? was axiomatized in a way not requiring the de Morgan duality by [Blute, Cockett & Seely (1996)](#BluteCockettSeely96):
+-- {: .num_defn}
###### Definition
Let $LinTypes$ be a [[linearly distributive category]] with tensor product $\otimes$ and cotensor product $\parr$. A (!,?)-modality on $LinTypes$ consists of:
1. a $\otimes$-monoidal comonad ! and a $\parr$-comonoidal monad ?
1. ? is a !-strong monad, and ! is a ?-strong comonad
1. all free !-coalgebras are naturally commutative $\otimes$-comonoids, and all free ?-algebras are naturally commutative $\parr$-monoids.
=--
Here a functor $F$ is [[strong functor|strong]] with respect to a [[lax monoidal functor]] $G$ if there is a [[natural transformation]] of the form $F A \otimes G B \to F(A\otimes G B)$ satisfying some natural axioms, and we similarly require compatibility of the monad and comonad structure transformations. [BCS96](#BluteCockettSeely96) showed that if $LinTypes$ is in fact $\ast$-autonomous, it follows from the above definition that $?A = \big(!(A^*)\big)^*$, as expected.
## Examples
### Relation to Chu construction
\begin{theorem}
Suppose $F \colon M \rightleftarrows C : G$ is a linear-nonlinear adjunction (Def. \ref{LinearNonlinearAdjunction}), where $C$ is closed symmetric monoidal with finite limits and colimits, and $\bot\in C$ is an object. Then there is an induced linear-nonlinear adjunction $M \rightleftarrows Chu(C,\bot)$ where $Chu(C,\bot)$ is the [[Chu construction]] $Chu(C,\bot)$, which is $\ast$-autonomous with finite limits and colimits. Hence $Chu(C,\bot)$ admits a !-modality.
\end{theorem}
\begin{proof}
The embedding of $C$ in $Chu(C,\bot)$ as $A \mapsto (A, [A,\bot], ev)$ is coreflective: the coreflection of $(B^+, B^-, e_B)$ is $(B^+, [B^+,\bot], ev)$. Moreover, this subcategory is closed under the tensor product of $Chu(C,\bot)$, i.e. the embedding $C\hookrightarrow Chu(C,\bot)$ is strong monoidal, hence the adjunction is a monoidal adjunction. Therefore, the composite adjunction $M \rightleftarrows C \rightleftarrows Chu(C,\bot)$ is again a linear-nonlinear-adjunction.
\end{proof}
Note that this theorem is a direct consequence of the work of [[Yves Lafont]] and [[Thomas Streicher]] on game semantics using the Chu construction, together with Benton's reformulation of linear logic in terms of monoidal adjunctions. Since a Chu construction is $\ast$-autonomous, this !-modality implies a dual ?-modality.
\begin{corollary}
If $C$ is a cartesian closed category with finite limits and colimits and $\bot\in C$ is an object, then there is a linear-nonlinear adjunction $C \rightleftarrows Chu(C,\bot)$, and hence $Chu(C,\bot)$ admits a !-modality.
\end{corollary}
\begin{proof}
Apply the previous theorem to the identity adjunction $C\rightleftarrows C$.
\end{proof}
Note that the !-modality obtained from the corollary is [[idempotent comonad|idempotent]], while that obtained from the theorem is idempotent if and only if the original one was. Other ways of constructing !-modalities, such as by cofree coalgebras, may produce examples that are not idempotent.
### Realization in linear homotopy type theory
{#RealizationInLinearHomotopyTypeTheory}
In [[dependent linear homotopy type theory]] the "linear-nonlinear adjunction" is naturally identified ([Ponto & Shulman (2012), Ex. 4.2](#PontoShulman12), see also [Schreiber (2014), Sec. 4.2](#Schreiber14)) with the [[stabilization]] [[adjoint functor|adjunction]] between [[homotopy types]] and [[stable homotopy types]] ([Riley (2022), Prop. 2.1.31](#Riley22Thesis)), whose [[left adjoint]] (forming [[suspension spectra]] $\Sigma^\infty_+$) is equivalently given (cf. [Riley (2022), Rem. 2.4.13](#Riley22Thesis)) by sending $B \,\colon\, Type$ to the linear [[dependent sum]] $\star_B \mathbb{1} \;\coloneqq\; (p_B)_! (p_B)^\ast \mathbb{1}$ over the monoidal unit in the context $B$. In terms of [quantum modal logic](necessity+and+possibility#ModalQuantumLogic) this is forming the "linear randomization" of the given classical homotopy type (its "[[motive]]"):
\[
\label{MotivizationInLHott}
\]
\begin{imagefromfile}
"file_name": "MotivizationInLHoTT-230825b.jpg",
"width": "460",
"unit": "px",
"margin": {
"top": -20,
"bottom": 20,
"right": 0,
"left": 10
}
\end{imagefromfile}
Curiously, this homotopy-theoretic realization of the exponential modality (beware the little subtlety of whether or not to reflect onto [[pointed homotopy types]])
$$
\Sigma^\infty \circ \Omega^\infty
\;\colon\;
Spectra \to Spectra
$$
has independently been argued to have properties of an [[exponential map]] in the context of [[Goodwillie calculus]], see [there](Goodwillie+calculus#GoodwillieDerivativeOfExponentialModality).
{#ComonadicityOfClassicalOverLinearHomotopyTypes} Moreover $\Sigma^\infty \dashv \Omega^\infty$ is a [[comonadic adjunction]] on [[simply connected homotopy type|simply connected]] [[homotopy types]] [[Blomquist & Harper 2016 Thm. 1.8](#BlomquistHarper16); [Hess & Kedziorek 2017 Thm. 3.11](#HessKedziorek17)], meaning that simply-connected classical (but [[pointed homotopy types|pointed]]) homotopy types are identified with the $\Sigma^\infty \Omega^\infty$-[[modales]] among [[stable homotopy types]].
## Modal term calculi
Girard's original presentation of linear logic involved rules that explicitly assumed the presence of $!$ on hypotheses or on entire contexts, such as [[dereliction rule| dereliction]], [[weakening rule|weakening]] and [[contraction rule|contraction]]:
$$\frac{\Gamma, A \vdash B}{\Gamma, !A \vdash B} \qquad \frac{\Gamma \vdash B}{\Gamma, !A\vdash B} \qquad \frac{\Gamma,!A,!A \vdash B}{\Gamma, !A\vdash B}$$
and "promotion":
$$ \frac{!\Gamma \vdash A}{!\Gamma\vdash !A} $$
If this is translated into a [[natural deduction]] style term calculus, the resulting rules are more complicated than those of most type formers. This can be avoided using [[adjoint type theory]] with two context zones, one "nonlinear" one where contraction and weakening are permitted (and [[admissible rule|admissible]]) and one "linear" one where they are not, with $!$ as a modality relating the two zones.
Such a "modal" presentation of linear logic was first introduced by [Girard 1993](#Girard93) and then developed by [Plotkin 1993](#Plotkin93), [Wadler 1993](#Wadler93), [Benton 1995](#Benton95), [Barber 1996](#Barber96).
This presentation also generalizes naturally to [[dependent linear type theory]], with the nonlinear type theory being dependent, and the linear types depending on the nonlinear ones but nothing depending on linear types. In this context, the $!$-modality decomposes into "context extension" and a "dependent sum".
## Related concepts
* [[exponential map]]
[[!include logic symbols -- table]]
## References
### General
The notion of the exponential modality originates in [[linear logic]] with
* {#Girard1987} [[Jean-Yves Girard]], _Linear logic_, Theoretical Computer Science **50** 1 (1987) [<a href="https://doi.org/10.1016/0304-3975(87)90045-4">doi:10.1016/0304-3975(87)90045-4</a>, [pdf](http://iml.univ-mrs.fr/~girard/linear.pdf)]
A streamlined statement of the [[inference rules]] and the observation that these make the exponential a [[comonad]] is due to:
* {#Seely89} [[R. A. G. Seely]], §2 of: *Linear logic, $\ast$-autonomous categories and cofree coalgebras*, in *Categories in Computer Science and Logic*, Contemporary Mathematics **92** (1989) [[[SeelyLinearLogic.pdf:file]], [ps.gz](http://www.math.mcgill.ca/rags/nets/llsac.ps.gz), [ISBN:978-0-8218-5100-5](https://bookstore.ams.org/conm-92)]
* {#dePaiva89} [[ Valeria de Paiva]], §2 of: *The Dialectica Categories*, in *Categories in Computer Science and Logic*, Contemporary Mathematics **92** (1989) [[ISBN:978-0-8218-5100-5](https://bookstore.ams.org/conm-92), [doi:10.1090/conm/092](https://doi.org/10.1090/conm/092)]
Review:
* [[Anne Sjerp Troelstra]], §12 in: *Lectures on Linear Logic* (1992) [[ISBN:0937073776](https://web.stanford.edu/group/cslipublications/cslipublications/site/0937073776.shtml)]
Brief survey in a context of [[computer science]]/[[linear type theory]]:
* {#MihályiNovitzká13} Daniel Mihályi, Valerie Novitzká, Section 2.2 of: *What about Linear Logic in Computer Science?*, Acta Polytechnica Hungarica **10** 4 (2013) 147-160 [[pdf](http://acta.uni-obuda.hu/Mihalyi_Novitzka_42.pdf), [[MihalyiNovitzka-LinearLogic.pdf:file]]]
Further on the semantics of exponential conjunction as a [[comonad]]:
* {#deP1988} [[Valeria de Paiva]], *The Dialectica Categories*, PhD thesis, technical report 213, Computer Laboratory, University of Cambridge (1991) [[pdf](https://www.cl.cam.ac.uk/techreports/UCAM-CL-TR-213.pdf), [[dePaivaDialectica.pdf:file]]]
* {#BBP92} [[Nick Benton]], [[Gavin Bierman]], [[Valeria de Paiva]], §8 of: *Term assignment for intuitionistic linear logic*, Technical report 262, Computer Laboratory, University of Cambridge (August 1992) [[pdf](https://www.cl.cam.ac.uk/techreports/UCAM-CL-TR-262.pdf), [[BentonBiermanDePaiva-TermAssignment.pdf:file]]]
> (published abridged as [BBPH92](#BBPH92))
* {#BBPH92} [[Nick Benton]], [[Gavin Bierman]], [[Valeria de Paiva]], [[Martin Hyland]], *Linear $\lambda$-Calculus and Categorical Models Revisited*, in *Computer Science Logic. CSL 1992*, Lecture Notes in Computer Science **702**, Springer (1993) [[doi:10.1007/3-540-56992-8_6](https://doi.org/10.1007/3-540-56992-8_6)]
> (abridged version of [BBP92](#BBP92))
* {#BluteCockettSeely96} [[R. F. Blute]] , [[J. R. B. Cockett]], [[R. A. G. Seely]], *! and ? -- Storage as tensorial strength*, Mathematical Structures in Computer Science **6** 4 (1996) 313-351 [[doi:10.1017/S0960129500001055](https://doi.org/10.1017/S0960129500001055)]
* {#HylandSchalk01} [[Martin Hyland]] and Andreas Schalk, _Glueing and orthogonality for models of linear logic_, [pdf](http://www.cs.man.ac.uk/~schalk/publ/gomll.pdf)
and its [resolution](monad#CategoryOfAdjunctionResolutionsOfAMonad) by a [[monoidal adjunction]] between the linear and a classical (intuitionistic) type system:
* {#Bierman94} [[Gavin Bierman]], *On Intuitionistic Linear Logic*, Cambridge (1994) [[[Bierman-LinearLogic.pdf:file]], [pdf](https://www.dropbox.com/s/hdxgubjljb96rmf/Biermanthesis.pdf?dl=0)]
* {#Benton95} [[Nick Benton]], *A mixed linear and non-linear logic: Proofs, terms and models*, in *Computer Science Logic. CSL 1994*, Lecture Notes in Computer Science **933** (1995) 121-135 [[doi:10.1007/BFb0022251](https://doi.org/10.1007/BFb0022251), [[BentonLinearLogic.pdf:file]]]
Review:
* {#Melliès02} [[Paul-André Melliès]], *Categorical models of linear logic revisited* (2002) [[hal:00154229](https://hal.science/hal-00154229)]
* {#Melliès09} [[Paul-André Melliès]], *Categorical semantics of linear logic*, in *[Interactive models of computation and program behaviour](https://smf.emath.fr/publications/modeles-interactifs-de-calcul-et-de-comportement-de-programme)*, Panoramas et synthèses **27** (2009) 1-196 [[web](https://smf.emath.fr/publications/semantique-categorielle-de-la-logique-lineaire), [pdf](https://www.irif.fr/~mellies/papers/panorama.pdf), [[Mellies-CategoricalSemanticsLinear.pdf:file]]]
Construction of such comonads based on cofree comonoids:
* Mellies and Tabareau and Tasson, *An explicit formula for the free exponential modality of linear logic*. Mathematical Structures in Computer Science, 28(7), 1253-1286. doi:[10.1017/S0960129516000426](https://doi.org/10.1017/S0960129516000426)
* [[Sergey Slavnov]], *On Banach spaces of sequences and free linear logic exponential modality*, Math. Struct. Comp. Sci. 29 (2019) 215-242, [arXiv:1509.03853](https://arxiv.org/abs/1509.03853)
The relation to [[Fock space]] is discussed in:
* {#BlutePanangadenSeely94} [[Richard Blute]], [[Prakash Panangaden]], [[R. A. G. Seely]], _Fock Space: A Model of Linear Exponential Types_ (1994) ([web](http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.27.6825), [[BPSLinear.pdf:file]])
* {#Fiore07} [[Marcelo Fiore]], _Differential Structure in Models of Multiplicative Biadditive Intuitionistic Linear Logic_, Lecture Notes in Computer Science Volume 4583, 2007, pp 163-177 ([pdf](http://www.cl.cam.ac.uk/~mpf23/papers/Types/diff.pdf))
* {#Vicary07} [[Jamie Vicary]], _A categorical framework for the quantum harmonic oscillator_, International Journal of Theoretical Physics December 2008, Volume 47, Issue 12, pp 3408-3447 ([arXiv:0706.0711](http://arxiv.org/abs/0706.0711))
> (in the context of [[quantum information theory in terms of dagger-compact categories]])
The interpretation of $\Omega^\infty \Sigma^\infty_+$ as an exponential in the context of [[Goodwillie calculus]] is due to
* {#AroneKankaanrinta95} [[Gregory Arone]], Marja Kankaanrinta, _The Goodwillie tower of the identity is a logarithm_, 1995 ([web](http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.53.8306))
based on
* {#AroneMahowald98} [[Gregory Arone]], [[Mark Mahowald]], _The Goodwillie tower of the identity functor and the unstable periodic homotopy of spheres_, 1998 ([pdf](http://hopf.math.purdue.edu/Arone-Mahowald/ArMahowald.pdf))
On the [[modal type theory]]-approach to a term calculus for the $!$-modality:
* {#Girard93} [[Jean-Yves Girard]]. *On the unity of logic* Annals of Pure and Applied Logic **59** (1993) 201-217 [<a href="https://doi.org/10.1016/0168-0072(93)90093-S">doi:10.1016/0168-0072(93)90093-S</a>]
* {#Plotkin93} [[Gordon Plotkin]], *Type theory and recursion*, in: Proceedings of the Eigth Symposium of Logic in Computer Science, Montreal , IEEE Computer Society Press (1993) 374 [[doi:10.1109/LICS.1993.287571](https://doi.org/10.1109/LICS.1993.287571)]
* {#Wadler93} [[Philip Wadler]], *A syntax for linear logic*, in: *Ninth International Coference on the Mathematical Foundations of Programming Semantics*, Lecture Notes in Computer Science **802** Springer (1993) [[doi:10.1007/3-540-58027-1_24](https://doi.org/10.1007/3-540-58027-1_24)]
* [Benton 1995](#Benton95)
* {#Barber96} Andrew Barber, *Dual Intuitionistic Linear Logic*, Technical Report ECS-LFCS-96-347, University of Edinburgh (1996), [[web](http://www.lfcs.inf.ed.ac.uk/reports/96/ECS-LFCS-96-347/), [pdf](http://www.lfcs.inf.ed.ac.uk/reports/96/ECS-LFCS-96-347/ECS-LFCS-96-347.pdf), [[Barber-DualIntLinLogic.pdf:file]]]
A [[quantum programming language]] based on this linear/non-linear type theory adunction is [[QWIRE]]:
* [[Jennifer Paykin]], [[Robert Rand]], [[Steve Zdancewic]], *QWIRE: a core language for quantum circuits*, POPL 2017: Proceedings of the 44th ACM SIGPLAN Symposium on Principles of Programming LanguagesJanuary 2017 Pages 846–858 ([doi:10.1145/3009837.3009894](https://doi.org/10.1145/3009837.3009894))
theoretical background:
* [[Jennifer Paykin]], *Linear/non-Linear Types For Embedded Domain-Specific Languages*, 2018 ([upenn:2752](https://repository.upenn.edu/edissertations/2752))
applied to [[verified programming]] after implementation in [[Coq]]:
* [[Robert Rand]], [[Jennifer Paykin]], [[Steve Zdancewic]], *QWIRE Practice: Formal Verification of Quantum Circuits in Coq*, EPTCS 266, 2018, pp. 119-132 ([arXiv:1803.00699](https://arxiv.org/abs/1803.00699))
and using ambient [[homotopy type theory]]:
* [[Jennifer Paykin]], [[Steve Zdancewic]], *A HoTT Quantum Equational Theory*, [talk at QPL2019](http://qpl2019.org/a-hott-quantum-equational-theory/) ([arXiv:1904.04371](https://arxiv.org/abs/1904.04371))
### In dependent linear type theory
Discussion of the exponential modality via [[stabilization]] in [[dependent linear homotopy type theory]]:
* {#PontoShulman12} [[Kate Ponto]], [[Mike Shulman]], Ex. 4.2 in: *Duality and traces in indexed monoidal categories*, Theory and Applications of Categories **26** 23 (2012) [[arXiv:1211.1555](http://arxiv.org/abs/1211.1555), [tac:26-23](http://www.tac.mta.ca/tac/volumes/26/23/26-23abs.html), [blog](http://golem.ph.utexas.edu/category/2011/11/traces_in_indexed_monoidal_cat.html)]
* {#Schreiber14} [[Urs Schreiber]], Sec. 4.2 of: *[[schreiber:Quantization via Linear homotopy types]]* [[arXiv:1402.7041](http://arxiv.org/abs/1402.7041)]
* {#Riley22Thesis} [[Mitchell Riley]], §2.1.2 in: *A Bunched Homotopy Type Theory for Synthetic Stable Homotopy Theory*, PhD Thesis (2022) [[doi:10.14418/wes01.3.139](https://doi.org/10.14418/wes01.3.139)]
The $\Omega^\infty \Sigma^\infty$ [[comonadic functor|comonadicity]] of simply connected pointed spaces over spectra:
* {#BlomquistHarper16} Jacobson R. Blomquist, John E. Harper, Thm. 1.8 in: *Suspension spectra and higher stabilization* [[arXiv:1612.08623](https://arxiv.org/abs/1612.08623)]
* {#HessKedziorek17} [[Kathryn Hess]], [[Magdalena Kedziorek]], Thm. 3.11 in: *The homotopy theory of coalgebras over simplicial comonads*, Homology, Homotopy and Applications **21** 1 (2019) [[arXiv:1707.07104](https://arxiv.org/abs/1707.07104), [doi:10.4310/HHA.2019.v21.n1.a11](https://dx.doi.org/10.4310/HHA.2019.v21.n1.a11)]
category: logic
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$\infty$-category > history | https://ncatlab.org/nlab/source/%24%5Cinfty%24-category+%3E+history | < [[$\infty$-category]]
[[!redirects $\infty$-category/history]]
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't Hooft anomaly | https://ncatlab.org/nlab/source/%27t+Hooft+anomaly |
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### Context
#### Quantum systems
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#### Physics
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=--
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#Contents#
* table of contents
{:toc}
## Idea
As described in ['t Hooft (1980)](#tHooft1980), a [[global symmetry]] of a [[quantum field theory]] is said to have a **'t Hooft anomaly** if it is non-anomalous as a global symmetry but has a [[quantum anomaly]] if one attempts to turn it into a [[gauge symmetry]].
Informally, a global symmetry described by the [[action]] of a [[group]] $G$ may be "coupled" to a [[gauge potential]] $A$ (a [[connection on a bundle]]). Denoting the [[partition function]] coupled to such a connection as $Z(A)$, the partition function of the gauged theory is, modulo normalization factors, $Z=\sum_A Z(A).$ If the symmetry has a 't Hooft anomaly, then perfoming a [[gauge transformation]] on all connections results in a nontrivial phase multiplying each partition function $Z_g=\sum_A \phi(A,g) Z(g\cdot A)$ where each phase $\phi(A,g)$ depends on the particular background field and gauge transformation, so that generally $Z\neq Z_g$. This is problematic since the partition function is supposed to be gauge invariant.
For $G$ a [[finite group]], and when the $n$-dimensional spacetime $\Sigma$ is the boundary of an $(n+1)$-dimensional space $X$, this situation may be remedied by "coupling" the $n$-dimensional theory with symmetry $G$ to a $(n+1)$-dimensional [[topological quantum field theory]] (a [[Dijkgraaf-Witten theory]] classified by $H^{n+1}(G,U(1))$) on $X$ such that the phase contributions of a gauge transformation of both cancel each other. This is known as *anomaly inflow* (see e.g. [Freed, Hopkins, Lurie, and Teleman (2009)](#FHLT09) and [Freed (2014)](#Freed14)).
Any [[generalized global symmetry]] is also thought to potentially exhibit 't Hooft anomalies described by a TQFT. In the literature, this is referred to as the *Anomaly TFT*, an [[invertible field theory]] (cf. [Freed 2014](#Freed14)), but little is known about what this TQFT is supposed to be for cases not equivalent to group-like cases (for which the TQFT is DW).
## References
{#References}
* {#tHooft1980} [[Gerard 't Hooft]], *Naturalness, Chiral Symmetry, and Spontaneous Chiral Symmetry Breaking* In: G. 'Hooft et al., *Recent Developments in Gauge Theories*, NATO Advanced Study Institutes Series **59**, Springer (1980) [[doi:10.1007/978-1-4684-7571-5_9](https://doi.org/10.1007/978-1-4684-7571-5_9)]
* {#FHLT09} [[Daniel Freed]], [[Michael Hopkins]], [[Jacob Lurie]], [[Constantin Teleman]], *[[Topological Quantum Field Theories from Compact Lie Groups]]*, in P. R. Kotiuga (ed.) *A celebration of the mathematical legacy of Raoul Bott* AMS (2010) [[arXiv:0905.0731](https://arxiv.org/abs/0905.0731), [ISBN:978-0-8218-4777-0](https://bookstore.ams.org/view?ProductCode=CRMP/50)]
* {#Freed14} [[Daniel Freed]]. *Anomalies and Invertible Field Theories*, talk at [StringMath2013](http://scgp.stonybrook.edu/events/event-pages/string-math-2013) [[arXiv:1404.7224](https://arxiv.org/abs/1404.7224)]
[[!redirects 't Hooft anomalies]]
|
't Hooft coupling | https://ncatlab.org/nlab/source/%27t+Hooft+coupling |
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#Contents#
* table of contents
{:toc}
## Idea
In [[Yang-Mills theory]]/[[QCD]] with [[coupling constant]] $g_{YM}$ and with $N_c$ [[colour charge|colours]], the _’t Hooft coupling_ is the expression
$$
\lambda
\;\coloneqq\;
g_{YM}^2 N_c
\,.
$$
## Properties
### In the large $N$ limit
The study of [[Yang-Mills theory]] in the [[large N limit]] but at fixed ’t Hooft coupling controls the [[AdS/CFT correspondence]]:
[[large N limit]] $\;N_c \to \infty$ at fixed [['t Hooft coupling]] $\lambda = g_{YM}^2 N_c$
| $\lambda \lt 1$ | $\lambda \gt 1$ |
|--|--|
| [[perturbative quantum field theory|small coupling limit]] of [[Yang-Mills theory]] | low-energy [[supergravity]] limit of [[string theory]] on asymptotically [[AdS spacetime]]
(e. g. [Aharony-Gubser-Maldacena-OoguriOz 99, p. 11, p. 60](#AharonyGubserMaldacenaOoguriOz99), [Nastase 07, Section 8](#Nastase07))
(...)
## Related concepts
* [[large N limit]]
* [[AdS/CFT]]
## References
The original article
* [[Gerard 't Hooft]], _A Planar Diagram Theory for Strong Interactions_, Nucl. Phys. B72 (1974) 461 ([spire:80491](http://inspirehep.net/record/80491), <a href="https://doi.org/10.1016/0550-3213(74)90154-0">doi:10.1016/0550-3213(74)90154-0</a>)
Review:
* {#AharonyGubserMaldacenaOoguriOz99} [[Ofer Aharony]], [[Steven Gubser]], [[Juan Maldacena]], [[Hirosi Ooguri]], [[Yaron Oz]], _Large $N$ Field Theories, String Theory and Gravity_, Phys. Rept. **323** 183-386 (2000) $[$<a href="https://doi.org/10.1016/S0370-1573(99)00083-6">doi:10.1016/S0370-1573(99)00083-6</a>, [arXiv:hep-th/9905111](http://arxiv.org/abs/hep-th/9905111)$]$
* {#Nastase07} [[Horatiu Nastase]], _Introduction to AdS-CFT_ ([arXiv:0712.0689](http://arxiv.org/abs/0712.0689)) |
't Hooft double line notation | https://ncatlab.org/nlab/source/%27t+Hooft+double+line+notation |
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###Context###
#### Lie theory
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#### Quantum Field Theory
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#### String theory
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[[!include string theory - contents]]
=--
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#Contents#
* table of contents
{:toc}
## Idea
### Double line notation
What is called _’t Hooft double line notation_ (following [’t Hooft 74](#tHooft74)) is the observation that for the [[fundamental representations]] $V$ of [[semisimple Lie algebras]] $\mathfrak{g}$. the [[Lie algebra weight system|Lie algebra weight]] of a $\mathfrak{g}$-[[Yang-Mills theory]] [[Feynman diagram]] with internal/[[virtual particle|virtual]] [[gluon]] lines is equivalent to one without any [[virtual particle|virtual]] [[gluon]]-lines, obtained by:
{#JacobiIdentity} 1) using the [[Jacobi identity]] to replace all internal gluon vertices by ([[linear combinations]] of) quark-gluon vertices
<center>
<img src="https://ncatlab.org/nlab/files/QuarkGluonJacobiIdentity.jpg" width="580">
</center>
{#MetricContractionOfFundamentalLieAction} 2) replacing any remaining internal gluon line together with the [[quark]] [[interaction]] [[vertices]] at its ends (equivalently: an [[M2-brane 3-algebra]] [[tensor]])
by (a [[linear combination]] of) _double_ [[quark]]-lines, according to the following rules:
<center>
<img src="https://ncatlab.org/nlab/files/MetricActionContraction.jpg" width="600">
</center>
{#Example} For example, for $\mathfrak{g} = \mathfrak{so}(N)$ a [[special orthogonal Lie algebra]], the ’t Hooft double line notation of a single trivalent vertex inside a single Wilson loop is the following:
<center>
<img src="https://ncatlab.org/nlab/files/tHooftDoubleLineTrivalentVertex.jpg" width="640">
</center>
Here we are using the [[string diagram]]/[[Penrose notation]] from _[[metric Lie representations]]_.
> graphics from [[schreiber:Differential Cohomotopy implies intersecting brane observables|Sati-Schreiber 19c]]
{#Attribution} **Attribution.** In [’t Hooft 74](#tHooft74) this is observed for the [[unitary Lie algebra]] $\mathfrak{g} = \mathfrak{u}(N)$ (see also [Bar-Natan 95 (34)](#BarNatan95), [Chmutov-Duzhin-Mostovoy 11, p. 177](#ChmutovDuzhinMostovoy11), in which case only the first summand of the expressions [above](#MetricContractionOfFundamentalLieAction) appears. The generalization to arbitrary [[semisimple Lie algebras]] is observed in [Cvitanović 76, Fig. 14](#Cvitanovic76), reviewed for [[special unitary Lie algebra|su(N)]], [[special orthogonal Lie algebra|so(N)]] and [[symplectic Lie algebra|sp(N)]] in [Cvitanović 08 (9.57), (10.13) and (12.9)](#Cvitanovic08) and for [[special orthogonal Lie algebra|so(N)]] in [Chmutov-Duzhin-Mostovoy 11, 6.2.6](#ChmutovDuzhinMostovoy11).
For the case $\mathfrak{so}(N)$ see also
[Bar-Natan 95, Section 6.3](#BarNatan95) (with an eye towards [[Vassiliev knot invariants]]) and
[Cicuta 82](#Cicuta82), [Ita-Nieder-Oz 02, Figure 3](#ItaNiederOz02), [McGreevy-Swingle 08, Figure 10](#McGreevySwingle08) (with more discussion of the [[large N limit]]). The case of [[general linear Lie algebra|gl(N)]] is considered in [Bar-Natan 96, Section 2.2](#BarNatan96), [Chmutov-Duzhin-Mostovoy 11, 6.2.5](#ChmutovDuzhinMostovoy11), and the case [[special linear Lie algebra|sl(N)]] in [Chmutov-Duzhin-Mostovoy 11, 6.1.8](#ChmutovDuzhinMostovoy11), [Jackson-Moffat 19, Section 14.4](#JacksonMoffat19).
### Surface notation
Furthermore, one may regard the resulting double-line diagrams as a [[ribbon graph]] thickening of the original [[Feynman diagrams]], and thus as [[surfaces]] [[manifold with boundary|with boundary]] and with markings on the [[boundary]].
In the case $\mathfrak{g} = \mathfrak{so}(N)$ this means that a single [[virtual particle|virtual]] [[gluon]] line is represented by the [[formal linear combination]] of a strip (marked [[disk]]) and a twisted strip:
<center>
<img src="https://ncatlab.org/nlab/files/tHooftDoubleLineToSurfaces.jpg" width="600">
</center>
Using this on the reduction of internal gluon vertices by the [[Jacobi identity]] as [above](#JacobiIdentity) one finds that a single [[gluon]] vertex turns into the following linear combination of marked surfaces:
<center>
<img src="https://ncatlab.org/nlab/files/tHooftSurfacesForGluonVertex.jpg" width="700">
</center>
{#SurfaceExample} For example:
<center>
<img src="https://ncatlab.org/nlab/files/RisingSuntHooftConstruction.jpg" width="620">
</center>
> graphics from [[schreiber:Differential Cohomotopy implies intersecting brane observables|Sati-Schreiber 19c]]
This gives [[open string]] [[worldsheets]]. Regarding them as such in the case of [[Chern-Simons theory]] exhibits [[Chern-Simons theory as an open topological string theory]] ([Witten 92, see Figures 1 & 2](#Witten92)) even for [[small N limit|small N]].
For [[AdS/CFT duality]] relating [[super Yang-Mills theory]] to _[[closed strings|closed]]_ [[string theory]] and open to _closed_ [[topological string theory]] ([Gopakumar-Vafa 98](#GopakumarVafa98)) there is an operation of gluing in [[faces]] to turn these [[open strings]] into [[closed string]] [[worldsheets]], see
[Gaiotto-Rastelli 03, Section 1.1](#GaiottoRastelli03)
and see [Marino 04, Section III, p. 14](#Marino04) for a clear statement).
Here [[open/closed string duality]] plays a subtle role in interpreting the [['t Hooft double line notation]] of [[gauge theory]] [[Feynman diagrams]] in the [[large N limit]] alternatively as [[open string]] or as [[closed string]] [[worldsheets]], see also [Gopakumar 04](#Gopakumar04).
### As a surface-valued weight system
After averaging over the $n!$ permutations of the ordering of the $n$ external vertices (those on the Wilson line) of a Jacobi diagram, this construction respects the [[STU relation]] on [[Jacobi diagrams]] and hence gives a [[weight system]] with values in marked surfaces ([Bar-Natan 95, Theorem 10 with Theorem 8](#BarNatan95)):
<center>
<img src="https://ncatlab.org/nlab/files/AtHooftConstrictionIII.jpg" width="650">
</center>
> graphics from [[schreiber:Differential Cohomotopy implies intersecting brane observables|Sati-Schreiber 19c]]
This means that [[stringy weight systems]] pull back to ordinary [[weight systems]] along this map, corresponding to taking their [[worldline formalism|point-particle limit]]. Under this map [[stringy weight systems span classical Lie algebra weight systems]].
\linebreak
Of course, the ’t Hooft double line construction applies not just to [[Jacobi diagrams]] but also to [[Sullivan chord diagrams]]:
<center>
<img src="https://ncatlab.org/nlab/files/ClosingHorizontalChordsToSullivanChordsIII.jpg" width="800">
</center>
> from [[schreiber:Differential Cohomotopy implies intersecting brane observables|Sati-Schreiber 19c]]
## Applications
### Large $N$ Limit and holography
One upshot of this double-line reformulation is that it makes the dependence of the [[Feynman amplitude]] on the [[natural number]] $N$ fully explicit, since a double line diagram simply contributes one factor of $N$ for each closed [[quark]] line (this being the [[trace]] over the [[fundamental representation]]).
In fact, by regarding the resulting double-line diagrams as a [[ribbon graph]] thickenings the original [[Feynman diagrams]], and regarding these [[ribbon graphs]] in turn as [[surfaces]], the $N$-dependence is proportional to the [[genus of a surface|genus]] of these surfaces.
One finds this way that in the _[[large N limit]]_, i.e. the [[limit of a sequence|limit]] where $N \to \infty$, precisely only the [[planar graphs]] contribute to the [[Yang-Mills theory]] [[Feynman amplitudes]], corresponding to [[surfaces]] of [[genus]] 0 (i.e. [[tree level]] [[string scattering amplitudes]] see at _[[planar limit]]_ for more on this), while $1/N$-corrections correspond to higher [[worldsheet]] [[topology]].
Last not least, these [[surfaces]] have the interpretation of [[open string]] [[worldsheets]] of a [[string theory]] which is "[[duality in string theory|dual]]" to the original ([[super Yang-Mills theory|super]]) in the sense made precise by [[AdS-CFT duality]].
### Classification of weight systems
Later the same double line technique was used (without any reference to the earlier physics articles(?)) in [Bar-Natan 95, Section 6](#BarNatan95) for discussion of the classification of [[Lie algebra weight systems]] and [[stringy weight systems]] with an eye towards discussion of [[Vassiliev knot invariants]].
## Related concepts
* [[large N limit]], [[small N limit]]
* [[AdS/CFT correspondence]]
## References
### General
The original article is:
* {#tHooft74} [[Gerard ’t Hooft]], _A Planar Diagram Theory for Strong Interactions_, Nucl. Phys. B72 (1974) 461 ([spire:80491](http://inspirehep.net/record/80491), <a href="https://doi.org/10.1016/0550-3213(74)90154-0">doi:10.1016/0550-3213(74)90154-0</a>)
reviewed in
* [[Gerard ’t Hooft]], _Large $N$_, workshop lecture ([hep-th/0204069](http://arxiv.org/abs/hep-th/0204069))
* Markus Gross, _Large $N$_, 2006 ([[GrossLargeN.pdf:file]])
Generalization to arbitrary [[semisimple Lie algebras]] ([[semisimple Lie groups]]) is due to:
* {#Cvitanovic76} [[Predrag Cvitanović]], _Group theory for Feynman diagrams in non-Abelian gauge theories_, Phys. Rev. D14 (1976) 1536-1553 ([doi:10.1103/PhysRevD.14.1536](https://doi.org/10.1103/PhysRevD.14.1536), [spire:108133](http://inspirehep.net/record/108133), [[CvitanovicWeights76.pdf:file]])
with a textbook account in
* {#Cvitanovic08} [[Predrag Cvitanović]], _Group Theory: Birdtracks, Lie's, and Exceptional Groups_, Princeton University Press July 2008 ([PUP](https://press.princeton.edu/books/paperback/9780691202983/group-theory), [birdtracks.eu](http://birdtracks.eu/), [pdf](http://www.birdtracks.eu/version9.0/GroupTheory.pdf))
Discussion in the context of [[Vassiliev invariants]] and the abstract classification of [[Lie algebra weight systems]] and [[stringy weight systems]]:
* {#BarNatan95} [[Dror Bar-Natan]], Section 6 of: _On the Vassiliev knot invariants_, Topology Volume 34, Issue 2, April 1995, Pages 423-472 (<a href="https://doi.org/10.1016/0040-9383(95)93237-2">doi:10.1016/0040-9383(95)93237-2</a>, [pdf](https://www.math.toronto.edu/drorbn/papers/OnVassiliev/OnVassiliev.pdf))
* {#BarNatan96} [[Dror Bar-Natan]], _Vassiliev and Quantum Invariants of Braids_, Geom. Topol. Monogr. 4 (2002) 143-160 ([arxiv:q-alg/9607001](https://arxiv.org/abs/q-alg/9607001))
of which textbook accounts are in
* {#ChmutovDuzhinMostovoy11} [[Sergei Chmutov]], [[Sergei Duzhin]], [[Jacob Mostovoy]], _Introduction to Vassiliev knot invariants_, Cambridge University Press, 2012 ([arxiv:1103.5628](http://arxiv.org/abs/1103.5628), [doi:10.1017/CBO9781139107846](https://doi.org/10.1017/CBO9781139107846))
* {#JacksonMoffat19} [[David Jackson]], [[Iain Moffat]], _An Introduction to Quantum and Vassiliev Knot Invariants_, Springer 2019 ([doi:10.1007/978-3-030-05213-3](https://link.springer.com/book/10.1007/978-3-030-05213-3))
Further discussion of the case of $\mathfrak{so}(N)$ in the context of the [[large N limit]]:
* {#Cicuta82} G.M. Cicuta, _Topological Expansion for $SO(N)$ and $Sp(2n)$ Gauge Theories_, Lett. Nuovo Cim. 35 (1982) 87 ([spire:177713](http://inspirehep.net/record/177713), [doi:10.1007/BF02754653](https://doi.org/10.1007/BF02754653))
* {#ItaNiederOz02} Harald Ita, Harald Nieder, [[Yaron Oz]], _Perturbative Computation of Glueball Superpotentials for $SO(N)$ and $USp(N)$_, JHEP 0301:018, 2003 ([arXiv:hep-th/0211261](https://arxiv.org/abs/hep-th/0211261))
* {#McGreevySwingle08} McGreevy, Swingle, _Large $N$ counting_, 2008 ([[GreevySwingle.pdf:file]])
On the [[logical equivalence]] between the [[four-colour theorem]] and a statement about transition from the [[small N limit]] to the [[large N limit]] for [[Lie algebra weight systems]] on [[Jacobi diagrams]] via the [['t Hooft double line construction]]:
* [[Dror Bar-Natan]], _Lie Algebras and the Four Color Theorem_, Combinatorica 17-1(1997) 43–52 ([arXiv:q-alg/9606016](https://arxiv.org/abs/q-alg/9606016), [doi:10.1007/BF01196130](https://doi.org/10.1007/BF01196130))
### For Chern-Simons theory
Discussion of [['t Hooft double line notation]] for [[Chern-Simons theory]], exhibiting [[Chern-Simons theory as topological string theory]]:
* {#Witten92} [[Edward Witten]], _Chern-Simons Gauge Theory As A String Theory_, Prog. Math. 133: 637-678, 1995 ([arXiv:hep-th/9207094](https://arxiv.org/abs/hep-th/9207094))
* S. Sinha, [[Cumrun Vafa]], _$SO$ and $Sp$ Chern-Simons at Large $N$_ ([arXiv:hep-th/0012136](https://arxiv.org/abs/hep-th/0012136))
### Open/closed string duality
On the role of [[open/closed string duality]] in interpreting the [[large N limit]] of the [['t Hooft double line notation]]:
* {#GopakumarVafa98} [[Rajesh Gopakumar]], [[Cumrun Vafa]], _On the Gauge Theory/Geometry Correspondence_, Adv. Theor. Math. Phys. 3 (1999) 1415-1443 ([arXiv:hep-th/9811131](https://arxiv.org/abs/hep-th/9811131))
* {#GaiottiRastelli05} [[Davide Gaiotto]], [[Leonardo Rastelli]], _A paradigm of open/closed duality: Liouville D-branes and the Kontsevich model_, JHEP 0507:053,2005 ([hep-th/0312196](https://arxiv.org/abs/hep-th/0312196))
> Nowadays we interpret $[$ the [['t Hooft double line notation]] $]$ quite literally as the perturbative expansion of an open string theory, either because the full open string theory is just equal to the gauge
theory (as e.g. for Chern-Simons theory [27]), or because we take an appropriate
low-energy limit (as e.g. for N = 4 SYM [31]).
> The general speculation [1] is that upon summing over the number of holes, (1.1) can be recast as the genus expansion for some closed string theory of coupling $g_s = g_{YM}^2$. This speculation is sometimes justified by appealing to the intuition that diagrams with
a larger and larger number of holes look more and more like smooth closed Riemann surfaces. This intuition is perfectly appropriate for the double-scaled matrix models, where the finite N theory is interpreted as a discretization of the closed Riemann surface; to recover the continuum limit, one must send $N\to \infty$ and tune $t$ to the
critical point $t_c$ where diagrams with a diverging number of holes dominate.
> However, in AdS/CFT, or in the Gopakumar-Vafa duality [2], $t$ is a free parameter, corresponding on the closed string theory side to a geometric modulus. The intuition described above clearly goes wrong here.
> A much more fitting way in which the open/closed duality may come about in these cases is for each fatgraph of genus g and
with h holes to be replaced by a closed Riemann surface of the same genus g and with h punctures: each hole is filled and replaced by a single closed string insertion.
* {#Gopakumar04} [[Rajesh Gopakumar]], _Free Field Theory as a String Theory?_, Comptes Rendus Physique 5 (2004) 1111-1119 ([hep-th/0409233](https://arxiv.org/abs/hep-th/0409233))
* {#Marino04} [[Marcos Marino]], _Chern-Simons Theory and Topological Strings_, Rev. Mod. Phys. 77:675-720, 2005 ([arXiv:hep-th/0406005](https://arxiv.org/abs/hep-th/0406005))
[[!redirects 't Hooft double line notations]]
[[!redirects 't Hooft double line construction]]
[[!redirects 't Hooft double line constructions]]
|
't Hooft-Polyakov monopole | https://ncatlab.org/nlab/source/%27t+Hooft-Polyakov+monopole |
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### Context
#### Physics
+-- {: .hide}
[[!include physicscontents]]
=--
=--
=--
#Contents#
* table of contents
{:toc}
## Idea
A type of [[monopole]] solution of [[Yang-Mills theory]] with [[Higgs field]] included.
## Properties
### In string theory
In [[string theory]], in the context of [[intersecting D-brane models]], the 't Hooft-Polyakov monopole has been identified with the [[black brane|black]] "half" [[NS5-brane]] coincident on an [[O8-plane]] ([Hanany-Zaffaroni 99](#HananyZaffaroni99)). See there at _[Intersection of D6s with O8s](intersecting+D-brane+model#IntersectionOfD6WithO8)_.
## Related concepts
* [[Yang monopole]]
* [[Dirac monopole]]
## References
The original articles are
* [[Gerard 't Hooft]], _Magnetic Monopoles in Unified Gauge Theories_, Nucl.Phys. B79 (1974) 276-284 ([spire](http://inspirehep.net/record/89705?ln=en))
* [[Alexander Polyakov]], _Particle Spectrum in the Quantum Field Theory_, JETP Lett. 20 (1974) 194-195 ([spire](http://inspirehep.net/record/90679/))
Interpretation in [[string theory]] in terms of "half" [[NS5-branes]] at [[O8-planes]] is due to
* {#HananyZaffaroni99} [[Amihay Hanany]], [[Alberto Zaffaroni]], _Monopoles in String Theory_, JHEP 9912 (1999) 014 ([arXiv:hep-th/9911113](https://arxiv.org/abs/hep-th/9911113))
See also
* Wikipedia _['t Hooft-Polyakov monopole](https://en.wikipedia.org/wiki/'t_Hooft–Polyakov_monopole)_
[[!redirects 't Hooft-Polyakov monopoles]] |
(-1)-category | https://ncatlab.org/nlab/source/%28-1%29-category | +-- {: .rightHandSide}
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### Context ###
#### Higher category theory
+-- {: .hide}
[[!include higher category theory - contents]]
=--
=--
=--
## Idea
As a degenerate case of the general notion of *[[n-category]]*, _$(-1)$-categories_ may be understood as [[truth value]]. Compare the concept of [[0-category]] (a [[set]]) and [[(−2)-category]] (which is trivial). The point of $(-1)$-categories (a kind of [[negative thinking]]) is that they complete some patterns in the [[periodic table]] of $n$-categories. (They also shed light on the theory of [[homotopy group]]s and [[n-stuff]].)
For example, there should be a $0$-category of $(-1)$-categories; this is the set of truth values, classically
$$
(-1)Cat := \{true, false\}
\,.
$$
Similarly, $(-2)$-categories form a $(-1)$-category (specifically, the true one).
If we equip the category of $(-1)$-categories with the monoidal structure of [[logical conjunction|conjunction]] (the logical AND operation), then a [[enriched category|category enriched]] over this is a [[partial order|poset]]; an enriched groupoid is a [[set]]. Notice that this doesn\'t fit the proper patterns of the [[periodic table]]; we see that $(-1)$-categories work better as either $0$-[[0-poset|poset]]s or as $(-1)$-[[(-1)-groupoid|groupoid]]s. Nevertheless, there is no better alternative for the term '$(-1)$-category'.
For an introduction to $(-1)$-categories and $(-2)$-[[(-2)-category|categories]] see [page 11](http://arxiv.org/PS_cache/math/pdf/0608/0608420v2.pdf#page=11) and [page 34](http://arxiv.org/PS_cache/math/pdf/0608/0608420v2.pdf#page=34) of
* John C. Baez, Michael Shulman, _[[Lectures on n-Categories and Cohomology]]_ ([arXiv](http://arxiv.org/abs/math.CT/0608420)).
$(-1)$-categories and $(-2)$-categories were discovered (or invented) by [[James Dolan]] and [[Toby Bartels]]. To witness these concepts in the process of being discovered, read the discussion here:
* John Baez, Toby Bartels, David Corfield and James Dolan,
[Property, structure and stuff](http://math.ucr.edu/home/baez/qg-spring2004/discussion.html). See also [[stuff, structure, property]] for more on that material.
[[!redirects (-1)-category]]
[[!redirects (-1)-categories]]
[[!redirects (−1)-category]]
[[!redirects (−1)-categories]]
|
(-1)-functor | https://ncatlab.org/nlab/source/%28-1%29-functor |
As a $(-1)$-[[(-1)-category|category]] is simply a [[truth value]], so a __$(-1)$-functor__ is simply [[implication]]. See also $n$-[[n-functor|functor]].
[[!redirects (-1)-functor]]
[[!redirects (-1)-functors]]
[[!redirects (−1)-functor]]
[[!redirects (−1)-functors]]
|
(-1)-groupoid | https://ncatlab.org/nlab/source/%28-1%29-groupoid |
+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Higher category theory
+--{: .hide}
[[!include higher category theory - contents]]
=--
=--
=--
#Contents#
* table of contents
{:toc}
## Definition
A __$(-1)$-groupoid__ or **[[homotopy n-type|(-1)-type]]** is a [[truth value]], or equivalently an [[n-truncated object of an (infinity,1)-category|(-1)-truncated object]] in [[∞Grpd]]. By [[excluded middle]], this is either [[the]] empty groupoid (false) or the [[terminal category|terminal groupoid]] (true, the [[point]]).
## Remarks
Compare the concept of [[0-groupoid]] (a [[set]]) and [[(-2)-groupoid]] (which is trivial). The point of $(-1)$-groupoids is that they complete some patterns in the [[periodic table]] of $n$-categories. (They also shed light on the theory of [[homotopy group]]s and [[n-stuff]].)
For example, there should be a $0$-[[0-category|category]] of $(-1)$-groupoids; a $0$-category is also a set, and this set is the set of [[truth value|truth values]]: classically
$$
(-1)Grpd := \{\bot, \top\}
$$
Actually, since for other values of $n$, [[n-groupoid]]s form not just an $(n+1)$-category but an $(n+1,1)$-category, we should expect the $0$-category of $(-1)$-groupoids to be a $(0,1)$-category, or $1$-[[1-poset|poset]]. This simply means a [[partial order|poset]], and indeed truth values do always form a poset, classically ($\bot \leq \top$).
If we equip the category of $(-1)$-groupoids with the [[monoidal category|monoidal structure]] of [[logical conjunction|conjunction]] (the logical AND operation), then a [[enriched category|groupoid enriched]] over this is a [[symmetric proset]], and a category enriched over it is a [[preorder|proset]]. Up to [[equivalence of categories]], these are the same as a [[set]] (a $0$-[[0-groupoid|groupoid]]) and a [[partial order|poset]] (a (0,1)-[[1-poset|category]]); this fits the patterns of the periodic table.
See [[(-1)-category]] for more on this sort of [[negative thinking]].
## Related concepts
[[!include homotopy n-types - table]]
[[!redirects (-1)-groupoid]]
[[!redirects (-1)-groupoids]]
[[!redirects (-1)-groupoid]]
[[!redirects (-1)-groupoids]]
[[!redirects (-1)-type]]
[[!redirects (-1)-types]]
[[!redirects (-1)-type]]
[[!redirects (-1)-types]]
|
(-1)-poset | https://ncatlab.org/nlab/source/%28-1%29-poset | There is just one __$(-1)$-poset__, namely the [[point]]. Compare the concepts of $0$-[[0-poset|poset]] (a [[truth value]]) and $1$-[[1-poset|poset]] (a [[partial order|poset]]). Compare also with $(-2)$-[[(-2)-category|category]] and $(-2)$-[[(-2)-groupoid|groupoid]], which mean the same thing for their own reasons.
The point of $(-1)$-posets is that they complete some patterns in the [[periodic table]]s and complete the general concept of $n$-[[n-poset|poset]]. For example, there should be a $0$-[[0-poset|poset]] $(-1)\Pos$ of $(-1)$-posets; a $0$-poset is simply a truth value, and $(-1)\Pos$ is the [[true]] truth value.
As a category, $(-1)\Pos$ is a [[monoidal category]] in a unique way, and a [[enriched category|category enriched]] over this should be (at least up to equivalence) a $0$-poset, which is a truth value; and indeed, a category enriched over $(-1)\Pos$ is a category in which any two objects are isomorphic in a unique way, which is [[equivalence of categories|equivalent]] to a truth value.
See [[(−1)-category]] for references on this sort of [[negative thinking]].
[[!redirects (−1)-poset]]
|
(-2)-category | https://ncatlab.org/nlab/source/%28-2%29-category | There is just one _$(-2)$-category_, namely the truth value [[True]]. Compare the concepts of [[(−1)-category]] (a [[truth value]] in general) and [[0-category]] (a [[set]]). The point of $(-2)$-categories is that they complete some patterns in the [[periodic table]] of $n$-categories. (They also shed light on the theory of [[homotopy group]]s and [[n-stuff]].)
For example, there should be a $(-1)$-category of $(-2)$-categories; this is the true truth value. The category of $(-2)$-categories is a [[monoidal category]] in a unique way; then a [[enriched category|category enriched]] over this is a $(-1)$-category; such is necessarily an enriched groupoid. If you think of a $(-1)$-category as a [[0-poset]], then this makes a $(-2)$-category a [[(−1)-poset]]. If you think of a $(-1)$-category as a [[(−1)-groupoid]], then this makes a $(-2)$-category a [[(−2)-groupoid]].
For an introduction to $(-1)$-[[(-1)-category|categories]] and $(-2)$-categories see [page 11](http://arxiv.org/PS_cache/math/pdf/0608/0608420v2.pdf#page=11) of
* John C. Baez, Michael Shulman, _[[Lectures on n-Categories and Cohomology]]_ ([arXiv](http://arxiv.org/abs/math.CT/0608420)).
$(-1)$-categories and $(-2)$-categories were discovered (or invented) by [[James Dolan]] and [[Toby Bartels]]. To witness these concepts in the process of being discovered, read the discussion here:
* John Baez, Toby Bartels, David Corfield and James Dolan,
[Property, structure and stuff](http://math.ucr.edu/home/baez/qg-spring2004/discussion.html). See also [[stuff, structure, property]] for more on that material.
[[!redirects (-2)-category]]
[[!redirects (-2)-categories]]
[[!redirects (−2)-category]]
[[!redirects (−2)-categories]]
|
(-2)-groupoid | https://ncatlab.org/nlab/source/%28-2%29-groupoid |
+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Higher category theory
+--{: .hide}
[[!include higher category theory - contents]]
=--
=--
=--
#Contents#
* table of contents
{:toc}
## Definition
A **$(-2)$-groupoid** or **[[homotopy n-type|(-2)-type]]** is a [[n-truncated object of an (infinity,1)-category|(-2)-truncated object]] in [[∞Grpd]].
There is, up to equivalence, just one $(-2)$-groupoid, namely the [[point]].
## Remarks
Compare the concepts of $(-1)$-[[(-1)-groupoid|groupoid]] (a [[truth value]]) and $0$-[[0-groupoid|groupoid]] (a [[set]]). Compare also with $(-2)$-[[(-2)-category|category]] and $(-1)$-[[(-1)-poset|poset]], which mean the same thing for their own reasons.
The point of $(-2)$-groupoids is that they complete some patterns in the [[periodic table]]s and complete the general concept of $n$-[[n-groupoid|groupoid]]. For example, there should be a $(-1)$-[[(-1)-groupoid|groupoid]] $(-2)\Grpd$ of $(-2)$-groupoids; a $(-1)$-groupoid is simply a truth value, and $(-2)\Grpd$ is the [[true]] truth value.
As a category, $(-2)\Grpd$ is a [[monoidal category]] in a unique way, and a [[enriched category|groupoid enriched]] over this should be (at least up to equivalence) a $(-1)$-groupoid, which is a truth value; and indeed, a groupoid enriched over $(-2)\Grpd$ is a groupoid in which any two objects are isomorphic in a unique way, which is [[equivalence of categories|equivalent]] to a truth value.
See [[(-1)-category]] for references on this sort of [[negative thinking]].
## Related concepts
* [[true]]
[[!include homotopy n-types - table]]
[[!redirects (?2)-groupoid]]
[[!redirects (-2)-type]]
[[!redirects (-2)-types]]
|
(0,1)-category | https://ncatlab.org/nlab/source/%280%2C1%29-category |
+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### $(0,1)$-Category theory
+-- {: .hide}
[[!include (0,1)-category theory - contents]]
=--
=--
=--
# Contents
* table of contents
{: toc}
## Idea
{#Idea}
Following the general concept of *[[(n,r)-category|$(n,r)$-category]]*, a *$(0,1)$-category* is a [[category]] whose [[hom-objects]] are [[(-1)-groupoids]], hence which for every [[pair]] of [[objects]] $a,b$ have either no morphism $a \to b$ or an [[essentially unique]] one.
More in detail, recall that:
* an [[(n,1)-category]] is an [[(∞,1)-category]] such that every [[hom-space|hom ∞-groupoid]] is [[n-truncated|(n-1)-truncated]].
* a [[0-truncated]] [[∞-groupoid]] is equivalently a set;
* a [[(-1)-truncated]] [[∞-groupoid]] is either [[contractible]] or [[empty set|empty]].
Therefore:
+-- {: .num_remark}
###### Remark
**([[relation between preorders and (0,1)-categories]])**
\linebreak
An $(0,1)$-category is [[equivalence of categories|equivalently]] a [[proset]] ([[relation between preorders and (0,1)-categories|hence]] a [[poset]]).
We may without restriction assume that every hom-$\infty$-groupoid is just a set. Then since this is [[(-1)-truncated]] it is either empty or the singleton. So there is at most one morphism from any object to any other.
=--
## Extra stuff, structure, property
* A $(0,1)$-category with the structure of a [[site]] is a [[(0,1)-site]]: a [[posite]].
* A $(0,1)$-category with the structure of a [[topos]] is a [[(0,1)-topos]]: a [[Heyting algebra]].
* A $(0,1)$-category with the structure of a [[Grothendieck topos]] is a [[Grothendieck (0,1)-topos]]: a [[frame]] or [[locale]].
* A $(0,1)$-category which is also a [[groupoid]] (that is, every morphism is an isomorphism) is a $(0,0)$-category (which may think of as either a $0$-category or as a $0$-groupoid), which is the same as a [[set]] (up to equivalence) or a [[symmetric proset]] (up to isomorphism).
## Related concepts
* [[relation between preorders and (0,1)-categories]]
* [[(-2)-groupoid]]
* [[(-1)-groupoid]]
* [[0-groupoid]]
* [[1-groupoid]]
* [[2-groupoid]]
* [[3-groupoid]]
* [[4-groupoid]]
* [[n-groupoid]]
* [[∞-groupoid]]
* [[(0,0)-category]]
* [[(1,0)-category]]
* [[(n,0)-category]]
* [[(∞,0)-category]]
* [[0-category]]
* [[1-category]]
* [[2-category]]
* [[3-category]]
* [[4-category]]
* [[n-category]]
* [[∞-category]]
* **(0,1)-category**
* [[(1,1)-category]]
* [[(2,1)-category]]
* [[(n,1)-category]]
* [[(∞,1)-category]]
* [[(1,2)-category]]
* [[(∞,2)-category]]
* [[(∞,n)-category]]
* [[(n,r)-category]]
[[!redirects (0,1)-category]]
[[!redirects (0,1)-categories]]
|
(0,1)-category theory | https://ncatlab.org/nlab/source/%280%2C1%29-category+theory |
+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### $(0,1)$-Category theory
+--{: .hide}
[[!include (0,1)-category theory - contents]]
=--
=--
=--
#Contents#
* table of contents
{:toc}
## Idea
In the context of [[higher category theory]] / [[(n,r)-categories]], a [[poset]] is equivalently regarded as a [[(0,1)-category]].
$(0,1)$-categories play a major role in [[logic]], where their objects are interpreted as [[propositions]], their morphisms as [[implications]] and [[limits]]/[[products]] and [[colimits]]/[[coproducts]] as [[logical conjunctions]] _[[and]]_ and _[[or]]_, respectively.
[[Stone duality|Dually]], $(0,1)$-categories play a major role in [[topology]], where they are interpreted as [[categories of open subsets]] of a [[topological spaces]], or, more generally, of [[locales]].
Clearly, much of [[category theory]] simplifies drastically when restricted to $(0,1)$-categories, but it is often most useful to make the parallel explicit.
## Related concepts
[[!include table of category theories]]
|
(0,1)-category theory - contents | https://ncatlab.org/nlab/source/%280%2C1%29-category+theory+-+contents |
**[[(0,1)-category theory]]**: [[logic]], [[order theory]]
**[[(0,1)-category]]**
* [[relation between preorders and (0,1)-categories]]
* [[proset]], [[partially ordered set]] ([[directed set]], [[total order]], [[linear order]])
* [[top]], [[true]],
* [[bottom]], [[false]]
* [[monotone function]]
* [[implication]]
* [[filter]], [[interval]]
* [[lattice]], [[semilattice]]
* [[meet]], [[logical conjunction]], [[and]]
* [[join]], [[logical disjunction]], [[or]]
* [[compact element]]
* [[lattice of subobjects]]
* [[complete lattice]], [[algebraic lattice]]
* [[distributive lattice]], [[completely distributive lattice]], [[canonical extension]]
* [[hyperdoctrine]]
* [[first-order hyperdoctrine|first-order]], [[Boolean hyperdoctrine|Boolean]], [[coherent hyperdoctrine|coherent]], [[tripos]]
**[[(0,1)-topos]]**
* [[Heyting algebra]]
* [[regular element]]
* [[Boolean algebra]]
* [[frame]], [[locale]]
## Theorems
* [[Stone duality]]
|
(0,1)-presheaf | https://ncatlab.org/nlab/source/%280%2C1%29-presheaf | +-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### (0,1)-Category theory
+--{: .hide}
[[!include (0,1)-category theory - contents]]
=--
#### Topos theory
+--{: .hide}
[[!include topos theory - contents]]
=--
=--
=--
#Contents#
* table of contents
{:toc}
## Idea
A **(0,1)-presheaf** is a [[presheaf]] with values in the [[(0,1)-category]]
of [[truth values]]. A 0-[[truncated]] [[(∞,1)-presheaf]].
## Definition
A **(0,1)-presheaf** on a [[poset]] or [[proset]] $P$ is an [[antitone function|antitone]] [[predicate]]
$$F:P \rightarrow \Omega$$
from $P$ to the poset $\Omega$ of truth values, or equivalently, a [[monotone]] predicate
$$F:P^\op \rightarrow \Omega$$
from the [[opposite poset]] of $P$ to $\Omega$.
More generally, for a poset $S$, a **S-valued (0,1)-presheaf** on $P$ is just an antitone
$$f:P \rightarrow S$$
so (0,1)-presheaves are just antitones.
## (0,1)-category of (0,1)-presheaves
The [[(0,1)-category]] of a (0,1)-presheaf on a [[(0,1)-site]] forms a [[(0,1)-topos]]. In traditional order theoretic language, the poset (or proset) of (0,1)-presheaves on a [[posite]] forms a [[locale]].
## Related concepts
* [[lower set]]
* **(0,1)-presheaf**, [[(0,1)-sheaf]]
* [[presheaf]]
* [[(2,1)-presheaf]]
[[presheaf of groupoids]]
[[(2,1)-sheaf]], [[2-sheaf]], [[indexed category]], [[stack]]
* [[(∞,1)-presheaf]]
* [[(∞,n)-presheaf]]
[[!redirects (0,1)-presheaves]]
[[!redirects proposition-valued presheaf]]
[[!redirects proposition-valued presheaves]] |
(0,1)-topos | https://ncatlab.org/nlab/source/%280%2C1%29-topos |
+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### $(0,1)$-Category theory
+--{: .hide}
[[!include (0,1)-category theory - contents]]
=--
#### Topos Theory
+--{: .hide}
[[!include topos theory - contents]]
=--
=--
=--
#Contents#
* automatic table of contents goes here
{:toc}
## Idea
The notion of $(0,1)$-topos is that of [[topos]] in the context of [[(0,1)-category theory]].
The notion of $(0,1)$-topos is essentially equivalent to that of [[Heyting algebra]]; similarly, a [[Grothendieck topos|Grothendieck]] $(0,1)$-topos is a [[locale]].
Notice that every $(1,1)$-[[Grothendieck topos]] comes from a [[localic groupoid]], i.e. a [[groupoid]] [[internal category|internal to]] locales, hence a groupoid internal to $(0,1)$-toposes. See [[classifying topos of a localic groupoid]] for more.
## Related concepts
[[!include flavors of higher toposes -- list]]
## References
* [[John Baez|J. C. Baez]], [[Mike Shulman|M. Shulman]], _Lectures on n-categories and cohomology_ , pp.1-68 in J. C. Baez, P. May (eds.), _Towards Higher Categories_, Springer Heidelberg 2010. ([preprint](http://math.ucr.edu/home/baez/cohomology.pdf); section 5.3)
* [[Jacob Lurie]], _[[Higher Topos Theory]]_ , Princeton UP 2009. (section 6.4.2)
[[!redirects Grothendieck (0,1)-topos]]
[[!redirects 0-topos]]
[[!redirects Grothendieck 0-topos]]
[[!redirects (0,1)-toposes]].
|
(1,0)-category | https://ncatlab.org/nlab/source/%281%2C0%29-category | +-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### Higher category theory
+-- {: .hide}
[[!include higher category theory - contents]]
=--
=--
=--
By the general rules of $(n,r)$-[[(n,r)-category|categories]], a $(1,0)$-category is an $\infty$-[[infinity-category|category]] such that
* any $j$-morphism is an [[equivalence]], for $j \gt 0$;
* any two parallel $j$-morphisms are equivalent, for $j \gt 1$.
You can start from any notion of $\infty$-category, strict or weak; up to [[equivalence of categories|equivalence]], the result is always the same as a [[groupoid]].
[[!redirects (1,0)-categories]] |
(1,1)-category | https://ncatlab.org/nlab/source/%281%2C1%29-category |
+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Higher category theory
+--{: .hide}
[[!include higher category theory - contents]]
=--
=--
=--
In the pattern of [[(n,r)-categories]] a **$(1,1)$-category** is just an ordinary [[category]].
[[!redirects (1,1)-categories]]
|
(1,1)-dimensional Euclidean field theories and K-theory | https://ncatlab.org/nlab/source/%281%2C1%29-dimensional+Euclidean+field+theories+and+K-theory |
+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Functorial quantum field theory
+--{: .hide}
[[!include functorial quantum field theory - contents]]
=--
#### Super-Geometry
+--{: .hide}
[[!include supergeometry - contents]]
=--
=--
=--
+-- {: .standout}
This is a sub-entry of [[geometric models for elliptic cohomology]] and [[A Survey of Elliptic Cohomology]]
See there for background and context.
This entry here indicates how 1-dimensional [[FQFT]]s (the [[superparticle]]) may be related to [[topological K-theory|topological]] [[K-theory]].
=--
> **raw material**: this are notes taken more or less verbatim in a seminar -- needs polishing
Previous:
* [[Axiomatic field theories and their motivation from topology]].
Next
* [[(2,1)-dimensional Euclidean field theories and tmf]]
#Contents#
* table of contents
{:toc}
## $(1,1)d$ EFTs
recall the **commercial for supergeometry** with which we ended [[Axiomatic field theories and their motivation from topology|last time]]: the grading introduced by supergeometry makes it possible to have push-forward diagrams of the kind:
$$
\array{
(0|1)TFTs^n(X)/\simeq &\leftarrow& H^n_{dR}(X)
\\
\downarrow && \downarrow
\\
(0|1)TFT^0(X)/\simeq &\leftarrow& H^0_{dR}(pt)
}
$$
**Example** of 1-EFT
$$
\sigma_1(M^n) = E : 1-EB \to tV
$$
$$
pt \mapsto \Gamma M
$$
$$
(pt \stackrel{[0,t]}{\to})
\mapsto
e^{- t \Delta}
$$
**Example** of $(1|1)-EFT$ associated to a [[Spin structure|spin manifold]], there is the [[spinor bundle]]
$$
S = S^+ \oplus S^-
$$
a $\mathbb{Z}/2$-graded [[vector bundle]] and on this there is the [[Dirac operator]]
$$
D : \Gamma(S) \to \Gamma(S)
$$
where $\Gamma(S) = \Gamma(S^+) \oplus \Gamma(S^-)$. So we can write
$$
D =
\left(
\array{
0 & D_-
\\
D+ & 0
}
\right)
$$
$$
\sigma_{1|1}(M) : Bord_{1|1} \to TV
$$
$$
\mathbb{R}^{0|1} \mapsto E(\mathbb{R}^{0|1}) = \Gamma(S)
$$
there is an involution $invol : \mathbb{R}^{0|1} \to \mathbb{R}^{0|1}$. It maps to
$$
invol \mapsto grading involution
$$
we have the following [[moduli space]] of super [[interval]]s (super 1d-bordisms)
$$
\mathbb{R}^{1|1}_+ \simeq
\{super intervals I_{t,\theta}\}/\sim
$$
and these are mapped by the EFT as
$$
I_{t,\theta} \mapsto e^{-t D^2 + \theta D}
$$
(here we are implicitly working in the [[topos]] of [[sheaf|sheaves]] on the category of [[supermanifold]]s and these equations have to be interpreted in that topos-logic, mapping [[generalized element]]s to [[generalized element]]s).
So we have for $E$ a $1|1$ EFT a _reduced_ non-susy field theory
$$
\array{
(1|1)EBord &\stackrel{E}{\to}& TV
\\
\uparrow & \nearrow_{E_{red}}
\\
EBord_1^{spin}
}
$$
**Definition** $E \in (1|1)EFT$, the [[partition function]] $Z_E$ of $E$ is the function
$$
Z_E : \mathbb{R}_+ \to \mathbb{C}
$$
$$
t \mapsto Z_{E_{red}}(t) = E_{red}(S^1_t)
$$
that sends a length to the value of the EFT on the circle of that circumferene.
**Example** Consider from above the EFT
$$
E = \sigma_{1|1}(M)
$$
look at its reduced part
$$
z_E(t) = E_{red}(S^1_t)
$$
notice that by the above this assigns
$$
[0,t] \stackrel{E_{red}}{\mapsto} e^{-t D^2}
$$
$$
S^1_t \mapsto str(e^{-t D^2})
=
tr(e^{-t D^2})|_{even} - tr(e^{-t D^2})|_{odd}
$$
where on the right we have the [[super trace]].
This evaluates to
$$
str(e^{-t D^2})
=
\sum_{\lambda \in Spec(D^2)}
e^{-t \lambda}
sdim E_{\lambda}
$$
where the [[super dimension]] of the [[eigenspace]] $E_\lambda$ is
$$
dim E^+_\lambda - dim E^-_\lambda
$$
and this vanishes for $\lambda \neq 0$ since there $D : E_\lambda^+ \stackrel{\simeq}{\to} E_\lambda^-$
is an [[isomorphism]].
So further in the computation we have
$$
\cdots = dim ker D_+ - dim coker D_+
=
\hat A(M)
$$
where the last step is the [[Atiyah-Singer index theorem]].
So **due to supersymmetry** , the [[partition function]] has two very special properties:
* it is constant -- in that it does not depend on $t$,
* it takes integer values $\in \mathbb{N} \subset \mathbb{R}$.
**recall** from $V \to X$ a [[vector bundle]] [[connection on a bundle|with connection]] $\nabla$ we get a 1d EFT
$$
E_{(V,\nabla)} \in 1d EFT(X)
$$
given by the assignment
$$
E_{(V,\nabla)} : 1s EB(X) \to TV
$$
$$
(x : pt \to X) \mapsto V_x = fiber of V over x
$$
a morphism is an [[interval]] $[0,t]$ of length $t$ equipped with a map $\gamma : [0,t] \to X$, this is sent to the [[parallel transport]] associated with the [[connection on a bundle]]
$$
\gamma \mapsto (V_{\gamma_x} \to V_{\gamma_y})
$$
Now refine this example to super-dimension $(1|1)$:
**example** of a $(1|1)$-EFT over $X$ consider
$$
EBord_{(1|1)} \to EBord_{1}(X) \stackrel{E_{(V,\nabla)}}{\to}
TV
$$
given by the assignment
$$
(\Sigma^{(1|1)} \to X)(
\mapsto
(\Sigma^{(1|1)}_{red} \to X)
\mapsto
parallel transport as before
$$
so we just forget the super-part and consider the same [[parallel transport]] as before.
now to [[K-theory]]:
$
KO^0(X) =
$ [[Grothendieck group]] of real [[vector bundle]]s over $X$
$$
KO^{-n}(pt) =
\left\{
\array{
\mathbb{Z} & n = 0 mod 4
\\
\mathbb{Z}_2 & n = 1,2 mod 8
\\
0 & otherwise
}
\right.
$$
there is a [[Bott element]] $\beta \in KO^{-8}(pt)$
such that
$$
KO^*(pt) \stackrel{\simeq_{\mathbb{Q}}}{\to}
\mathbb{Z}[u,u^{-1}]
$$
$$
\beta \mapsto u^2
$$
now the **push-forward in [[topological K-theory]]**
$$
p : X^n \to pt
$$
for $X$ a closed [[spin structure]] manifold
then there exists an embedding $X \hookrightarrow S^{n+m}$. Let $\nu$ be the [[normal bundle]] to this embedding.
then we define
$$
\int_X : KO^k(X) \to KO^{k-n}(pt)
$$
as follows:
let $D(\nu)$ be the [[disk bundle]] and $S(\nu)$ be the [[sphere bundle]] of $\nu$. Then the [[Thom bundle]] is
$$
T(\nu) := D(\nu)/S(\nu)
$$
we get a map
$$
S^{n+m} \stackrel{C}{\to} T(\nu) := D(\nu)/S(\nu)
$$
involving the [[Thom isomorphism]]
$$
C(X) = \left\{
\array{
X & if x \in D(\nu)
\\
* & otherwise
}
\right.
$$
then we set
$$
\array{
KO^k(X)
&& \stackrel{\int_X}{\to}&& KO^{k-n}(pt)
\\
& {}_{Thom iso}\searrow
&&&
\downarrow^{\simeq}_{suspension}
\\
&& \tilde KO^{k+m}(T(\nu))
&\stackrel{C^*}{\to}&
}
$$
***
now start with $X^n$ again a [[spin structure|spin]] [[manifold]]
then
**theorem** (Stolz-Teichner): we have the horizontal isomorphism in the following diagram:
$$
\array{
&& [E_{(V,\nabla)}]&& \stackrel{}{\leftarrow}
&& [V^+ - V^-]
\\
1 \in
&&(1|1)EFT^0(X)/_{conc}
&&\stackrel{\simeq}{\to}&& KO^0(X) && \ni 1
\\
\downarrow &&\downarrow^{quantization}
&&&& \downarrow^{\int_X} && \downarrow
\\
\sigma_{(1|1)}(X)
&&EFT^{-n}(pt)/_{conc}
&&\stackrel{\simeq}{\to}&& KO^{-n} && \alpha(X)
\\
&\searrow&&{}_{partition func}\searrow&&
\swarrow_{\simeq} && \swarrow_{Atiyah's \alpha invariant}
\\
&&&&
(\mathbb{Z}[u,u^{-1}])^{-n}
\\
&&&&
index D = \hat A(X) u^{n/4}
}
$$
**question** if we don't divide out [[concordance]], do we get [[differential K-theory]] on the right?
**answer** presumeably, but not worked out yet
## Related concepts
* [[supersymmetric quantum mechanics]]
* [[Euclidean quantum field theory]]
* [[spectral triple]]
* [[spectral action]]
* [[higher category theory and physics]]: <a href="http://ncatlab.org/nlab/show/higher+category+theory+and+physics#SpecStandModAndGravity">Spectral standard model and gravity</a>
* **(1,1)-dimensional Euclidean field theories and K-theory**
* [[(2,1)-dimensional Euclidean field theories and tmf]]
* [[2-spectral triple]]
## References
{#References}
* [[Stephan Stolz]], [[Peter Teichner]], _[[What is an elliptic object?]]_ in _Topology, geometry and quantum field theory_ , London Math. Soc. LNS 308, Cambridge Univ. Press (2004), 247-343. ([pdf](http://web.me.com/teichner/Math/Reading_files/Elliptic-Objects.pdf))
* Pokman Cheung, _Supersymmetric field theories and cohomology_ ([arXiv:0811.2267](http://arxiv.org/abs/0811.2267))
* {#Stolz} [[Stefan Stolz]] (notes by Arlo Caine), _Supersymmetric Euclidean field theories and generalized cohomology_ Lecture notes (2009) ([pdf](http://www.cpp.edu/~jacaine/pdf/Lectures_complete.pdf))
* [[Stefan Stolz]], [[Peter Teichner]], _Supersymmetric Euclidean field theories and generalized cohomology_ , in [[Hisham Sati]], [[Urs Schreiber]] (eds.), _[[schreiber:Mathematical Foundations of Quantum Field and Perturbative String Theory]]_ Proceedings of Symposia in Pure Mathematics, AMS (2011) |
(1,2)-topos | https://ncatlab.org/nlab/source/%281%2C2%29-topos | +-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Topos Theory
+--{: .hide}
[[!include topos theory - contents]]
=--
=--
=--
#Contents#
* table of contents
{:toc}
## Idea
The notion of *(1,2)-topos* should be the notion of [[higher toposes]] among [[(1,2)-categories]] or [[2-posets]].
## Examples
\begin{example}\label{The12ToposOfPosets}
The [[(1,2)-category]] [[Pos]] of [[posets]] and monotone maps should be the archetypal $(1,2)$-topos.
The poset of [[truth values]]
$$
\big( \bot \to \top \big)
\;\in\;
Pos
$$
should play the role of the "sub-poset classifier" in $Pos$, the (1,2)-analog of the [[subobject classifier]] in a [[1-topos]].
Here, morphisms into it
classify [[monic]] [[fibrations]] of posets, namely [[sieves]] (e.g. [Exp. 9.26 here](https://www.andrew.cmu.edu/course/80-413-713/notes/chap09.pdf#page=31))
\end{example}
## Related concepts
[[!include flavors of higher toposes -- list]]
|
(2,1)-algebraic theory of E-infinity algebras | https://ncatlab.org/nlab/source/%282%2C1%29-algebraic+theory+of+E-infinity+algebras | [[!redirects (2,1)-algebraic theeory of E-infinity algebras]]
+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Higher algebra
+--{: .hide}
[[!include higher algebra - contents]]
=--
=--
=--
#Contents#
* table of contents
{:toc}
## Idea
The [[(∞,1)-algebraic theory]] whose [[∞-algebra over an (∞,1)-algebraic theory|algebras]] are [[E-∞ algebra]]s is the [[(2,1)-category]] of [[span]]s of [[finite set]]s.
## Definition
+-- {: .num_defn}
###### Definition
Let
$$
2Comm := Span(FinSet)
$$
be the [[(2,1)-category]] of [[span]]s of [[finite set]]s:
* [[object]]s are finite sets;
* [[morphism]]s are [[span]]s $X_1 \leftarrow Y \to X_1$ in [[FinSet]];
* [[2-morphism]]s are diagrams
$$
\array{
&& Y
\\
& \swarrow && \searrow
\\
X_0 &&\downarrow^{\mathrlap{\simeq}}&& X_1
\\
& \nwarrow && \nearrow
\\
&& Y'
}
$$
in [[FinSet]] with the vertical morphism an [[isomorphism]].
=--
+-- {: .num_prop}
###### Observation
The [[homotopy category of an (infinity,1)-category|homotopy category]] of $2Comm$ is the category $Comm$ that is the [[Lawvere theory]] of commutative [[monoid]]s.
=--
+-- {: .proof}
###### Proof
The Lawvere theory of commutative monoids has as objects the free commutative monoids $F[k]$ on $k \in \mathbb{N}$ generators, and as morphisms monoid homomorphisms.
By the [[free functor|free property]], morphisms
$$
f : F[k] \to F[l]
$$
are in natural bijection to $k$-tuples of elements of $F[l]$. Such elements in turn are sums $a_1 + a_1 + \cdots + a_1 + a_2 + a_2 + \cdots + a_2 + a_3 + \cdots$ of copies of the $l$ generators, hence are in bijection to sequences of natural numbers $(n_{1}, \cdots, n_l)$. Hence morphisms $f : F[k] \to F[l]$ are in bijection to $k \times l$-[[matrices]] with entries in the [[natural number]]s.
One checks that under this identification composition of morphisms corresponds to matrix multiplication.
=--
+-- {: .num_remark}
###### Remark
For instance the spans
$$
\{1,2\} \stackrel{id}{\leftarrow} \{1,2\} \to \{1\}
$$
and
$$
\{1,2\} \stackrel{\simeq}{\leftarrow} \{2,1\} \to \{1\}
$$
describe the operation
$$
(a,b) \mapsto a + b
$$
and the operation
$$
(a,b) \mapsto b + a
\,,
$$
respectively. Clearly, in $Comm$ both these operations are identified. In $2Comm$ however they the are only equivalent
$$
\array{
&& \{1,2\}
\\
& {}^{\mathllap{id}}\swarrow && \searrow
\\
\{1,2\} &&\downarrow^{\mathrlap{\simeq}}&& \{1\}
\\
& {}_{\mathllap{\simeq}}\nwarrow && \nearrow
\\
&& \{2,1\}
}
\,.
$$
=--
## Properties
+-- {: .num_lemma}
###### Observation
Let $Comm$ be the ordinary [[Lawvere theory]] of commutative monoids. There is a forgetful 2-functor
$$
2Comm \to Comm
\,.
$$
This exhibits $2Comm$ as being like $Comm$ but with some additional auto-2-morphisms of the morphism of $Comm$.
=--
This is discussed in ([Cranch, beginning of section 5.2](#Cranch)).
+-- {: .num_prop}
###### Proposition
The $(\infty,1)$-category $2Comm$ has finite [[product]]s. The products of objects $A, B$ in $2Comm$ is their [[coproduct]] $A \coprod B$ in [[FinSet]].
=--
This appears as ([Cranch, prop. 4.7](#Cranch)).
+-- {: .num_prop}
###### Proposition
An [[(∞,1)-category]] with [[(∞,1)-limit|(∞,1)-product]] is naturally an algebra over the $(2,1)$-theory $2Comm$.
=--
This appears as ([Cranch, theorem 4.26](#Cranch)).
+-- {: .num_theorem}
###### Theorem
An algebra over the $(2,1)$-theory $2Comm$ in [[(∞,1)Cat]] is
a [[symmetric monoidal (∞,1)-category]].
=--
This appears as ([Cranch, theorem 5.3](#Cranch)).
+-- {: .num_theorem}
###### Theorem
There is a $(2,1)$-algebraic theory $E_\infty$ whose algebras
are at least in parts like [[E-∞ algebra]]s.
=--
This is ([Cranch](#Cranch)), prop. 6.12, theorem 6.13 and section 8.
## Examples {#Examples}
### Free algebras
The free algebra over $2Comm$ in [[∞Grpd]] on a single generator is $2Comm(*, -) : 2Comm \to \infty Grpd$. Its underlying [[∞-groupoid]] is therefore
$$
2Comm(*,*) = Core(FinSet)
\,,
$$
the [[core]] groupoid of the category [[FinSet]]. This is equivalent to
$$
\cdots \simeq \coprod_{n \in \mathbb{N}} \mathbf{B} \Sigma_n
\,,
$$
where $\Sigma_n$ is the [[symmetric group]] on $n$ elements and $\mathbf{B}\Sigma_n$ its one-object [[delooping]] groupoid.
Notice that this is indeed the free [[E-∞-algebra]], on the nose so if we use the [[Barratt-Eccles operad]] $P$ as our model for the [[E-∞-operad]]: that has $P_n = \mathbf{E} \Sigma_n$. The free [[algebra over an operad]] is given by $\coprod_{n \in \mathbb{N}} P_n/\Sigma_n$, which here is $\cdots = \coprod_{n \in \mathbb{N}} \mathbf{E}\Sigma_n/\Sigma_n = \coprod_n \mathbf{B} \Sigma_n$.
## References
* [[James Cranch]], _Algebraic Theories and $(\infty,1)$-Categories_ ([arXiv](http://arxiv.org/abs/1011.3243))
{#Cranch}
[[!redirects (2,1)-algebraic theory of E-∞ algebras]]
[[!redirects (2,1)-algebraic theory of E-∞-algebras]] |
(2,1)-category | https://ncatlab.org/nlab/source/%282%2C1%29-category |
+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### 2-Category theory
+--{: .hide}
[[!include 2-category theory - contents]]
=--
#### Higher category theory
+--{: .hide}
[[!include higher category theory - contents]]
=--
=--
=--
#Contents#
* table of contents
{:toc}
## Idea
By the general rules of $(n,r)$-[[(n,r)-category|categories]], a __$(2,1)$-category__ is an $\infty$-[[infinity-category|category]] such that
* any $j$-morphism is an [[equivalence]], for $j \gt 1$;
* any two parallel $j$-morphisms are equivalent, for $j \gt 2$.
You can start from any notion of $\infty$-category, strict or weak; up to [[equivalence of categories|equivalence]], the result can always be understood as a [[locally groupoidal 2-category|locally groupoidal]] $2$-[[2-category|category]].
## Models
So, a (2,1)-category is in particular modeled by
* a [[2-category]] in which all [[2-morphisms]] are invertible;
* an [[(∞,1)-category]] that is 2-[[truncated]].
## Properties
The [[oidification]] of a [[monoidal groupoid]] is a (2,1)-category.
## Related concepts
* [[hom-groupoid]]
* [[strict (2,1)-category]]
* [[equivalence of (2,1)-categories]]
* [[concrete (2,1)-category]]
* [[monoidal (2,1)-category]], [[symmetric monoidal (2,1)-category]]
* [[monoidal groupoid]]
## References
The special case of [[strict (2,1)-categories]], motivated from the [[homotopy 2-category]] of [[topological spaces]]:
* [[Peter H. H. Fantham]], [[Eric J. Moore]], *Groupoid Enriched Categories and Homotopy Theory*, Canadian Journal of Mathematics **35** 3 (1983) 385-416 ([doi:10.4153/CJM-1983-022-8](https://doi.org/10.4153/CJM-1983-022-8))
[[!include oidification - table]]
[[!redirects (2,1)-categories]]
|
(2,1)-dimensional Euclidean field theories and tmf | https://ncatlab.org/nlab/source/%282%2C1%29-dimensional+Euclidean+field+theories+and+tmf | +-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Functorial quantum field theory
+--{: .hide}
[[!include functorial quantum field theory - contents]]
=--
#### Super-Geometry
+--{: .hide}
[[!include supergeometry - contents]]
=--
=--
=--
+-- {: .standout}
This is a sub-entry of [[geometric models for elliptic cohomology]] and [[A Survey of Elliptic Cohomology]]
See there for background and context.
This entry here indicates how 2-dimensional [[FQFT]]s may be related to [[tmf]].
=--
> **raw material**: this are notes taken more or less verbatim in a seminar -- needs polishing
Previous:
* [[Axiomatic field theories and their motivation from topology]].
* [[(1,1)-dimensional Euclidean field theories and K-theory]]
recall the big diagram from the end of the [[(1,1)-dimensional Euclidean field theories and K-theory|previous entry]].
The **goal** now is to replace everywhere [[topological K-theory]] by [[tmf]].
previously we had assumed that $X$ has [[spin structure]]. Now we assume [[String structure]].
So we are looking for a diagram of the form
$$
\array{
1
&&
(2|1)EFT^0(X)/\sim
&& \stackrel{\simeq conjectural}{\leftarrow}&&
tmf^0(X)
&& \ni 1
\\
\downarrow && \downarrow^{quantization}
&&&&
\downarrow^{\int_X}
&&
\downarrow
\\
\sigma_{(2|1)(X)}&& (2|1)EFT^{-n}(X)/\sim
&&\stackrel{\simeq conjectural }{\leftarrow}&&
tmf^{-n}(pt)
&&
\\
&\searrow & \searrow &&& \swarrow& \swarrow
\\
&&&& mf^{-n}
\\
&&&&
index^{S^1}(D_{L X})
= W(X)
}
$$
the vertical maps here are due to various theorems by various people -- except for the "physical quantization" on the left, that is used in physics but hasn't been formalized
the **horizontal maps are the conjecture we are after** in the Stolz-Teichner program: The top horizontal map will involve making the notion of $(2|1)$EFT _local_ by refining it to an _extended_ [[FQFT]]s. This will not be considered here.
we will explain the following items
* the [[ring]] $mf^\bullet$ of [[integral modular form]]s
$$
mf^\bullet \simeq \mathbb{Z}[c_4, c_6, \Delta, \Delta^{-1}]/(c_4^{3}- c_6^{2} - 1728 \Delta)
$$
one calls $w = -\frac{n}{2}$ the _weight_ . We have degree of $\Delta$ is $deg(\Delta) = -24$, hence $w(\Delta) = 12$.
* $W(X)$ is the [[Witten genus]]
$$
W(X) = \sum_{k \in \mathbb{Z}} a_k \cdot q^k
\,,
a_k \in \mathbb{Z}
$$
where $a_k = index(D_X \otimes E_k)$ where $E_k$ is some explicit vector bundle over $X$.
## modular forms ##
**definition** An **(integral) [[modular form]]** of weight $w$ is a [[holomorphic function]] on the [[upper half plane]]
$$
f : (\mathbb{R}^2)_+ \hookrightarrow \mathbb{C}
$$
(complex numbers with strictly positive imaginary part)
such that
1. if $A = \left( \array{a & b \\ c& d}\right) \in SL_2(\mathbb{Z})$ acting by $A : \tau \mapsto = \frac{a \tau + b }{c \tau + d}$ we have
$$
f(A(\tau)) = (c \tau + d)^w f(\tau)
$$
**note** take $A = \left( \array{1 & 1 \\ 0& 1}\right)$ then we get that $f(\tau + 1) = f(\tau)$
1. $f$ has at worst a pole at $\{0\}$ (for _weak_ modular forms this condition is relaxed)
it follows that $f = f(q)$ with $q = e^{2 \pi i \tau}$ is a meromorphic funtion on the open disk.
1. **integrality** $\tilde f(q) = \sum_{k = -N}^\infty a_k \cdot q^k$ then $a_k \in \mathbb{Z}$
by this definition, modular forms are not really functions on the upper half plane, but functions on a [[moduli space]] of [[torus|tori]]. See the following definition:
if the weight vanishes, we say that modular form is a **[[modular function]]** .
**definition (2|1)-dim [[partition function]]**
Let $E$ be an EFT
$$
(2|1)EFT^0 \stackrel{S}{\to}
2 EFT \ne E
$$
$$
E \mapsto E_{red}
$$
then the [[partition function]] is the map $Z_E : \mathbb{C} \to \mathbb{R}$
$$
Z_E : \tau \mapsto E_{red}(T_\tau)
$$
where
$$
T_\tau := \mathbb{C}/{\mathbb{Z} \times \mathbb{Z} \cdot \tau}
$$
is thee standard torus of modulus $\tau$.
then the central **theorem** that we are after here is
**therorem (Stolz-Teichner)** (after a suggestion by [[Edward Witten]])
There is a precise definition of $(2|1)$-EFTs $E$ such that the [[partition function]] $Z_E$ is an integral [[modular function]]
(so this is really four theorems: the function is holomorphic, integral, etc.)
moreover, every [[integral modular function]] arises in this way.
## Related concepts
A concrete relation between [[2d SCFT]] and [[tmf]] is the lift of the [[Witten genus]] to the [[string orientation of tmf]]. See there fore more.
## References
* [[Stephan Stolz]], [[Peter Teichner]], _[[What is an elliptic object?]]_ in _Topology, geometry and quantum field theory_ , London Math. Soc. LNS 308, Cambridge Univ. Press (2004), 247-343. ([pdf](http://web.me.com/teichner/Math/Reading_files/Elliptic-Objects.pdf))
* Pokman Cheung, _Supersymmetric field theories and cohomology_ ([arXiv:0811.2267](http://arxiv.org/abs/0811.2267))
* [[Stefan Stolz]], [[Peter Teichner]], _Supersymmetric Euclidean field theories and generalized cohomology_ , in [[Hisham Sati]], [[Urs Schreiber]] (eds.), _[[schreiber:Mathematical Foundations of Quantum Field and Perturbative String Theory]]_
A hint supporting the conjectured relation of [[2d SCFT]] to [[tmf]], vaguely in line with the lift of the [[Witten genus]] to the [[string orientation of tmf]]:
* [[Davide Gaiotto]], [[Theo Johnson-Freyd]], _Holomorphic SCFTs with small index_ ([arXiv:1811.00589](https://arxiv.org/abs/1811.00589))
|
(2,1)-functor | https://ncatlab.org/nlab/source/%282%2C1%29-functor |
+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### 2-Category theory
+--{: .hide}
[[!include 2-category theory - contents]]
=--
#### $(\infty,1)$-Category theory
+--{: .hide}
[[!include quasi-category theory contents]]
=--
=--
=--
#Contents#
* table of contents
{:toc}
## Idea
The concept of _$(2,1)$-functors_ is that of the natural kind of [[morphisms]] between [[(2,1)-categories]].
If [[(2,1)-categories]] are regarded as special cases of [[2-categories]], then $(2,1)$-functors are equivalently the [[2-functors]] between [[(2,1)-categories]].
If [[(2,1)-categories]] are regarded as special cases of [[(∞,1)-categories]], then $(2,1)$-functors are equivalently the [[(∞,1)-functors]] between [[(2,1)-categories]].
## Examples
* The construction of [[groupoid convolution algebras]] constitutes a $(2,1)$-functor from [[differentiable stacks]] with proper maps between them and the [[opposite 2-category|opposite]] $(2,1)$-category of [[C*-algebras]] with [[Hilbert bimodules]] between them. For details see [there](#category+algebra#GroupoidConvolutionIs2Functor).
[[!redirects (2,1)-functors]]
|
(2,1)-presheaf | https://ncatlab.org/nlab/source/%282%2C1%29-presheaf |
+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### 2-Category theory
+--{: .hide}
[[!include 2-category theory - contents]]
=--
#### $(\infty,1)$-Topos Theory
+--{: .hide}
[[!include (infinity,1)-topos - contents]]
=--
=--
=--
#Contents#
* table of contents
{:toc}
## Idea
A **(2,1)-presheaf** is a [[presheaf]] with values in the [[(2,1)-category]] [[Grpd]]. A 2-[[truncated]] [[(∞,1)-presheaf]].
Sometimes this is also called a **prestack**. Other times a prestack is more specifically taken to be a [[separated (infinity,1)-presheaf|separated (2,1)-presheaf]]: a $(2,1)$-presheaf such that the functors into its [[descent]] objects are [[full and faithful functor]]s.
The [[∞-stackification]] of a $(2,1)$-presheaf is a certain [[2-sheaf]] or [[stack]].
## Related concepts
* [[(0,1)-presheaf]]
* [[presheaf]]
* **(2,1)-presheaf**
[[presheaf of groupoids]]
[[(2,1)-sheaf]], [[2-sheaf]], [[indexed category]], [[stack]]
* [[(∞,1)-presheaf]]
* [[(∞,n)-presheaf]]
[[!redirects (2,1)-presheaves]]
[[!redirects prestack]]
[[!redirects prestacks]]
[[!redirects pre-stack]]
[[!redirects pre-stacks]]
[[!redirects groupoid-valued presheaf]]
[[!redirects groupoid-valued presheaves]] |
(2,1)-sheaf | https://ncatlab.org/nlab/source/%282%2C1%29-sheaf |
+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Locality and descent
+--{: .hide}
[[!include descent and locality - contents]]
=--
#### 2-Category theory
+--{: .hide}
[[!include 2-category theory - contents]]
=--
#### $(\infty,1)$-Topos Theory
+--{: .hide}
[[!include (infinity,1)-topos - contents]]
=--
=--
=--
#Contents#
* table of contents
{:toc}
## Idea
A $(2,1)$-sheaf is a [[sheaf]] with values in [[groupoid]]s. This is traditionally called a [[stack]].
## Definition
Let $C$ be a [[(2,1)-site]]. Write [[Grpd]] for the [[(2,1)-category]] of [[groupoid]]s, [[functor]]s and [[natural isomorphism]]s.
A **$(2,1)$-sheaf** on $C$ is equivalently
* a [[2-functor]] $C^{op} \to Grpd$ that satisfies [[descent]];
* a [[2-sheaf]] with values in [[groupoid]]s;
* a [[1-truncated]] [[(∞,1)-sheaf]] on $C$.
## The $(2,1)$-category of $(2,1)$-sheaves
The [[(2,1)-category]] of a $(2,1)$-sheaves on a [[(2,1)-site]] forms a [[(2,1)-topos]].
There are [[model category]] [[presentable (infinity,1)-category|presentations]] of this $(2,1)$-topos. See [[model structure for (2,1)-sheaves]].
## Related concepts
* [[sheaf]]
* [[2-sheaf]] / **$(2,1)$-sheaf** / [[stack]]
* [[(∞,1)-sheaf]] / [[∞-stack]]
* [[(∞,n)-sheaf]]
* [[descent]]
* [[cover]]
* [[cohomological descent]]
* [[descent morphism]]
* [[monadic descent]],
* [[Sweedler coring]]
* [[higher monadic descent]]
* [[descent in noncommutative algebraic geometry]]
[[!include homotopy n-types - table]]
[[!redirects (2,1)-sheaves]]
[[!redirects (2,1)-category of (2,1)-sheaves]] |
(2,1)-sheafification | https://ncatlab.org/nlab/source/%282%2C1%29-sheafification | [[!redirects (2,1)-sheafififcation]]
+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### $(\infty,1)$-Topos Theory
+--{: .hide}
[[!include (infinity,1)-topos - contents]]
=--
=--
=--
#Contents#
* table of contents
{:toc}
## Idea
The higher analogue of [[sheafification]].
A [[stack]]/[[2-sheaf]] [[2-topos]] on a [[site]]/[[2-site]] $C$ is
$$
St(C) \stackrel{\overset{L}{\leftarrow}}{\hookrightarrow} PSt(C)
\,.
$$
The [[left adjoint]] $L$ to the inclusion of [[stack]]s into all [[2-functor]]s on $C^{op}$ is _stackification_ .
## Related concepts
* [[sheafification]]
* **$(2,1)$-sheafification**
* [[(∞,1)-sheafification]]
[[!redirects stackification]]
[[!redirects stackifications]]
|
(2,1)-site | https://ncatlab.org/nlab/source/%282%2C1%29-site |
+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### $(\infty,1)$-Topos Theory
+-- {: .hide}
[[!include (infinity,1)-topos - contents]]
=--
=--
=--
# Contents
* table of contents
{:toc}
## Definition
A _$(2,1)$-site_ is an [[(∞,1)-site]]. whose underlying [[(∞,1)-category]] is a [[(2,1)-category]]. Equivalently, it is a [[2-site]] whose underlying [[2-category]] is a [[(2,1)-category]].
## Properties
The [[(2,1)-category]] of [[(2,1)-sheaves]] on a (2,1)-site is a Grothendieck-[[(2,1)-topos]].
The [[(∞,1)-category of (∞,1)-sheaves]] on a $(2,1)$-site is an [[n-localic (∞,1)-topos|2-localic (∞,1)-topos]].
## Related concepts
* [[site]]
* [[2-site]], **(2,1)-site**
* [[(2,1)-sheaf]]
* [[(∞,1)-site]]
* [[model site]], [[simplicial site]]
[[!redirects (2,1)-site]]
[[!redirects (2,1)-sites]]
|
(2,1)-topos of (2,1)-sheaves | https://ncatlab.org/nlab/source/%282%2C1%29-topos+of+%282%2C1%29-sheaves |
## Idea
A [[(2,1)-topos]] of [[(2,1)-sheaves]]
## Related concepts
* [[sheaf topos]]
* [[(infinity,1)-category of (infinity,1)-sheaves]]
|
(2|1)-dimensional Euclidean field theory | https://ncatlab.org/nlab/source/%282%7C1%29-dimensional+Euclidean+field+theory | [[!redirects (2,1)-dimensional Euclidean field theory]]
[[!redirects (2,1)-dimensional Euclidean field theory]]
[[!redirects (2,1)-dimensional Euclidean field theory]]
+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Quantum field theory
+--{: .hide}
[[!include functorial quantum field theory - contents]]
=--
#### Supergeometry
+--{: .hide}
[[!include supergeometry - contents]]
=--
=--
=--
+-- {: .standout}
This is a sub-entry of [[geometric models for elliptic cohomology]] and [[A Survey of Elliptic Cohomology]]
See there for background and context.
This entry here is about the definition of $(2|1)$-dimensional [[super-cobordism]] categories where cobordisms are [[Euclidean supermanifold]]s, and about $the (2|1)$-dimensional [[FQFT]]s given by functors on these.
=--
Previous:
* [[Axiomatic field theories and their motivation from topology]]
* [[(1,1)-dimensional Euclidean field theories and K-theory]]
* [[(2,1)-dimensional Euclidean field theories and tmf]]
* [[bordism categories following Stolz-Teichner]]
#Contents#
* tic
{:toc}
#Idea#
Previously we had defined smooth categories of [[Riemannian cobordism]]s. Now we pass from [[Riemannian manifold]]s to [[Euclidean supermanifold]]s and define the corresponding smooth [[cobordism category]]. Then we define $(d|\delta)$-dimensional Euclidean field theories to be smooth representations of these categories.
As described at [[(2,1)-dimensional Euclidean field theories and tmf]], the idea is that $(2|1)$-dimensional Euclidean field theories are a geometric model for [[tmf]] [[cohomology theory]]. While there is no complete proof of this so far, in the next and final session
* [[modular forms from partition functions]]
it will be shows that the claim is true at least for the [[cohomology ring]] over the point: the [[partition function]] of a $(2|1)$-dimensional EFT is a modular form. Hence $(2|1)$-dimensional EFTs do yield the correct [[cohomology ring]] of [[tmf]] over the point.
#Details#
Let [[SDiff]] be the [[category]] of [[supermanifold]]s.
We will define a [[stack]]/[[fibered category]] on $SDiff$ called $E Bord_{2|1}$ whose morphisms are smooth families of (2|1)-dimensional [[super-cobordism]]s, and a [[stack]]/[[fibered category]] $sTV^{fam}$ of topological super vector bundles.
So recall
* [[supergeometry - contents|supergeometry]].
**question**: What is the right notion of Riemannian or Euclidean structure on [[super-cobordism]]s?
**strategy**: From the [[path integral]] perspective we need some structure on $\Sigma$ such that the "space" of maps $Maps(\Sigma,X)$ naturally carries some measure that allows to perform a [[path integral]].
This perspective suggests certain generalizations of the notion of [[Riemannian manifold]] to [[supermanifold]]s which may be a little different than what one might have thought of naively.
We want to define [[Euclidean supermanifold]]s as a generalization of [[Riemannian manifold]] with _flat_ Riemannian metric.
notice that there is a canonical bijection between
* flat [[Riemannian metric]]s on a $d$-dimensional [[manifold]] $X$
* a maximal [[atlas]] on $X$ consisting of charts such that all transition functions belong the the **Euclidean group** or **Galileo group**
$$
Eucl(\mathbb{R}^d)
:=
\mathbb{R}^d \rtimes O(\mathbb{R}^d)
$$
(rigid translations and rotations)
In analogy to that we define:
Similarly a **Euclidean structure** on a $(d|\delta)$-dimensional [[supermanifold]] is defined using the Euclidean [[super Lie group]] given by the [[semidirect product]]
$$
Eucl(\mathbb{R}^{d|\delta})
:=
\mathbb{R}^{d|\delta}
\rtimes
Spin(\mathbb{R}^d)
$$
where the [[Spin]] group acts on the translations in $\mathbb{R}^{d|\delta}$ in a way to be specified.
first recall the notion of
* [[complex supermanifold]]
**goal** replace the standard Euclidean group $(\mathbb{R}^d, Eucl(\mathbb{R}^d))$ by the [[super Euclidean group]]
$(X,G)$ where $X$ is a suitable [[supermanifold]]
and $G$ a suitable [[super Lie group]].
This leads to the notion of
* [[Euclidean supermanifold]].
The morphisms of the category $E Bord_{(2|1)}$ will be [[cobordism]]s that are [[Euclidean supermanifold]]s.
**goal** define the [[fibered category]]
$$
\array{
E Bord_{d|\delta}^{sfam}
\\
\downarrow
\\
cSDiff
}
$$
where $cSDiff$ is the category of [[complex supermanifold]]s.
The objects of this fibered category are
$$
\array{
Y^{\pm} &\to& Y &\leftarrow& Y^c
\\
& \searrow & \downarrow & \swarrow
\\
&& S
}
$$
where $Y \to S$ is a family of [[Euclidean supermanifold]]s of dimension $(d|\delta)$.
For the non-super, non-family version of **Euclidean bordism** we require that the core $Y^c$ is totally geodesic in $Y$.
now for the superversion we require that there exist charts (in the open atlas) of $Y \to S$ covering all of $Y^c$ such that
$$
\array{
&& S
\\
& \nearrow && \nwarrow
\\
Y \supset_{open} U &&\stackrel{\phi}{\to}&&
V \subset S \times \mathbb{R}^{d|\delta}_{cs}
\\
\downarrow^{\supset} &&&& \downarrow^{\supset}
\\
Y^c \supset U \cap Y^c
&&\stackrel{\simeq}{\to}&&
V \cap S \times \mathbb{R}^{d-1|\delta}
\subset S \times \mathbb{R}^{d-1|\delta}_{cs}
}
$$
next, a **Euclidean superbordism** from $Y_0 \to S$ to $Y_1 \to S$ is a diagram
$$
\array{
Y_1
&\stackrel{i_1}{\to}&
\Sigma
&\stackrel{i_0}{\leftarrow}& Y_1
\\
& \searrow & \downarrow & \swarrow
\\
&& S
}
$$
where $i_0, i_1$ are morphisms (of families of $(X,G)$-spaces) satisfying the (+)-condition and the (c)-condition described at [[bordism categories following Stolz-Teichner]].
Now a morphism in $E Bord^{sfam}_{d|\delta}$ from $Y_0 \to S_0$ to $Y_1 \to S_1$ is a bordism fitting into a diagram
$$
\array{
\Sigma &\stackrel{i_1}{\leftarrow}&
f^* Y_1 &\to& Y_1
\\
\uparrow^{i_0} &\searrow& \downarrow && \downarrow
\\
Y_1 &\to & S_0 &\stackrel{f}{\to}& S1
}
$$
and we identify bordisms $\Sigma, \Sigma'$ if they are isometric -- namely isomorphic in the category of [[Euclidean supermanifold]]s -- "rel boundary".
**definition** A **$(d|\delta)$-dimensional Euclidean field theory** is a [[symmetric monoidal functor]]
$$
E \in Fun_{csDiff}^\otimes(E Bord_{(d|\delta)}^{sfam},
TV^{sfam})
$$
of [[symmetric monoidal category|symmetric monoidal]] [[fibered category|fibered cateories]] (i. [[symmetric monoidal functor]] as well as [[cartesian functor]] ) over the category $cSDiff$ of [[complex supermanifold]]s.
**Definition** (roughly) $TV^{sfam}$ is the category of families of topological vector spaces parameterized by [[complex supermanifold]]s.
Recall that the ordinary category $TV$ is the category of complete [[Hausdorff space|Hausdorff]], locally convex [[topological vector space]].
define the [[projective tensor product]] of two such $V, W \in TV$. This is a certain completion of the algebraic tensor product $V \otimes_{alg} W$ with respect to the projective topology on $V \otimes_{alg} W$.
This will be the coarsest [[topology]] (the one with the least open sets) making the following maps $f'$
$$
\array{
V \otimes_{alg} W &\to& Z
\\
& \nearrow_{f'}
\\
V \times W
}
$$
continuous.
**Remark**
$$
\array{
C^\infty(M \times N)
&\leftarrow&
C^\infty(M) \otimes_{alg} C^\infty(N)
\\
& {}_{\simeq}\nwarrow & \downarrow^{\subset}
\\
&& C^\infty(M) \otimes C^\infty(N)
}
$$
**Definition** the objects of $TV^{sfam}$ are pairs $(S,V)$ for $S$ a [[supermanifold]] and $V$ is a [[sheaf]] of locally complex $\mathbb{Z}_2$-graded [[topological vector space]] with the structure of a sheaf of modules of the [[structure sheaf]] $O_S$.
**goal** define the [[partition function]] of of a $(2|1)$-dimensional Euclidean field theory.
**definition** Let $E$ be an EFT as above.
We may think of $\mathbb{R}_+ \times h$ (positive axis times upper half plane) as moduli space of Euclidean tori, where for $(\ell, \tau) \in \mathbb{R}_+ \times h$ we get a torus (regarded as a [[cobordism]]) denoted $T_{\ell,\tau}$. This is the torus given by the lattice spanned by $(1,0)$ and $\ell Re(\tau) + Im(\tau)$ in the upper half plane. Then for the ordinary EFT we would define
$$
Z_E : \mathbb{R}_+ \times h \to \mathbb {C}
$$
$$
(\ell,\tau) \mapsto E(T_{\ell,\tau})
$$
For the superversion we put
$$
Z_{E} := Z_{E_{red}}
$$
where
$$
\array{
&& E Bord_{2|1}^{sfam}
\\
& {}^{\rho}\nearrow
&&
\searrow^{E}
\\
E Bord_{2, Spin}^{fam}
&\stackrel{E_{red}}{\to}&
TV^{fam}
&
\hookrightarrow
&
TV^{sfam}
}
$$
# Examples #
## explicit description of $E Bord_{1}^{fam}$ ##
See [[bordism categories following Stolz-Teichner]].
The category $E Bord_1^{fam}$ is generated from
* the _family of right elbows__
$$
\array{
1-E Bord_{\mathbb{R}_+}^{fam}(\emptyset, pt \coprod pt)
& \ni& R & := \mathbb{R}_+ \times \mathbb{R}
\\
&& \downarrow
\\
&& S := \mathbb{R}_+
}
$$
* the point-family of the left elbox
$$
\array{
L_0
\\
\downarrow
\\
S := pt
}
$$
* the family of intervals in $E Bord^{fam}_{\mathbb{R}^+}(pt,pt)$
$$
\array{
I
\\
\downarrow
\\
\mathbb{R}_{\geq 0}
}
$$
Because:
$
E \in Fun^{\otimes}_{Diff}(E Bord^{fam}, TV^{fam})
$
is determined by
$$
E(pt) =: V \in TV
$$
$$
E(L_0) =: \lambda : V \otimes V \to \mathbb{R}
$$
$$
E(R) =: \rho \in TV_{\mathbb{R}^+}(\mathbb{R}, V \otimes V) \simeq C^\infty(\mathbb{R}_+, V \otimes V)
$$
$$
E(I) =: e^{-t H} \in C^\infty(\mathbb{R}_{\geq}, End(V))
$$
forms a _smooth_ semigroup under composition generated by
$$
H \in End(V)
$$
(the [[Hamiltonian operator]])
$$
\array{
V \otimes V &&\stackrel{\lambda}{\hookrightarrow}&&
End(V)
\\
& {}_{\rho}\nwarrow && \nearrow_{e^{- t H}}
\\
&& \mathbb{R}_+
}
$$
so due to smoothness the data collapses to the infinitesimal data
$$
(V, \lambda, H)
$$
**example -- ordinary quantum mechanics** Let $M$ be a [[Riemannian manifold]]. Then set
* $H:= \Delta$ the corresponding [[Laplace operator]];
* $V := C^\infty(M) \subset L^2(M)$;
* $\lambda$ is the restriction of the $L^2(M)$ inner product to $V$
where $e^{-t H}$ is "trace class" in the non-standard sense described above in that it makes the above diagram commute.
So everything as known from standard [[quantum mechanics]] textbooks, except that we don't use the full [[Hilbert space]] of states, but just the [[Frechet space]] of smooth functions.
## explicit description of $E Bord_{2}$ ##
The category $E Bord_{2}_{oriented}^{fam}$ has the following generators:
objects are generated from
* the **circle** $K_\ell := \mathbb{R}^2/\mathbb{Z}\cdot \ell$ of length $\ell \gt 0$ (with collars!! that's why it looks like a cylinder of circumference $\ell$)
notice that we may think of $\ell $ as parameteriing translation by $\ell$ in $\mathbb{R}^2 \rtimes SO(2) = Eucl_{or}(\mathbb{R}^2)$
and the circle with $(+)$/$(-)$-collars reversed
morphisms are generated from
* **cylinders** $C_{\ell,\tau}$ which as a manifold is $\simeq \mathbb{R}^2/\mathbb{Z}\cdot \ell$ where $\tau$ parameterizes the embedding of the outgoing circle: the incoming circle is embedded in the canonical way (the identity map on the cylinder, really), while the outgoing circle is embedded by translating the cylinder by $\ell \cdot Re(\tau)$ upwards and rotated by $\ell \cdots Im(\tau)$.
* **right elbows** which are the same as the cylinder, except that now the second circle is embedded after reflection so that it becomes an ingoing circle.
* the **thin left elbow** $L_\ell$, similar to the above, with arbitrary $\ell$ but $\tau = 0$
* the **torus** $T_\tau$ obtained from the cylinder by gluing incoming and outgoing
**notice** the **pair of pants** is not a morphism in the category at all! since, recall, we require all bordisms to be _flat_ and all boundaries to be _geodesics_ . There is no way to put such a flat metric on the trinion.
satisfying the relations
$$
L_\ell \circ R_{\ell, \tau}
=
T_{\ell, \tau}
$$
as for the non-family version, but now also with the new relations
$$
T_{\ell', \tau'} = T_{\ell, \tau}
$$
whenever $\ell' = \ell \cdot|c \tau + d|$ and $\tau' = \frac{a \tau + b }{c \tau + d}$
for $\left(\array{a & b \\ c & d }\right) \in SL_2(\mathbb{Z})$.
Notice that $SL_2(\mathbb{Z})$ is generated by
* translation $(\ell, \tau) \mapsto (\ell, \tau + 1)$
* S-matrix $(\ell, \tau) \mapsto (\ell \cdot |\tau|, - \frac{1}{\tau})$
and now there is **one more relation**
$$
T_{\ell, \tau}
=
T_{\ell |\tau|, - \frac{1}{\tau}}
$$
as usual write $q := e^{2 \pi i \tau}$ which is on the pointed unit disk since $\tau$ is half plane since $\tau$
## explicit description of $E Bord_{2}^{fam}$ ##
thwe category $E Bord_{2, oriented}^{fam}$ is generated from
objects:
* $\array{K \\ \downarrow \\ S = \mathbb{R}_+}$
* morphisms
$$
\array{
L \\ \downarrow \\ \mathbb{R}_+
}
$$
$$
\array{
R \\ \downarrow \\ \mathbb{R}_+ \times h/\mathbb{Z}
}
$$
$$
\array{
C \\ \downarrow \\ \mathbb{R}_+ \times (h \cup \mathbb{R})/\mathbb{Z}
}
$$
subject to the relations
... as before (homework 3, problem 4).. and the furhter one:
for
$$
\array{
T && \alpha^* T
\\
\downarrow && \downarrow
\\
\mathbb{R}_+ \times h
&\stackrel{\alpha}{\leftarrow}&
\mathbb{R}_+ \times h
\\
\\
(\ell\cdot |\tau|, -\frac{1}{\tau})
&\stackrel{}{\lt\leftarrow}&
(\ell, \tau)
}
$$
the relation is
$$
\alpha^* T = T
\,.
$$
## References
* [[Stephan Stolz]], [[Peter Teichner]], *[[What is an elliptic object?]]* in: _Topology, geometry and quantum field theory_, London Math. Soc. LNS **308**, Cambridge Univ. Press (2004) 247-343 [[pdf](https://math.berkeley.edu/~teichner/Papers/Oxford.pdf), [[Stolz-Teichner_EllipticObject.pdf:file]]]
* {#Stolz} [[Stefan Stolz]] (notes by Arlo Caine), _Supersymmetric Euclidean field theories and generalized cohomology_ Lecture notes (2009) ([pdf](http://www.nd.edu/~jcaine1/pdf/Lectures_complete.pdf))
* [[Stefan Stolz]], [[Peter Teichner]], _Supersymmetric field theories and generalized cohomology_ , in: [[Hisham Sati]], [[Urs Schreiber]] (eds.), *[[Mathematical Foundations of Quantum Field and Perturbative String Theory]]*, Symposia in Pure Mathematics (2011) [[arXiv:1108.0189](http://arxiv.org/abs/1108.0189)]
* [[Daniel Berwick-Evans]], *How do field theories detect the torsion in topological modular forms?* [[arXiv:2303.09138](https://arxiv.org/abs/2303.09138)]
* [[Daniel Berwick-Evans]], *How do field theories detect the torsion in topological modular forms?*, talk at *[QFT and Cobordism](https://nyuad.nyu.edu/en/events/2023/march/quantum-field-theories-and-cobordisms.html)*, [[CQTS]] (Mar 2023) [[web](Center+for+Quantum+and+Topological+Systems#BerwickEvansMar23)]
[[!redirects (2,1)-dimensional Euclidean field theories]] |
(bo, ff) factorization system | https://ncatlab.org/nlab/source/%28bo%2C+ff%29+factorization+system |
+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### Factorization systems
+--{: .hide}
[[!include factorization systems - contents]]
=--
#### Category theory
+-- {: .hide}
[[!include category theory - contents]]
=--
=--
=--
#Contents#
* table of contents
{:toc}
## Definition
There is an [[orthogonal factorization system]] on the category [[StrCat]], whose left class is the class of [[bo functor|bijective-on-objects functors]], or "bo functors" and whose right class is the class of [[full and faithful functor|full and faithful functors]], or "ff functors".
This means that each functor $f$ decomposes as a composition of the form $j e$, where $e$ is bijective on objects and $j$ fully faithful; and if
$$\array{
A &\overset{u}{\longrightarrow}& C
\\
\mathllap{{}^{e}}\big\downarrow &&\big\downarrow \mathrlap{{}^{j}}
\\
B &\underset{v}\longrightarrow& D
}$$
is a commutative diagram with $e$ bijective on objects and $j$ fully faithful, then there is a unique functor $h \colon B\to C$ such that $h e = u$ and $j h = v$. The object through which $f$ factors is called the [[full image]] of $f$.
In fact, this can be generalized to a square commuting up to invertible [[natural transformation]], in which case one still concludes that $h e = u$ but that $j h \cong v$, with the isomorphism composing with $e$ to give the original isomorphism. This means that this is an [[enhanced factorization system]].
## Properties
This factorization system can be constructed using [[generalized kernels]].
For [[essentially surjective functors]], one can relax both the commuting and the uniqueness to obtain a [[factorization system in a 2-category]].
[[!redirects (bo,ff) factorization system]]
[[!redirects bo-ff factorization system]]
[[!redirects (bo,ff) factorization]]
[[!redirects bo-ff factorization]]
[[!redirects (bo,ff) factorisation system]]
[[!redirects bo-ff factorisation system]]
[[!redirects (bo,ff) factorisation]]
[[!redirects bo-ff factorisation]]
|
(classical) axiom of multiple choice | https://ncatlab.org/nlab/source/%28classical%29+axiom+of+multiple+choice |
+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### Foundations
+-- {: .hide}
[[!include foundations - contents]]
=--
=--
=--
# The axiom of multiple choice
* table of contents
{: toc}
This article is about a [[classical mathematics|classical]] [[set theory]] [[axiom]]. Some literature instead uses this name for an unrelated weakening of [[axiom of choice|AC]]. For that notion, see _[[axiom of multiple choice]]_.
## Idea
The **axiom of multiple choice** (AMC) weakens the [[axiom of choice]] by allowing [[choice functions]] to choose [[finite sets]], rather than particular [[elements]].
##Statement
Recall that one common statement of the [[axiom of choice]] is:
:For every [[set]] $S$ of [[inhabited set|non-empty sets]], there is a [[function]] $f$ defined on $S$ such that $\forall X\in S$, $f(X)\in X$.
Such an $f$ is called a __choice function__ for $S$.
The axiom of multiple choice weakens the axiom of choice by allowing choice functions to pick out finite [[subsets]], rather than finite sets. It says:
:For every set $S$ of non-empty sets, there is a function $f$ defined on $S$ such that $\forall X\in S$, $f(X)\subseteq X$ and $f(X)$ is finite and non-empty.
## Relationships to other axioms
The axiom of multiple choice is [[logical equivalence|equivalent]] to the axiom of choice modulo [[ZF]] set theory. However, it is strictly weaker in [[ZFA]] and other similar set theories. AMC holds in any [[permutation model]] with finite supports where each atom has a finite orbit. For a detailed proof, see [Jech's "The Axiom of Choice"](https://www.gwern.net/docs/math/1973-jech-theaxiomofchoice.pdf), chapter 9.
The [[axiom of multiple choice|constructive axiom by the same name]] is not historically related, and the two axioms are independent. Any permutation model will satisfy [[SVC]], which [Rathjen](#Rathjen) proves implies the constructive axiom, but this AMC can fail in a permutation model.
## References
* Jech, _The Axiom of Choice_ (1973), ISBN : 0444104844 (New York)
* A. Lévy. Axioms of multiple choice. Fundamenta mathematicae, vol. 50 no. 5 (1962), pp. 475–483
The constructive axiom by the same name is discussed in:
* {#Rathjen} Rathjen, "Choice principles in constructive and classical set theories"
category: foundational axiom
|
(co)isotropic subspaces - table | https://ncatlab.org/nlab/source/%28co%29isotropic+subspaces+-+table |
| type of [[subspace]] $W$ of [[inner product space]] | condition on [[orthogonal]] space $W^\perp$ | |
|--|--|--|
| [[isotropic subspace]] | $W \subset W^\perp $ | |
| [[coisotropic subspace]] | $W^\perp \subset W$ | |
| [[Lagrangian subspace]] | $W = W^\perp$ | (for [[symplectic form]]) |
| [[symplectic vector space|symplectic space]] | $W \cap W^\perp = \{0\}$ | (for [[symplectic form]])|
|
(d,d) Klein space | https://ncatlab.org/nlab/source/%28d%2Cd%29+Klein+space |
+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Geometry
+--{: .hide}
[[!include higher geometry - contents]]
=--
#### Differential geometry
+--{: .hide}
[[!include synthetic differential geometry - contents]]
=--
=--
=--
> not to be confused with [[Klein geometry]]
#Contents#
* table of contents
{:toc}
## Idea
The [[Euclidean space|flat space]] $\mathbb{R}^{d,d}$ of $2d$ dimensions equipped with a [[pseudo-Riemannian metric]] of $(d,d)$ signature.
## Related concepts
* [[Klein geometry]]
* [[pseudo-Riemannian manifold]]
* [[smooth Lorentzian space]]
## References:
* [[John Barrett]], [[Gary W. Gibbons]], M. J. Perry, [[Christopher N. Pope]], P. J. Ruback, *Kleinian Geometry and the $N=2$ Superstring*, Int. J. Mod. Phys. **A9** (1994) 1457-1494 [[doi:10.1142/S0217751X94000650](https://doi.org/10.1142/S0217751X94000650), [arXiv:hep-th/9302073](https://arxiv.org/abs/hep-th/9302073)] |
(dense,closed)-factorization | https://ncatlab.org/nlab/source/%28dense%2Cclosed%29-factorization |
+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### Topos Theory
+-- {: .hide}
[[!include topos theory - contents]]
=--
=--
=--
#Contents#
* table of contents
{:toc}
## Idea
In [[topology|point set topology]], every subspace $X$ of a space $A$ has a unique largest subspace in which $X$ is [[dense subspace|dense]], namely simply the closure $\overline{X}$. Using maps, this amounts to say that $X\hookrightarrow A$ factors as $X\hookrightarrow\overline{X}$ followed by $\overline{X}\hookrightarrow A$.
The **(dense,closed)-factorization** generalizes this idea from topology to [[topos theory]]. It can be viewed as a way to associate to every [[subtopos]] $Sh_j(\mathcal{E})\hookrightarrow\mathcal{E}$ a closure $\overline{Sh_{j}(\mathcal{E})}$.
## Statement
A [[geometric embedding]] of [[elementary toposes]]
$$
Sh_j(\mathcal{E}) \hookrightarrow \mathcal{E}
$$
factors as
$$
Sh_j(\mathcal{E})
\hookrightarrow
Sh_{c(ext(j))}(\mathcal{E})
\hookrightarrow
\mathcal{E}
$$
where $ext(j)$ (the "exterior" of $j$) denotes the $j$-closure of $\emptyset \rightarrowtail 1$ and
$$
\bar j \coloneqq c(ext(j))
$$
the [[closed subtopos|closed topology]] that corresponds to the [[subterminal object]] $ext(j)$.
Here the first inclusion exhibits a [[dense subtopos]] and the second a [[closed subtopos]].
## Remark
$Sh_{c(ext(j))}(\mathcal{E})$ can be viewed as the 'best approximation' of $Sh_j(\mathcal{E})$ by a closed subtopos and therefore might be called the _closure_ $Cl(Sh_j(\mathcal{E}))$ of $Sh_j(\mathcal{E})$.[^sga4]
[^sga4]: ([SGA4](#SGA4), p.461) uses the term '_l'adhérence_' for it.
Its complement, the [[open subtopos]] $Ext(Sh_j(\mathcal{E}))$ corresponding to the [[subterminal object]] $ext(j)$ deserves in turn to be called the _exterior_ of $Sh_j(\mathcal{E})$.
## The (dominant,closed)-factorization
The (dense,closed)-factorization is a special case for inclusions of a slightly more general factorization which attaches to a general [[geometric morphism]] the closure of its image.
Recall that an inclusion is [[dense subtopos|dense]] precisely if it is a [[dominant geometric morphism]], hence the following is pertinent for the (dense,closed)-factorization as well.
+-- {: .num_prop #dense-closed}
###### Proposition
Let $i:Sh_{c(U)}(\mathcal{E})\hookrightarrow\mathcal{E}$ be dominant and a [[closed subtopos|closed inclusion]] at the same time. Then $i$ is an isomorphism.
=--
**Proof**: Recall that $X\in\mathcal{E}$ are in the [[closed subtopos]] precisely when they satisfy $X\times U\cong U$ with $U$ the [[subterminal object]] associated to $i$. But $i$ is dominant, or what comes to the same for inclusions: dense, hence $\emptyset$ is in $Sh_{c(U)}(\mathcal{E})$ and therefore $\emptyset\times U\cong U$ . From this follows $U\cong\emptyset$, which in turn implies that all $X\in\mathcal{E}$ are in $Sh_{c(U)}(\mathcal{E})$ . $\qed$
+-- {: .num_prop #dominant-closed}
###### Proposition
Let $f:\mathcal{F}\to\mathcal{E}$ be a geometric morphism. Then $f$ factors as a [[dominant geometric morphism]] $d$ followed by a [[closed subtopos|closed inclusion]] $c$.
=--
**Proof**: Let $i\circ d_1$ be the [[(geometric surjection, embedding) factorization system|surjection-inclusion factorization]] of $f$. Since $d_1$ is surjective, it is dominant (cf. [this proposition](dominant+geometric+morphism#dominant_surjection)). Then we use the (dense,closed)-factorization to factor $i$ into $c\circ d_2$. Since both $d_i$ are dominant, so is $d:=d_2\circ d_1$ and $c\circ d$ yields the demanded factorization of $f$. $\qed$
## Related entries
* [[dense]]
* [[dense subtopos]]
* [[dominant geometric morphism]]
* [[(geometric surjection, embedding) factorization system]]
## References
* {#SGA4}[[M. Artin]], [[A. Grothendieck]], [[J. L. Verdier]], _Théorie des Topos et Cohomologie Etale des Schémas ([[SGA4]])_, LNM **269** Springer Heidelberg 1972. (Exposé IV 9.3.4-9.4., pp.456ff)
* {#Johnstone}[[Peter Johnstone]], _[[Sketches of an Elephant]] vol.I_ , Oxford UP 2002. (around Lemma A 4.5.19, p. 219)
* {#Caramello09} [[Olivia Caramello]], _Lattices of theories_ , [arXiv:0905.0299](http://arxiv.org/abs/0905.0299) (2009). (section 8)
|
(epi, mono) factorization system | https://ncatlab.org/nlab/source/%28epi%2C+mono%29+factorization+system |
+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### Factorization systems
+--{: .hide}
[[!include factorization systems - contents]]
=--
#### Category theory
+-- {: .hide}
[[!include category theory - contents]]
=--
=--
=--
#Contents#
* table of contents
{:toc}
## Definition
An **epi-mono factorization system** is an [[orthogonal
factorization system]] in which the left class is the class of [[epimorphisms]] and the right class is the class of [[monomorphisms]]. Such a factorization system exists on any (elementary) [[topos]], and indeed on any [[pretopos]]. It provides the factorization through the [[image]] of any morphism.
## Properties
Note that any category which admits an epi-mono factorization system is necessarily [[balanced category|balanced]]. This excludes many commonly occurring categories. More common are ([[strong epimorphism|strong epi]], mono) and (epi, strong mono) factorization systems; the former exists in any [[regular category]] and the latter in any [[quasitopos]], as well as in other categories such as [[Top]].
The epi-mono factorization system in a topos is the special case of the [[n-connected/n-truncated factorization system]] in an [[(∞,1)-topos]] for the case that $(n = -1)$ and restricted to [[0-truncated]] [[object]]s.
## Related concepts
* **(epi, mono) factorization system**
* [[(eso+full, faithful) factorization system]]
* [[n-connected/n-truncated factorization system]].
## References
For instance:
* {#Borceux94} [[Francis Borceux]], vol 1, section 4.4. of: _[[Handbook of Categorical Algebra]]_, Cambridge University Press (1994)
[[!redirects (epi, mono) factorization system]]
[[!redirects (epi,mono)-factorization]]
[[!redirects (epi,mono) factorization]]
[[!redirects (epi, mono)-factorization]]
[[!redirects (epi, mono) factorization]]
[[!redirects (epi, mono)-factorization system]]
[[!redirects (epi, mono)-factorization systems]]
[[!redirects (epi,mono) factorization system]]
[[!redirects (epi,mono) factorization systems]]
[[!redirects (epi,mono)-factorization system]]
[[!redirects (epi,mono)-factorization systems]]
[[!redirects epi/mono factorization system]]
[[!redirects epi/mono factorization systems]]
[[!redirects epi-mono factorization system]]
[[!redirects epi-mono factorization systems]]
[[!redirects (effective epi, mono) factorization system]]
[[!redirects (effective epi, mono) factorization systems]]
[[!redirects (effective epi,mono) factorization system]]
[[!redirects (effective epi,mono) factorization systems]]
[[!redirects epi-mono factorization]]
[[!redirects epi-mono factorizations]]
|
(eso and full, faithful) factorization system | https://ncatlab.org/nlab/source/%28eso+and+full%2C+faithful%29+factorization+system | [[!redirects (eso+full, faithful) factorization system]]
+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### Factorization systems
+--{: .hide}
[[!include factorization systems - contents]]
=--
#### Category theory
+-- {: .hide}
[[!include category theory - contents]]
=--
=--
=--
#Contents#
* table of contents
{:toc}
## Definition
In the [[2-topos]] [[Cat]], the [[pair]] of [[classes]] of [[morphisms]]
* left class: [[essentially surjective and full functors]]
* right class: [[faithful functors]]
forms a [[factorization system in a 2-category]]. This factorization system can also be restricted to the [[(2,1)-topos]] [[Grpd]].
In fact, an analogous factorization system exists in any [[2-exact 2-category]] and any [[(2,1)-exact (2,1)-category]], including any [[Grothendieck 2-topos]] or [[(2,1)-topos]]; see [[michaelshulman:full morphism|here]].
## Properties
* When restricted to [[Grpd]], this is the special case of the [[n-connected/n-truncated factorization system]] in the [[(∞,1)-topos]] [[∞Grpd]] for the case that $(n = 0)$ and restricted to [[1-truncated]] [[objects]].
* For $f : X \to Y$ a functor between groupoids, its factorization is through a groupoid $im_2 f$ which is, up to equivalence, given as follows;
* [[objects]] are those of $X$;
* a [[morphism]] $[\phi] : x_1 \to x_2$ is an [[equivalence class]] of morphisms in $X$ where $[\phi] = [\phi']$ if $f(\phi) = f(\phi')$.
More on this is at _[infinity-image -- Of Functors between groupoids](infinity-image#NImagesOf1FunctorsBetweenGroupoids)_.
## Related concepts
* [[epi-mono factorization system]]
* [[(eso, fully faithful) factorization system]]
* **(eso+full, faithful)-factorization system**
* [[n-connected/n-truncated factorization system]].
[[!redirects essentially surjective and full/faithful factorization system]]
|
(eso, fully faithful) factorization system | https://ncatlab.org/nlab/source/%28eso%2C+fully+faithful%29+factorization+system |
+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### Factorization systems
+--{: .hide}
[[!include factorization systems - contents]]
=--
#### Category theory
+-- {: .hide}
[[!include category theory - contents]]
=--
=--
=--
#Contents#
* table of contents
{:toc}
## Definition
In the [[2-topos]] [[Cat]], the pair of classes of morphisms ([[essentially surjective functor|essentially surjective]] functors, [[fully faithful functors]]) form a [[factorization system in a 2-category]]. This factorization system can also be restricted to the [[(2,1)-topos]] [[Grpd]].
e.g. ([Dupont-Vitale 03, 7.9, example 2](#DupontVitale03))
In fact, an analogous factorization system exists in any regular 2-category and any (2,1)-exact (2,1)-category, including any Grothendieck 2-topos or (2,1)-topos; see [[michaelshulman:regular morphism|here]].
## Properties
* The factorization of a functor in this factorization system is the construction of its [[full image]].
* More on this for the case of functors between [[groupoids]] is at _[infinity-image -- Of Functors between groupoids](infinity-image#NImagesOf1FunctorsBetweenGroupoids)_.
## Related concepts
* [[(epi, mono) factorization system]]
* **(eso, fully faithful)-factorization system**
* [[(eso+full, faithful) factorization system]]
* [[ternary factorization system]]
## References
* {#DupontVitale03} M. Dupont, [[Enrico Vitale]], _Proper factorization systems in 2-categories_, Journal of pure and applied algebra, 179 (2003), pp.65-86
[[!redirects essentially surjective/fully faithful factorization system]]
[[!redirects essentially surjective-fully faithful factorization system]]
[[!redirects eso/ff factorization system]]
[[!redirects eso/fully faithful factorization system]]
[[!redirects eso-ff factorization system]]
[[!redirects eso-fully faithful factorization system]]
[[!redirects (eso, ff) factorization system]]
|
(g-2) anomaly | https://ncatlab.org/nlab/source/%28g-2%29+anomaly |
#Contents#
* table of contents
{:toc}
## Idea
For the moment see at _[[anomalous magnetic moment]]_ _[this section](https://ncatlab.org/nlab/show/anomalous+magnetic+moment#Anomalies)_
## Related concepts
* [[flavour anomaly]]
* [[Cabibbo anomaly]]
* [[V_cb puzzle]]
[[!redirects (g-2)-anomaly]]
|
(geometric surjection, embedding) factorization system | https://ncatlab.org/nlab/source/%28geometric+surjection%2C+embedding%29+factorization+system |
+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Topos Theory
+--{: .hide}
[[!include topos theory - contents]]
=--
=--
=--
#Contents#
* table of contents
{:toc}
## Idea
Every [[geometric morphism]] between [[toposes]] factors into a [[geometric surjection]] followed by a [[geometric embedding]]. This exhibits an [[image]] construction in the [[topos theory|topos-theoretic]] sense, and gives rise to a [[factorization system in a 2-category]] for [[Topos]].
## Statement
+-- {: .num_prop}
###### Proposition
There is a [[factorization system on a 2-category|factorization system]] on the [[2-category]] [[Topos]] whose left class is the
[[surjective geometric morphism]]s and whose right class is the [[geometric embeddings]].
Moreover, the factorization of a given geometric morphism $f : \mathcal{E} \to \mathcal{F}$ is, up to [[equivalence of categories|equivalence]], through the canonical surjection onto the [[topos of coalgebras]] $f^* f_* CoAlg(\mathcal{E})$ of the [[comonad]] $f^* f_* : \mathcal{E} \to \mathcal{E}$:
$$
\array{
\mathcal{E} &&\stackrel{f}{\to}&& \mathcal{F}
\\
& {}_{\mathllap{F}}\searrow && \nearrow
\\
&& f^* f_* CoAlg(\mathcal{E})
}
\,E.
$$
=--
This appears for instance as ([MacLaneMoerdijk, VII 4., theorem 6](#MacLaneMoerdijk)).
We use the following lemma
+-- {: .num_lemma #ConditionForSheafFactorization}
###### Lemma
Let $j$ be a [[Lawvere-Tierney topology]] on a [[topos]] $\mathcal{E}$ and write $i : Sh_j(\mathcal{E}) \to \mathcal{E}$ for the corresponding [[geometric embedding]].
Then a [[geometric morphism]] $f : \mathcal{F} \to \mathcal{E}$ factors through $i$ precisely if
* the [[direct image]] $f_*$ takes values in $j$-sheaves;
or, equivalently
* the [[inverse image]] $f^*$ sends $j$-[[dense monomorphism]]s to [[isomorphism]]s.
=--
This appears as ([MacLaneMoerdijk, VII 4. prop. 2](#MacLaneMoerdijk)).
+-- {: .proof}
###### Proof of the lemma
We first show the first statement, that for $f$ to factor it is sufficient for $f_*$ to take values in $j$-sheaves: in that case, set
$$
p_* := i^* f_*: \mathcal{F} \to Sh_j(\mathcal{E})
\,.
$$
Since by assumption the [[unit of an adjunction|unit]] map $x \to i_* i^* x$ is an [[isomorphism]] on the image of $f_*$ this indeed serves to factor $f_*$:
$$
i_* p_* \simeq i_* i^* f_* \simeq f_*
\,.
$$
The [[left adjoint]] to $p_*$ is then
$$
p^* \simeq f^* i_*
\,,
$$
because
$$
\begin{aligned}
\mathcal{F}(g^* E, F)
& \simeq \mathcal{F}(f^* i_* E, F)
\\
& \simeq \mathcal{E}(i_* E, f_* F)
\\
& \simeq \mathcal{E}(i_* E, i_* i^* f_* F)
\\
& \simeq Sh_j\mathcal{E} (E, i^* f_* F)
\\
& \simeq Sh_j(E, p_* F)
\end{aligned}
\,,
$$
where in the middle steps we used that $f_* F$ is a $j$-sheaf, by assumption, and that $i_*$ is full and faithful.
It is clear that $p^*$ is left exact, and so $(p^* \dashv p_*)$ is indeed a factorizing geometric morphism.
We now show that $f_*$ taking values in sheaves is equivalent to $f^*$ mapping dense monos to isos.
Let $u : U \hookrightarrow X$ be a $j$-[[dense monomorphism]] and $A \in \mathcal{E}$ any object. Consider the induced naturality square
$$
\array{
\mathcal{E}(X, f_* A) &\stackrel{\simeq}{\to}& \mathcal{F}(f^* X, A)
\\
{}^{\mathllap{\mathcal{E}(u, f_* A)}}\downarrow
&&
\downarrow^{\mathrlap{\mathcal{F}(f^* u, A)}}
\\
\mathcal{E}(U, f_* A)
&\stackrel{\simeq}{\to}&
\mathcal{F}(f^* U, A)
}
$$
of the adjunction [[natural isomorphism]]. If now $f_* A$ is a $j$-sheaf and $u$ a [[dense monomorphism]], then by definition the left vertical morphism is also an isomorphism and so is the right one. By the [[Yoneda lemma]] this being an iso for all $A$ is equivalent to $f^* u$ being an iso. And conversely.
=--
+-- {: .proof}
###### Proof of the proposition
Let $f : \mathcal{F} \to \mathcal{E}$ be any [[geometric morphism]].
We first discuss the existence of the factorization, then its uniqueness.
To construct the factorization, we shall give a [[Lawvere-Tierney topology]] on $\mathcal{E}$ and factor $f$ through the [[geometric embedding]] of the corresponding [[sheaf topos]].
Take the closure operator $\overline{(-)} : Sub(-)_{\mathcal{E}} \to Sub(-)_{\mathcal{E}}$ to be given by sending $U \hookrightarrow X$ to the [[pullback]]
$$
\array{
\overline{U} &\to& f_* f^* U
\\
\downarrow && \downarrow
\\
X &\to& f_* f^* X
}
\,,
$$
where the bottom morphism is the $(f^* \dashv f_*)$-[[unit of an adjunction|unit]]. One checks that this is indeed a closure operator by the fact that $f^*$ preserves both pullbacks and pushouts.
Notice that this implies that for two [[subobject]]s $U_1, U_2 \hookrightarrow X$ we have
\[
\label{ASubobjectRelation}
(U_1 \subset \overline{U_2})
\;\;\;
\Leftrightarrow
\;\;\;
(f^* U_1 \subset f^* U_2)
\]
Write $j$ for the corresponding [[Lawvere-Tierney topology]] and
$$
i : Sh_j(\mathcal{E}) \to \mathcal{E}
$$
for the corresponding [[geometric embedding]].
By lemma \ref{ConditionForSheafFactorization} we get a factorization through $I$ if $f^*$ sends $j$-[[dense monomorphism]]s to [[isomorphism]]s. But if $U \hookrightarrow X$ is dense so that $X \subset \overline{U}$ then, by (eq:ASubobjectRelation), $f^* X \subset f^* U$ and hence $f^* X = f^* U$.
Write
$$
f : \mathcal{F} \stackrel{p}{\to} Sh_j(\mathcal{E}) \stackrel{i}{\to}
\mathcal{E}
$$
for the factorization thus established. It remains to show that $p$ here is a [[geometric surjection]]. By one of the equivalent characterizations discussed there, this is the case if $p^*$ induces an injective homomorphism of subobject lattices.
So suppose that for subobjects $U_1, U_2 \subset X$ we have $p^* U_1 \simeq p^* U_2$. Observe that then also $f^* i_* U_1 \simeq f^* i_* U_2$, because
$$
\begin{aligned}
f^* i_* U_1 & \simeq p^* i^* i_* U_1
\\
& \simeq p^* U_1
\\
& \simeq p^* U_2
\\
& \simeq p^* i^* i_* U_2
\\
& \simeq f^* i:* U_2
\end{aligned}
$$
by the fact that $i_*$ is [[full and faithful functor|full and faithful]]. With (eq:ASubobjectRelation) it follows that also
$$
i_* U_1 \simeq \overline{i_* U_2}
$$
and hence
$$
\cdots \simeq i_* U_2
$$
by the very fact that $i_*$ includes $j$-sheaves in general, hence $j$-closed subobjects in particular. Finally since $i_*$ if a [[full and faithful functor]] this means that
$$
U_1 \simeq U_2
\,.
$$
So $p^*$ is indeed injective on subobjects and so $p$ is a [[geometric surjection]].
This establishes the existence of a surjection/embedding factorization. Next we discss that this is unique.
So consider two factorizations
$$
\array{
&& \mathcal{A}
\\
& {}^{\mathllap{p_1}}\nearrow &\Downarrow^\simeq& \searrow^{\mathrlap{i_1}}
\\
\mathcal{F}
&&\stackrel{f}{\to}&&
\mathcal{E}
\\
& {}_{\mathllap{p_2}}\searrow &\downarrow^{\simeq}& \nearrow_{\mathrlap{i_2}}
\\
&& \mathcal{B}
}
$$
into a geometric surjection followed by a geometric embedding.
We will now argue that $i_1$ factors -- essentially uniquely -- through $i_2$ in a way that makes
$$
\array{
&& \mathcal{A}
\\
& {}^{\mathllap{p_1}}\nearrow && \searrow^{\mathrlap{i_1}}
\\
\mathcal{F}
&&\downarrow^g&&
\mathcal{E}
\\
& {}_{\mathllap{p_2}}\searrow && \nearrow_{\mathrlap{i_2}}
\\
&& \mathcal{B}
}
$$
commute up to natural isomorphisms. By the same argument for the upside-down situation we find an essentially unique middle vertical morphism $h : \mathcal{B} \to \mathcal{A}$ the other way round. Then essential uniqueness of these factorizations implies that $g \circ h \simeq Id$ and $h \circ g \simeq Id$. This means that the original two factorizations are equivalent.
To find $g$ and $h$, use again that every [[geometric embedding]] (by the discussion there) is, up to equivalence, an inclusion of $j$-sheaves for some $j$. Find such a $j$ the bottom morphism and then use again lemma \ref{ConditionForSheafFactorization} that $i_1$ factors through $i_2$ -- essentially uniquely -- precisely if $i_1^*$ sends [[dense monomorphism]]s to isomorphisms.
To see that it does, let $IU \to X$ be a dense mono and consider the naturality square
$$
\array{
p_2^* i_2^* U &\stackrel{\simeq}{\to}& p_1^* i_1^* U
\\
\downarrow && \downarrow
\\
p_2^* i_2^* X &\stackrel{\simeq}{\to}& p_1^* i_1^* X
}
\,.
$$
Since $i_2^*(U \to X)$ is an iso by definition, the left vertical morphism is, and thus so is the right vertical morphism. But since $p_1$ is a [[geometric surjection]] we have (by the discussion there) that $p_1^*$ is [[conservative functor|conservative]], and hence also $i_1^* U \to i_1^* X$ is an isomorphism.
Hence $i_1$ factors via some $g$ through $i_2$ and the proof is completed by the above argument.
=--
## Examples
* For $f : X \to Y$ a [[continuous function]] between [[topological space]]s and $X \to im(f) \to Y$ its ordinary [[image]] factorization through an [[embedding]], the corresponding composite of geometric morphisms of [[sheaf topos]]es
$$
Sh(X) \to Sh(im(f)) \to Sh(Y)
$$
is a geometric surjection/geometric embedding factorization.
* For $\mathcal{E}$ any topos, $f : X \to Y$ any morphism in $\mathcal{E}$, and $X \to im(f) \to Y$ its [[image]] factorization, the corresponding composite of [[base change geometric morphisms]]
$$
\mathcal{E}/X \to \mathcal{E}/im(f) \to \mathcal{E}/Y
$$
is a geometric surjection/embedding factorization.
* For $f : C \to D$ any [[functor]] between [[categories]], write $C \to im(f) \to D$ for its [[essential image]] factorization. Then the induced composite <a href="http://ncatlab.org/nlab/show/geometric%20morphism#BetweenPresheafToposes">geometric morphism of presheaf toposes</a>
$$
[C^{op}, Set] \stackrel{}{\to} [im(f)^{op}, Set] \to
[D^{op}, Set]
$$
is a geometric surjection/embedding factorization.
See ([MacLaneMoerdijk, p. 377](#MacLaneMoerdijk)).
## Properties
### As idempotent approximation
A geometric morphism $f:\mathcal{F}\to\mathcal{E}$ induces via the adjunction $f^\ast\vdash f_\ast$ a [[monad]] on $\mathcal{E}$. Due to a general result by S. Fakir this induces an associated [[idempotent monad]] on $\mathcal{E}$ and this idempotent approximation coincides with the monad induced by $i^\ast\vdash i_\ast$ given by the inclusion $i$ from the factorization $f=i\circ q$.
For references and further details on the idempotent approximation see at [[idempotent monad]].
### A logical description
Let $T$ be a [[geometric theory]] over a signature $\Sigma$ and $f:\mathcal{E}\to Set[T]$ a geometric morphism to its [[classifying topos]]. Then by the general properties of a classifying topos, $f$ corresponds to a certain $T$-model $M$ in $\mathcal{E}$.
Notice that every geometric morphism $f$ between [[Grothendieck toposes]] is of this form for some geometric theory $T$ and hence corresponds to some model $M$ ! This model permits to attach a geometric theory to $f$ as well:
The **theory of M** $Th(M)$ consists of all geometric sequents $\sigma$ over $\Sigma$ such that $M\models \sigma$.
Then the following holds ([Caramello 2009](#Caramello09), p.57):
+-- {: .num_prop}
###### Proposition
The topos occurring in the middle of the surjection-embedding factorization of $f$ is precisely the classifying topos for $Th(M)$: $\mathcal{E}\twoheadrightarrow Set[Th(M)]\hookrightarrow Set[T]$.
=--
## Related entries
* [[open subtopos]]
* [[(dense,closed)-factorization]]
## References
* {#Johnstone77} [[Peter Johnstone]], _Topos Theory_ , Academic Press 1977 (Dover reprint 2014). (section 4.1, pp.103-107)
* {#Johnstone} [[Peter Johnstone]], _[[Sketches of an Elephant]] vol. I_, Oxford UP 2002. (section A4.2, pp.172ff)
* [[Saunders MacLane]], [[Ieke Moerdijk]], _[[Sheaves in Geometry and Logic]]_ , Springer Heidelberg 1994. (section VII.4)
{#MacLaneMoerdijk}
* [[Olivia Caramello|O. Caramello]], _Lattices of theories_ , arXiv:0905.0299v1 (2009). ([pdf](http://arxiv.org/pdf/0905.0299v1)) {#Caramello09}
[[!redirects geometric surjection/embedding factorization]]
[[!redirects geometric surjection/inclusion factorization]]
[[!redirects geometric surjection/embedding factorization system]]
[[!redirects geometric surjection/inclusion factorization system]]
[[!redirects (geometric surjection, inclusion) factorization system]]
|
(hyperconnected, localic) factorization system | https://ncatlab.org/nlab/source/%28hyperconnected%2C+localic%29+factorization+system |
+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### Factorization systems
+-- {: .hide}
[[!include factorization systems - contents]]
=--
#### Topos Theory
+-- {: .hide}
[[!include topos theory - contents]]
=--
=--
=--
#Contents#
* table of contents
{:toc}
## Idea
[[hyperconnected geometric morphism|Hyperconnected geometric morphisms]] are the left class of a 2-categorical [[orthogonal factorization system]] on the [[2-category]] [[Topos]] of [[toposes]]; the right class is the class of [[localic geometric morphisms]].
## References
* [[Peter Johnstone]], _Factorization theorems for geometric morphisms_ Cahiers, 22, no1 (1981) ([numdam](http://www.numdam.org/item?id=CTGDC_1981__22_1_3_0))
{#Johnstone}
[[!redirects (hyperconnected, localic) factorization system]]
[[!redirects (hyperconnected,localic) factorization system]] |
(infinity,0)-category | https://ncatlab.org/nlab/source/%28infinity%2C0%29-category |
+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### Higher category theory
+-- {: .hide}
[[!include higher category theory - contents]]
=--
=--
=--
#Contents#
* table of contents
{:toc}
## Idea
Following the terminology of *[[(n,r)-category|(n,r)-categories]]*, an __$(\infty,0)$-category__ is an [[∞-category]] in which every $j$-morphism (for $j \gt 0$) is an [[equivalence]].
So in an $(\infty,0)$-category _every_ morphism is an [[equivalence]]. Such [[∞-categories]] are usually called _[[∞-groupoid]]s_.
This is directly analogous to how a [[0-category]] is equivalent to a [[set]], a [[(1,0)-category]] is equivalent to a [[groupoid]], and so on. (In general, an [[(n,0)-category]] is equivalent to an [[n-groupoid]].)
The term "$(\infty,0)$-category" is rarely used, but does for instance serve the purpose of amplifying the generalization from [[Kan complex]]es, which are one model for [[∞-groupoid]]s, to [[quasi-category|quasi-categories]], which are a model for [[(∞,1)-categories]].
## References
On [[model categories]] [[presentable (infinity,1)-category|presenting]] $(\infty,0)$-categories, namely models for [[infinity-groupoids|$\infty$-groupoids]] (such as the [[model structure on simplicial groupoids]]) akin to corresponding models for [[(infinity,1)-categories|$(\infty,1)$-categories]] (such as the [[model structure on simplicial categories]]):
* [[Julia E. Bergner]], *Adding inverses to diagrams II: Invertible homotopy theories are spaces*, Homology, Homotopy Appl. **10** 2 (2008) 175-193 [[doi:10.4310/HHA.2008.v10.n2.a9](https://dx.doi.org/10.4310/HHA.2008.v10.n2.a9), [doi:0710.2254](https://arxiv.org/abs/0710.2254), erratum:[doi:10.4310/HHA.2012.v14.n1.a15](https://dx.doi.org/10.4310/HHA.2012.v14.n1.a15)]
[[!redirects (infinity,0)-categories]]
[[!redirects (∞,0)-category]]
[[!redirects (∞,0)-categories]]
|
(infinity,1)-bimodule | https://ncatlab.org/nlab/source/%28infinity%2C1%29-bimodule |
+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Higher algebra
+--{: .hide}
[[!include higher algebra - contents]]
=--
=--
=--
#Contents#
* table of contents
{:toc}
## Idea
## Definition
### $(\infty,1)$-Category of $(\infty,1)$-Bimodules and intertwiners
Write $BMod^\otimes$ for the [[(∞,1)-category of operators]] of the [[(∞,1)-operad]] [[operad for bimodules]]. Write
$$
\iota_{\pm} \colon Assoc \to BMod
$$
for the two canonical inclusions of the [[associative operad]] (as discussed at _[operad for bimodules - relation to the associative operad](#RelationToTheAssociativeOperad)_).
+-- {: .num_defn #NotationForWeaklyBiEnrichedInfinityCategory}
###### Definition (Notation)
For $p \colon \mathcal{C}^\otimes \to BMod^\otimes$ a [[fibration of (∞,1)-operads]], write
$$
\mathcal{C}^\otimes_{\pm}
\coloneqq
\mathcal{C}^\otimes \underset{BMod^\otimes}{\times}^\pm Assoc^\otimes
$$
for the two [[fiber products]] of $p$ with the inclusions $\iota_\pm$. The canonical [[projection]] maps
$$
\mathcal{C}^\otimes_{\pm}
\to
Assoc^\otimes
$$
exhibit these as two [[planar (∞,1)-operads]].
Finally write
$$
\mathcal{C}
\coloneqq
\mathcal{C}^\otimes \underset{BMod^\otimes}{\times} \{\mathfrak{n}\}
$$
for the [[(∞,1)-category]] over the object labeled $\mathfrak{n}$.
=--
([Lurie, notation 4.3.1.11](#Lurie)).
+-- {: .num_remark }
###### Remark
This exhibits $\mathcal{C}$ as equipped with [[weak tensoring]] over $\mathcal{C}_-$ and reverse weak tensoring over $\mathcal{C}_+$.
=--
The most familiar special case of these definitions to keep in mind is the following.
+-- {: .num_remark #MonoidalCategoryAsBitensoredOverItself}
###### Remark
For $\mathcal{C}^\otimes \to Assoc^\otimes$ a [[coCartesian fibration of (∞,1)-operads]], hence exhibiting $\mathcal{C}^\otimes$ as a [[monoidal (∞,1)-category]], [[pullback]] along the canonical map $\phi \colon BMod^\otimes \to Assoc^\otimes$ gives a fibration
$$
\phi^* \mathcal{C}^\otimes \to BMod^\otimes
$$
as in def. \ref{NotationForWeaklyBiEnrichedInfinityCategory} above. In the terminology there this exhibts $\mathcal{C}$ as weakly enriched (weakly [[tensoring|tensored]]) over itself from the left and from the right.
This is the special case for which bimodules are traditionally considered.
=--
([Lurie, example 4.3.1.15](#Lurie))
+-- {: .num_defn }
###### Definition
For $\mathcal{C}^\otimes \to BMod^\otimes$ a [[fibration of (∞,1)-operads]] we say that the corresponding [[(∞,1)-category]] of [[(∞,1)-algebras over an (∞,1)-operad]]
$$
BMod(\mathcal{C}) \coloneqq Alg_{/BMod}(\mathcal{C})
$$
is the **$(\infty,1)$-category of $(\infty,1)$-bimodules** in $\mathcal{C}$.
Composition with the two inclusions $\iota_{1,2}\colon Assoc BMod$ of the [[associative operad]] yields a [[fibration]] in the [[model structure for quasi-categories]] $BMod(\mathcal{C}) \to Alg(\mathcal{C}_-)\times Alg(\mathcal{C}_+)$. Then for $A_- \in Alg_{\mathcal{C}_-}$ and $A_+ \in Alg_{\mathcal{C}_+}$ two algebras the [[fiber product]]
$$
{}_A BMod_{B}(\mathcal{C})
\coloneqq
\{A\}
\underset{Alg(\mathcal{C}_-)}{\times}
BMod(\mathcal{C})
\underset{Alg(\mathcal{C}_-)}{\times}
\{B\}
$$
we call the **$(\infty,1)$-category of $A$-$B$-bimodules**.
=--
([Lurie, def. 4.3.1.12](#Lurie))
+-- {: .num_example}
###### Example
For the special case of remark \ref{MonoidalCategoryAsBitensoredOverItself} where the bitensored structure on $\mathcal{C}$ is induced from a monoidal structure $\mathcal{C}^\otimes \to Asoc^\otimes$, we have by the [[universal property]] of the [[pullback]] that
$$
BMod(\mathcal{C})
\simeq
{Alg_{BMod}}_{/Assoc}(\mathcal{C})
\simeq
\left\{
\array{
&& \mathcal{C}
\\
&{}^{\mathllap{(A,B,N)}}\nearrow& \downarrow
\\
BMod^\otimes &\to& Assoc^\otimes
}
\right\}
$$
=--
+-- {: .num_remark}
###### Remark
Let $\mathcal{C}$ be a [[1-category]], for simplicity.
Then a [[morphism]]
$$
(A_1,B_1,N_1) \to $(A_2,B_2,N_2)$
$$
in $BMod(\mathcal{C})$ is a pair $\phi_1 \colon A_1 \to A_1$, $\rho \colon B_1 \to B_2$ of algebra homomorphisms and a morphism $\kappa \colon N_1 \to N_2$ which is "linear in both $A$ and $B$" or "is an [[intertwiner]]" with respect to $\phi$ and $\rho$ in that for all $a \in A$, $b \in B$ and $n \in N$ we have
$$
\kappa(a \cdot n \cdot b) = \phi(a) \cdot \kappa(n)
\,.
$$
It is natural to depict this by the square diagram
$$
\array{
A_1 &\stackrel{N_1}{\to}& B_1
\\
{}^{\mathllap{\phi}}\downarrow
&
\Downarrow^{\kappa}
&
\downarrow^{\mathrlap{\rho}}
\\
A_2 &\underset{N_2}{\to}& B_2
}
\,.
$$
This notation is naturally suggestive of the existence of the further [[horizontal composition]] by [[tensor product of (∞,1)-modules]], which we come to [below](#TensorProductOfBimodules).
On the other hand, a morphism $N_1 \to N_2$ in ${}_A BMod(\mathcal{C})_B$ is given by the special case of the above for $\phi = id$ and $\rho = id$.
=--
### Tensor products of $(\infty,1)$-Bimodules
{#TensorProductOfBimodules}
+-- {: .num_defn #NotationForTensS}
###### Definition (Notation)
Write $Tens^\otimes$ for the [[generalized (∞,1)-operad]]
discussed at _[[tensor product of ∞-modules]]_.
For $S \to \Delta^{op}$ an [[(∞,1)-functor]] (given as a map of simplicial sets from a [[quasi-category]] $S$ to the [[nerve]] of the [[simplex category]]), write
$$
Tens^\otimes_{S}
\coloneqq
Tens^\otimes \underset{\Delta^{op}}{\times} S
$$
for the [[fiber product]] in [[sSet]].
Moreover, for $\mathcal{C}^\otimes \to Tens^\otimes_S$ a [[fibration]] in the [[model structure for quasi-categories]] which exhibits $\mathcal{C}^\otimes$ as an $S$-[[family of (∞,1)-operads]], write
$$
Alg_S(\mathcal{C}) \hookrightarrow Fun_{Tens^\otimes_S}(Step_S, \mathcal{C}^\otimes)
$$
for the full [[sub-(∞,1)-category]] on those [[(∞,1)-functors]] which send inert morphisms to inert morphisms.
=--
([Lurie, notation 4.3.4.15](#Lurie))
### The $(\infty,2)$-Category of $(\infty,1)$-algebras and -bimodules
We discuss the generalization of the notion of bimodules to [[homotopy theory]], hence the generalization from [[category theory]] to [[(∞,1)-category theory]]. ([Lurie, section 4.3](#Lurie)).
Let $\mathcal{C}$ be [[monoidal (∞,1)-category]] such that
1. it admits [[geometric realization]] of [[simplicial objects in an (∞,1)-category]] (hence a [[left adjoint|left]] [[adjoint (∞,1)-functor]] ${\vert-\vert} \colon \mathcal{C}^{\Delta^{op}} \to \mathcal{C}$ to the constant simplicial object functor), true notably when $\mathcal{C}$ is a [[presentable (∞,1)-category]];
1. the [[tensor product]] $\otimes \colon \mathcal{C}\times \mathcal{C} \to \mathcal{C}$ preserves this geometric realization separately in each argument.
Then there is an
[[(∞,2)-category]] $Mod(\mathcal{C})$ which given as an [[(∞,1)-category object]] internal to [[(∞,1)Cat]] has
* $(\infty,1)$-category of objects
$$
Mod(\mathcal{C})_{[0]} \simeq Alg(\mathcal{C})
$$
the [[A-∞ algebras]] and [[∞-algebra]] [[homomorphisms]] in $\mathcal{C}$;
* $(\infty,1)$-category of morphisms
$$
Mod(\mathcal{C})_{[1]} \simeq BMod(\mathcal{C})
$$
the $\infty$-bimodules and bimodule homomorphisms ([[intertwiners]]) in $\mathcal{C}$
This is ([Lurie, def. 4.3.6.10, remark 4.3.6.11](#Lurie)).
Morover, the [[horizontal composition]] of bimodules in this [[(∞,2)-category]] is indeed the relative
[[tensor product of ∞-modules]]
$$
\circ_{A,B,C}
=
(-) \otimes_B (-)
\;\colon\;
{}_A Mod_{B} \times {}_{B}Mod_C \to {}_A Mod_C
\,.
$$
This is ([Lurie, lemma 4.3.6.9 (3)](#Lurie)).
Here are some steps in the construction:
The **idea** of the following constructions is this:
we start with a [[generalized (∞,1)-operad]]
$Tens^\otimes \to FinSet_* \times \Delta^{op}$ which is such that
the [[(∞,1)-algebras over an (∞,1)-operad]] over its fiber over $[k] \in \Delta^{op}$ is equivalently the collection of $(k+1)$-tuples of
[[A-∞ algebras]] in $\mathcal{C}$ together with a string of $k$ $\infty$-bimodules between them. Then we turn that into a [[simplicial object in an (∞,1)-category|simplicial object]] in [[(∞,1)Cat]]
$$
Mod(\mathcal{C}) \in ((\infty,1)Cat)^{\Delta^{op}}
\,.
$$
This turns out to be an [[internal (∞,1)-category]] object in [[(∞,1)Cat]], hence an [[(∞,2)-category]] whose object of objects is the category $Alg(\mathcal{C})$ of [[A-∞ algebras]] and [[homomorphisms]] in $\mathcal{C}$ between them, and whose object of morphisms is the category $BMod(\mathcal{C})$ of $\infty$-bimodules and [[intertwiners]].
+-- {: .num_defn}
###### Definition
Define $Mod(\mathcal{C}) \to \Delta^{op}$ as the map of [[simplicial sets]] with the [[universal property]] that for every other map of simplicial set $K \to \Delta^{op}$ there is a canonical bijection
$$
Hom_{sSet/\Delta^{op}}(K, Mod(\mathcal{C}))
\simeq
Alg_{Tens_K / Assoc}( \mathcal{C} )
\,,
$$
where
* on the left we have the hom-simplicial set in the [[slice category]]
* on the right we have the [[(∞,1)-category]] of [[(∞,1)-algebras over an (∞,1)-operad]] given by lifts $\mathcal{A}$ in
$$
\array{
&& \mathcal{C}^\otimes
\\
&{}^{\mathcal{A}}\nearrow& \downarrow
\\
Tens_K &\to& Assoc
}
\,.
$$
=--
This is ([Lurie, cor. 4.3.6.2](#Lurie)) specified to the case of ([Lurie, lemma 4.3.6.9](#Lurie)). Also ([Lurie, def. 4.3.4.19](#Lurie))
## References
The general theory in terms of [[higher algebra]] of [[(∞,1)-operads]] is discussed in section 4.3 of
* [[Jacob Lurie]], _[[Higher Algebra]]_
Specifically the homotopy theory of [[A-infinity bimodules]] is discussed in
* Volodymyr Lyubashenko, Oleksandr Manzyuk, _A-infinity-bimodules and Serre A-infinity-functors_ ([arXiv:math/0701165](http://arxiv.org/abs/math/0701165))
and section 5.4.1 of
* [[Boris Tsygan]], _Noncommutative calculus and operads_ in
Guillermo Cortinas (ed.) _Topics in Noncommutative geometry_, Clay Mathematics Proceedings volume 16
The generalization to [[(infinity,n)-modules]] is in
* {#Haugseng14} [[Rune Haugseng]], _The higher Morita category of $E_n$-algebras_ ([arXiv:1412.8459](http://arxiv.org/abs/1412.8459))
[[!redirects (infinity,1)-bimodules]]
[[!redirects ∞-bimodule]]
[[!redirects ∞-bimodules]]
[[!redirects (∞,1)-bimodule]]
[[!redirects (∞,1)-bimodules]]
[[!redirects (∞,1)-category of (∞,1)-bimodules]]
[[!redirects (infinity,1)-category of (∞,1)-bimodules]]
[[!redirects (∞,1)-categories of (∞,1)-bimodules]]
[[!redirects (infinity,1)-categories of (∞,1)-bimodules]]
[[!redirects infinity-bimodule]]
[[!redirects infinity-bimodules]] |
(infinity,1)-categorical hom-space | https://ncatlab.org/nlab/source/%28infinity%2C1%29-categorical+hom-space |
+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Homotopy theory
+--{: .hide}
[[!include homotopy - contents]]
=--
#### $(\infty,1)$-Category theory
+--{: .hide}
[[!include quasi-category theory contents]]
=--
#### Model category theory
+--{: .hide}
[[!include model category theory - contents]]
=--
#### Mapping space
+--{: .hide}
[[!include mapping space - contents]]
=--
=--
=--
#Contents#
* table of contents
{:toc}
## Idea
Where an ordinary [[category]] has a [[hom-set]], an [[(∞,1)-category]] has an [[∞-groupoid]] of morphisms between any two objects, a _hom-space_.
There are several ways to _present_ an [[(∞,1)-category]] $\mathbf{C}$ by an ordinary [[category]] $C$ equipped with some extra structure: for instance $C$ may be a [[category with weak equivalences]] or a [[model category]] or even a [[simplicial model category]]. In all of these presentations, given two objects $X, Y \in C$, there is a way to construct a [[simplicial set]] $\mathbb{R}\mathbf{C}(X,Y)$ that presents the hom-[[∞-groupoid]] $\mathbf{C}(X,Y)$. This simplicial set -- or rather its [[homotopy type]] -- is called the
_derived hom space_ or _homotopy function complex_ and denoted $\mathbf{R}Hom(X,Y)$ or similarly.
## Presentations
There are many ways to present an [[(∞,1)-category]] by [[category theory|category theoretic data]], and for each of these there are corresponding tools for explicitly computing the derived hom spaces.
The most basic data is that of a [[category with weak equivalences]]. Here the derived hom spaces can be constructed in terms of zig-zags of morphisms by a process called _[[simplicial localization]]_. This we discuss below in _[For a category with weak equivalences](#ForACategoryWithWeakEquivalences)_.
Particularly useful extra structure on a [[category with weak equivalences]] that helps with computing the derived hom spaces is the structure of a _[[model category]]_. Using this one can construct simplicial resolutions of objects -- called _framings_ -- that generalize [[cylinder objects]] and [[path objects]], and then construct the derived hom spaces in terms of direct morphisms between these resolutions. This we discuss below in _[For a model category](#Framings)_.
Still a bit more helpful structure on top of a bare model category is that of a [[simplicial model category]]. Here, after a choice of cofibrant and fibrant resolutions of opjects, the derived hom spaces are given already by the [[sSet]]-[[hom objects]]. This we discuss below in _[For a simplicial model category](#EnrichedHomsCofToFib)_.
### For a category with weak equivalences
{#ForACategoryWithWeakEquivalences}
Let $(C,W \subset Mor(C))$ be a [[category with weak equivalences]].
+-- {: .num_defn #ZigZagCategories}
###### Definition
Fix $n \in \mathbb{N}$. For $X,Y \in Obj(C)$, define a category $wMor_C^n(X,Y)$
* whose objects are [[zig-zag]]s of morphisms in $C$ of length $n$
$$
X = X_0 \leftarrow X_1 \to X_2 \leftarrow \cdots \to X_{n-1} \leftarrow X_n = Y
$$
such that each morphism going to the left, $X_{2k}\leftarrow X_{2k +1}$, is a [[weak equivalence]], an element in $W$;
* morphisms between such objects $(X,X_i,Y) \to (X',X'_i,Y')$ are collections of weak equivalences $(X_i \to X'_i)$ for all $0 \lt i \lt n $ such that all triangles and squares commute.
=--
+-- {: .num_defn #HammockLocalization}
###### Definition
Write $N(wMor_C^n(X,Y))$ for the [[nerve]] of this category, a [[simplicial set]].
The _[[hammock localization]]_ $L_W^H C$ of $C$ is the [[simplicially enriched category]] with objects those of $C$ and [[hom-objects]] given by the [[colimit]] over the length of these hammock hom-categories
$$
L^H C(X,Y) := \lim_{\to_n} N(wMor_C^n(X,Y))
\,.
$$
The [[Kan fibrant replacement]] of this simplicial set is the derived hom-space between $X$ and $Y$ of the $(\infty,1)$-category modeled by $(C,W)$.
=--
### For a model category
{#Framings}
The derived hom spaces of a model category $C$ may always be computed
in terms of simplicial resolutions with respect to the [[Reedy model structure]]
$[\Delta^{op}, C]_{Reedy}$. These resolutions are often called
_framings_ ([Hovey](#Hovey)). These constructions are originally due to
([Dwyer-Hirschhorn-Kan](#DHK)).
Let $C$ be any [[model category]].
+-- {: .num_prop }
###### Observation
There is an [[adjoint triple]]
$$
(const \dashv ev_0 \dashv (-)^{\times^\bullet})
:
C
\stackrel{\overset{const}{\longrightarrow}}{\stackrel{\overset{ev_0}{\longleftarrow}}{\underset{(-)^{\times^\bullet}}{\longrightarrow}}}
\,,
[\Delta^{op}, C]
\,,
$$
where
1. $const X : [n] \mapsto X$;
1. $ev_0 X_\bullet = X_0$;
1. $X^{\times^\bullet} : [n] \mapsto X^{\times^{n+1}}$.
=--
+-- {: .num_remark #CoDiscreteIsReedyFibrant}
###### Remark
For $X \in C$ fibrant, $X^{\times^\bullet}$ is fibrant in
the [[Reedy model structure]] $[\Delta^{op}, C]_{Reedy}$.
=--
+-- {: .proof}
###### Proof
The matching morphisms are in fact [[isomorphisms]].
=--
+-- {: .num_defn}
###### Definition
Let $C$ be a model category.
1. For $X \in C$ any object, a _simplicial frame_ on $X$ is a factorization of $const X \to X^{\times^\bullet}$ into a weak equivalence followed by a fibration in the [[Reedy model structure]] $[\Delta^{op}, C]_{Reedy}$.
1. A _right framing_ in $C$ is a functor $(-)_\bullet : C \to [\Delta^{op}, C]$
with a [[natural isomorphism]] $(X)_0 \simeq X$ such that $X_\bullet$ is a simplicial
frame on $X$.
Dually for _cosimplicial frames_.
=--
This appears as ([Hovey, def. 5.2.7](#Hovey)).
+-- {: .num_remark}
###### Remark
By remark \ref{CoDiscreteIsReedyFibrant} a simplicial frame $X_\bullet$ in the above is in particular fibrant in $[\Delta^{op}, C]_{Reedy}$.
=--
+-- {: .num_prop #SimplicialFunctionComplexes}
###### Proposition
For $X \in C$ cofibrant and $A \in C$ fibrant, there
are weak equivalences in $sSet_{Quillen}$
$$
Hom_C(X^\bullet, A)
\stackrel{\simeq}{\to}
diag Hom_C(X^\bullet, A_\bullet)
\stackrel{\simeq}{\leftarrow}
Hom_C(X, A_\bullet)
\,,
$$
(where in the middle we have the diagonal of the [[bisimplicial set]] $Hom(X^\bullet, A_\bullet)$).
=--
This appears as ([Hovey, prop. 5.4.7](#Hovey)).
Either of these simplicial sets is a model for the derived hom-space $\mathbb{R}Hom(X,A)$.
+-- {: .num_remark }
###### Remark
By developing these constructions further, one obtains
a canonical [[simplicial model category]]-resolution of
(left proper and combinatorial) model categories $C$, such that
the simplicial resolutions given by framings are just the
cofibrant$\to$fibrant $sSet$-hom objects as discussed
[below](#EnrichedHomsCofToFib).
This is discussed at _[Simplicial Quillen equivalent models](http://ncatlab.org/nlab/show/simplicial+model+category#SimpEquivMods)_.
=--
+-- {: .num_prop }
###### Proposition
Let $C$ be a model category, let $\mathrm{c}_\mathrm{w} C$ be the full subcategory of $[\Delta, C]$ spanned by the cosimplicial objects whose coface and codegeneracy operators are weak equivalences, and let $\mathrm{s}_\mathrm{w} C$ be the full subcategory of $[\Delta^{op}, C]$ spanned by the simplicial objects whose face and degeneracy operators are weak equivalences.
1. $const : C \to \mathrm{c}_\mathrm{w} C$ is the right half of an adjoint homotopical equivalence of [[homotopical category|homotopical categories]], and $const : C \to \mathrm{s}_\mathrm{w} C$ is the left half of an adjoint homotopical equivalence of homotopical categories.
2. The functor $\operatorname{diag} Hom_C : (\mathrm{c}_\mathrm{w} C)^{op} \times \mathrm{s}_\mathrm{w} C \to sSet$ admits a right [[derived functor]].
3. The induced functor $(\operatorname{Ho} C)^{op} \times \operatorname{Ho} C \to \operatorname{Ho} sSet$ is the derived hom-space functor.
=--
### For a simplicial model category
{#EnrichedHomsCofToFib}
We describe here in more detail properties of [[derived hom-functors]] (see there for more) in a [[simplicial model category]].
The crucial axiom used for this is the axiom of an [[enriched model category]] $C$ which says that
* the [[copower|tensor operation]]
$$
\cdot : C \times SSet \to C
$$
is a [[Quillen bifunctor]];
* or equivalently that for $X \to Y$ a cofibration
and $A \to B$ a fibration the induced morphism
$$
C(Y, A) \to C(X,A) \times_{C(X,B)} C(Y,B)
$$
is a fibration, which is acyclic if either
$X \to Y$ or $A \to B$ is.
First of all the first statement directly implies that for $\emptyset \in C$ the [[initial object]] and $A \in C$ any object, the simplicial set $C(\emptyset,A) = {*}$ is the terminal simplicial set (see also [this Prop.](powered+and+copowered+category#InTensoredCotensoredCategoryInitialObjectIsEnrichedInitial)): because for any simplicial set $S$
$$
\begin{aligned}
SSet(S,C(\emptyset, A))
& = Hom_C(\emptyset \cdot S, A)
\\
& = Hom_C(colim_{\emptyset} \cdot S, A)
\\
& = Hom_C(\emptyset, A)
\\
&= {*}
\end{aligned}
\,,
$$
where we use that the [[copower|tensor]] [[Quillen bifunctor]] is required to respect [[colimit]]s and that the empty colimit is the [[initial object]]. (All equality signs here denote [[isomorphisms]], to distinguish them from weak equivalences.)
Similarly one has for all $X$ that $C(X,{*}) = {*}$.
Using this, the second equivalent form of the enrichment axiom has as a special case the following statement.
+-- {: .num_lemma }
###### Lemma
In a [[simplicial model category]] $C$, for $X \in C$ cofibrant and $A \in C$ fibrant, the [[simplicial set]] $C(X,A)$ is a [[Kan complex]].
=--
+-- {: .proof}
###### Proof
We apply the [[enriched model category]] axiom to the cofibration $\emptyset \to X$ and the fibration $A \to {*}$ to obtain a fibration
$$
C(X,A) \to C(\emptyset, A) \times_{C(\emptyset,{*})} C(X,{*})
\,.
$$
The right hand is the [[pullback]] of the terminal simplicial set ${*} = \Delta^0$ to itself, hence is itself the point. So we have a fibration $ C(X,A) \to {*}$
and $C(X,A)$ is a fibrant object in the standard [[model structure on simplicial sets]], hence a [[Kan complex]].
.
=--
+-- {: .num_lemma }
###### Lemma
In a [[simplicial model category]] $C$, for $X \in C$ cofibrant and $f : A \to B$ a fibration, the morphism of [[simplicial set]]s $C(X,f) : C(X,A) \to C(X,B)$ is a [[Kan fibration]] that is a [[weak homotopy equivalence]] if $f$ is acyclic.
Dually, for $i : X \to Y$ a cofibration and $A$ fibrant, the morphism $C(i,A) : C(X,A) \to C(Y,A)$ is a cofibration of simplicial sets.
=--
+-- {: .proof}
###### Proof
This is as before. Explicity, consider the first case, the second one is the formal dual of that:
We enter the enrichment axiom with the morphisms $\emptyset \to X$ and $A \to B$ and find for the required [[pullback]] that
$$
C(\emptyset,A) \times_{C(\emptyset, B)} C(X,B)
=
{*} \times_{*} C(X,B)
=
C(X,B)
$$
and hence that $C(X,A) \to C(X,B)$ is, indeed, a fibration, which is acyclic if $A \to B$ is.
=--
+-- {: .num_proposition }
###### Proposition
Let $C$ be a [[simplicial model category]].
Then for $X$ a cofibant object and
$$
f : A \stackrel{\simeq}{\to} B
$$
a weak equivalence between fibrant objects, the [[enriched functor|enriched]] [[hom-object|hom-functor]]
$$
C(X,f) : C(X,A) \to C(X,B)
$$
is a [[weak homotopy equivalence]] of [[Kan complex]]es.
Similarly, for $A$ a fibrant object and $j : X \stackrel{\simeq}{\to} Y$ a weak equivalence between cofibrant objects, the morphism
$$
C(j,A) : C(X,A) \to C(Y,A)
$$
is a [[weak homotopy equivalence]] of [[Kan complex]]es.
=--
+-- {: .proof}
###### Proof
The second case is formally dual to the first, so we restrict attention to the first one.
By the above, the axioms of an [[enriched model category]] ensure that the above statement is true when $f$ is in addition a fibration. So we reduce the situation to that case.
This is possible because both $A$ and $B$ are assumed to be fibrant. This allows to apply the _factorization lemma_ that is described in great detail at [[category of fibrant objects]]. By this lemma, for every morphism $f : A \to B$ between fibrant objects there is a commutative diagram
$$
\array{
&& \mathbf{E}_f B
\\
& {}^{\mathllap{\in fib \cap W}}\swarrow && \searrow^{\mathrlap{\in fib}}
\\
A &&\stackrel{\simeq}{\to}&& B
}
$$
Since $f$ is assumed a weak equivalence it follows by [[category with weak equivalences|2-out-of-3]] that $\mathbf{E}_f B$ is also a weak equivalence.
Therefore by the above properties of simpliciall enriched categories we obtain a [[span]] of acyclic fibrations of [[Kan complex]]es
$$
C(X,A) \stackrel{\simeq}{\leftarrow}
C(X, \mathbf{E}_f B)
\stackrel{\simeq}{\to}
C(X,B)
\,.
$$
By the [[Whitehead theorem]] every weak equivalence of Kan complexes is a [[homotopy equivalence]], hence there is a weak equivalence
$$
C(X,A) \stackrel{\simeq}{\to} C(X,\mathbf{E}_f B) \stackrel{\simeq}{\to} C(X,B)
$$
that is homotopic to our $C(X,f)$. Therefore this is also a weak equivalence.
=--
### Comparison
Let $C$ be a [[model category]]. We discuss how its
simplicial function complexes from prop. \ref{SimplicialFunctionComplexes}
are related to the simplicial localization from def. \ref{ZigZagCategories} and def. \ref{HammockLocalization}.
Suppose now that $Q : C \to C$ is a [[cofibrant replacement functor]] and $R : C \to C$ a [[fibrant replacement functor]], $\Gamma^\bullet : C \to (cC)_c$ a [[cosimplicial resolution functor]] and $\Lambda_\bullet : C \to (sC)_f$ a [[simplicial resolution functor]] in the [[model category]] $C$.
+-- {: .num_theorem #DKTheorem}
###### Theorem
**(Dwyer--Kan)**
There are natural weak equivalences between the following equivalent realizations of this [[SSet]] [[hom-object]]:
$$
\array{
Mor_C(\Gamma^\bullet X, R Y)
&\stackrel{\simeq}{\to}&
diag Mor_C(\Gamma^\bullet X, \Lambda_\bullet Y)
&\stackrel{\simeq}{\leftarrow}&
Mor_C(Q X, \Lambda_\bullet Y)
\\
&&
\uparrow^\simeq
\\
&&
hocolim_{p,q \in \Delta^{op} \times \Delta^{op}}
Mor_C(\Gamma^p X, \Lambda_q Y)
\\
&&\downarrow^\simeq
\\
&&N wMor_C^3(X,Y)
\\
&&\downarrow^\simeq
\\
&&Mor_{L^H C}(X,Y)
}
\,.
$$
=--
The top row weak equivalences are those of prop. \ref{SimplicialFunctionComplexes}
### In a category of fibrant objects
{#InACategoryOfFibrantObjects}
There is also an explicit simplicial construction of the derived hom spaces for a homotopical category that is equipped with the structure of a [[category of fibrant objects]].
This is described in ([Cisinksi 10](#Cisinski10)) and ([Nikolaus-Schreiber-Stevenson 12, section 3.6.2](#NSS12)).
## Properties
### Hom-spaces of equivalences
{#SpacesOfEquivalences}
+-- {: .num_theorem #DKTheorem}
###### Theorem
For $C$ a [[simplicial model category]] and $X$ an object, the [[delooping]] of the [[automorphism ∞-group]]
$$
Aut_W(X) \subset \mathbb{R}Hom(X,X)
$$
has the [[homotopy type]] of the component on $X$ of the [[nerve]] $N(C_W)$ of the [[subcategory]] of weak equivalences:
$$
\mathbf{B} Aut_W(X) \simeq N(C_W)_X
\,.
$$
The equivalence is given by a finite sequence of [[zig-zags]] and is natural with respect to [[sSet]]-[[enriched functors]] of simplicial model categories that preserve weak equivalences and send a fibrant cofibrant model for $X$ again to a fibrant cofibrant object.
=--
This is [Dwyer-Kan 84, 2.3, 2.4](#DK84).
+-- {: .num_cor }
###### Corollary
For $C$ a model category, the simplicial set $N(C_W)$ is a model for the
[[core]] of the [[(∞,1)-category]] determined by $C$.
=--
+-- {: .proof}
###### Proof
That core, like every [[∞-groupoid]] is equivalent to the disjoint union over its connected components of the deloopings of the automorphism $\infty$-groups of any representatives in each connected component.
=--
## Related concepts
* [[hom-object]]
* [[hom-set]], [[hom-functor]]
* [[hom-category]]
* [[hom-space]], [[cocycle space]]
* [[simplicial mapping complex]]
[[!include homotopy-homology-cohomology]]
## References
For some original references by [[William Dwyer]] and [[Dan Kan]] see [[simplicial localization]]. For instance
* {#DK84} [[William Dwyer]], [[Dan Kan]], _A classification theorem for diagrams of simplicial sets_, Topology 23 (1984), 139-155.
On the derived [[function complexes]] in a [[projective model structure on simplicial presheaves]]:
* [[William Dwyer]], [[Daniel Kan]], *Function complexes for diagrams of simplicial sets*, Indagationes Mathematicae (Proceedings) **86** 2 (1983) 139-147 [<a href="https://doi.org/10.1016/1385-7258(83)90051-3">doi:10.1016/1385-7258(83)90051-3</a>, [pdf](https://core.ac.uk/download/pdf/82652265.pdf)]
Discussion in terms of [[quasi-categories]]:
* [[Jacob Lurie]], Section 1.2.2 of: _[[Higher Topos Theory]]_, Annals of Mathematics Studies 170, Princeton University Press 2009 ([pup:8957](https://press.princeton.edu/titles/8957.html), [pdf](https://www.math.ias.edu/~lurie/papers/HTT.pdf))
* [[Dan Dugger]], [[David Spivak]], *Mapping spaces in quasi-categories*, Algebraic & Geometric Topology **11** (2011) 263–325 [[arXiv:0911.0469](http://arxiv.org/abs/0911.0469), [doi:10.2140/agt.2011.11.263](http://dx.doi.org/10.2140/agt.2011.11.263)]
The theory of _framings_ is due to
* {#DHK} [[William Dwyer]], [[Philip Hirschhorn]], [[Dan Kan]], _Model categories and general abstract homotopy theory_, (1997) ([pdf](http://www.mimuw.edu.pl/~jacho/literatura/ModelCategory/DHK_ModelCateogories1.pdf))
and in parallel section 5 of
* {#Hovey} [[Mark Hovey]], _Model categories_ ([ps](http://math.unice.fr/~brunov/SecretPassage/Hovey-Model%20Categories.ps))
and in sections 16, 17 of
* [[Philip Hirschhorn]], _Model categories and their localization_ .
A useful quick review of the interrelation of the various constructions of derived hom spaces is page 14, 15 of
* [[Clark Barwick]], _On (enriched) left Bousfield localization of model categories_ ([arXiv](http://arxiv.org/abs/0708.2067))
Discussion of derived hom spaces for [[categories of fibrant objects]] is in
* {#Cisinski10} [[Denis-Charles Cisinski]], _Invariance de la K-théorie par equivalences dérivées_, J. K-theory, 6 (2010), 505–546.
and section 3.6.2 of
* {#NSS12} [[Thomas Nikolaus]], [[Urs Schreiber]], [[Danny Stevenson]], _[[schreiber:Principal ∞-bundles -- theory, presentations and applications|Principal ∞-bundles -- Presentations]]_ ([arXiv:1207.0249](http://arxiv.org/abs/1207.0249))
[[!redirects (infinity,1)-categorical hom-spaces]]
[[!redirects (∞,1)-categorical hom-space]]
[[!redirects (∞,1)-categorical hom-spaces]]
[[!redirects (infinity,1)-categorical hom space]]
[[!redirects (∞,1)-categorical hom space]]
[[!redirects (infinity,1)-categorial hom-space]]
[[!redirects (∞,1)-categorial hom-space]]
[[!redirects (infinity,1)-categorial hom space]]
[[!redirects (∞,1)-categorial hom space]]
[[!redirects quasi-categorical hom-space]]
[[!redirects quasi-categorical hom-spaces]]
[[!redirects derived hom space]]
[[!redirects derived hom-space]]
[[!redirects derived hom spaces]]
[[!redirects derived hom-spaces]]
[[!redirects hom-space]]
[[!redirects hom-spaces]]
[[!redirects hom-∞-groupoid]]
[[!redirects hom-infinity-groupoid]]
[[!redirects hom-(∞,0)-category]]
[[!redirects hom-(infinity,0)-category]]
[[!redirects hom ∞-groupoid]]
[[!redirects hom infinity-groupoid]]
[[!redirects hom (∞,0)-category]]
[[!redirects hom (infinity,0)-category]]
[[!redirects hom-∞-groupoids]]
[[!redirects hom-infinity-groupoids]]
[[!redirects hom-(∞,0)-categories]]
[[!redirects hom-(infinity,0)-categories]]
[[!redirects hom ∞-groupoids]]
[[!redirects hom infinity-groupoids]]
[[!redirects hom (∞,0)-categories]]
[[!redirects hom (infinity,0)-categories]]
[[!redirects homotopy function complex]]
[[!redirects homotopy function complexes]]
[[!redirects (∞,1)-hom (∞,1)-functor)]]
[[!redirects (∞,1)-categorical hom]]
[[!redirects (∞,1)-categorical hom-spaces]]
[[!redirects derived mapping space]]
[[!redirects derived mapping spaces]]
|
(infinity,1)-categorification | https://ncatlab.org/nlab/source/%28infinity%2C1%29-categorification |
+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### $(\infty,1)$-Category theory
+--{: .hide}
[[!include quasi-category theory contents]]
=--
=--
=--
#Contents#
* table of contents
{:toc}
## Idea
[[categorification|Categorification]] is a creative nonmechanical process in which categorical structures are promoted to $n$-categorical structures, for some $n\ge 2$.
(∞,1)-categorification is a special instance of this idea, in which the newly created $n$-morphisms are invertible for all $n\ge 2$.
The phrase [[higher structures]] also refers primarily to (∞,1)-categorification.
Sometimes more than one (∞,1)-categorification is possible,
as is the case for [[abelian groups]],
which can be categorified to [[module spectra|$\mathrm{H}\mathbb{Z}$-module spectra]] (as represented by [[simplicial abelian groups]]) or [[connective spectra]].
In addition to this creative choice of a [[Platonic form]] categorifying a given structure, another creative aspect is a choice of a specifc [[homotopy theory FAQ|model]] for the resulting object,
e.g., [[(∞,1)-categories]] can be modeled by [[relative categories]], [[simplicial categories]], [[quasicategories]], etc.
## Simplicial objects
For [[algebras over an algebraic theory]] $T$,
one can construct an (∞,1)-categorification
by passing to [[simplicial objects]] valued in algebras over $T$,
and equipping them with weak equivalences
induced by the forgetful functor to simplicial sets.
In some cases, the result can be different from the result of the animation
procedure described below, e.g., for the algebraic theory that
defines commutative monoids
we get commutative simplicial monoids, equivalently,
[[E-infinity algebras]] over the [[Eilenberg-MacLane spectrum]]
of the [[integers]],
whereas animation produces connective [[E-infinity ring spectra]].
## Animation
In some cases, there is an automatic (∞,1)-categorification. For example, the [[animation]] $\mathrm{Ani}(C)$ of a cocomplete category $C$ that is generated under colimits by its subcategory $C^{\mathrm{sfp}}$ of compact projective objects, is the (∞,1)-category freely generated by $C^{sfp}$ under sifted colimits. (See Kęstutis Česnavičius and Peter Scholze, Sec. 5.1.4.)
For example, the animation of the 1-category of modules over an ordinary [[ring]] $R$ is the (∞,1)-category of connective [[module spectra]] over the [[Eilenberg-MacLane spectrum|Eilenberg-MacLane ring spectrum]] $\mathrm{H}R$.
## Further Examples
In the table below, structures on the left are always understood up to an isomorphism, whereas on the right we explicitly indicate the notion of a weak equivalence used (except for [[Platonic forms]] such as [[(∞,1)-categories]]).
|categorical structure|(∞,1)-categorical structure|
|---|---|
|[[set]]|[[∞-groupoid]]|
|[[set]]|[[simplicial set]] up to a [[simplicial weak equivalence]]|
|[[groups]]|[[simplicial groups]] up to [[simplicial weak equivalences]]|
|[[algebras]] over [[algebraic theory]] $T$|simplicial algebras over $T$ up to [[simplicial weak equivalences]]|
|[[algebras]] over [[algebraic theory]] $T$|weak simplicial algebras over $T$ up to [[simplicial weak equivalences]]|
|[[abelian group]]|nonnegatively graded [[chain complex]] up to a [[quasi-isomorphism]]|
|[[abelian group]]|[[connective spectrum]] up to a [[weak equivalence]]|
|[[category]]|[[(∞,1)-category]]|
|[[category]]|[[relative category]] up to a [[Barwick-Kan equivalence]]|
|[[category]]|[[simplicial category]] up to a [[Dwyer-Kan equivalence]]|
|[[operad]]|[[(∞,1)-operad]]|
|[[operad]]|[[simplicial operad]] up to a [[weak equivalence]]|
|[[algebras over an operad]] $O$|simplicial algebras over a [[cofibrant resolution]] of $O$ up to a [[weak equivalence]]|
|[[Lie groupoid]]|[[Kan simplicial manifold]]|
|[[presheaf]]|[[simplicial presheaf]]|
|[[sheaf]]|[[(∞,1)-sheaf]]|
|[[sheaf]]|[[simplicial presheaf]] that satisfies the [[homotopy descent]] property|
|[[topos]]|[[(∞,1)-topos]]|
|[[elementary topos]]|[[elementary (∞,1)-topos]]|
|[[2-category]]|[[(∞,2)-category]]|
|[[n-category]]|[[(∞,n)-category]]|
[[!redirects homotopy coherent analog]]
[[!redirects homotopy coherent analogue]]
[[!redirects homotopy coherent analogs]]
[[!redirects homotopy coherent analogues]]
[[!redirects homotopy coherent]]
[[!redirects (∞,1)-categorification]]
[[!redirects (∞,1)-categorifications]]
[[!redirects (infinity,1)-categorifications]]
|
(infinity,1)-category | https://ncatlab.org/nlab/source/%28infinity%2C1%29-category |
+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### $(\infty,1)$-Category theory
+--{: .hide}
[[!include quasi-category theory contents]]
=--
#### Homotopy theory
+--{: .hide}
[[!include homotopy - contents]]
=--
#### Higher category theory
+--{: .hide}
[[!include higher category theory - contents]]
=--
#### $(\infty,1)$-topos theory
+--{: .hide}
[[!include (infinity,1)-topos - contents]]
=--
=--
=--
#Contents#
* table of contents
{:toc}
## Idea
According to the general pattern on [[(n,r)-category]], an $(\infty,1)$-category is a (weak) [[∞-category]] in which all $n$-morphisms for $n \geq 2$ are [[equivalences]]. This is the joint generalization of the notion of _[[category]]_ and _[[∞-groupoid]]_.
More precisely, this is the notion of _[[category]]_ up to [[coherence|coherent]] [[homotopy]]:
an $(\infty,1)$-category is equivalently
* an [[internal category in an (∞,1)-category|internal category]] in [[∞-groupoids]]/basic [[homotopy theory]] (as such usually modeled as a [[complete Segal space]]).
* a category [[enriched (infinity,1)-category
|homotopy enriched]] over [[∞Grpd]] (as such usually modeled as a [[Segal category]]).
Among all [[(n,r)-category|(n,r)-categories]], $(\infty,1)$-categories are special in that they are the simplest structures that at the same time:
* admit a [[higher category theory|higher version]] of [[category theory]] ([[limit]]s, [[adjunction]]s, [[Grothendieck construction]], etc, [[sheaf and topos theory]], etc.) : [[(infinity,1)-category theory]]
* and know everything about higher [[equivalences]].
Notably for understanding the collections of all [[(n,r)-category|(n,r)-categories]] for arbitrary $n$ and $r$, which in general is an $(n+1,r+1)$-category, the knowledge of the underlying $(n,1)$- (and hence $(\infty,1)$-)category already captures much of the information of interest: it allows to decide if two given $(n,r)$-categories are equivalent and allows to obtain new $(n,r)$-categories from existing ones by universal constructions.
The collection of all $(\infty,1)$-categories forms the [[(∞,2)-category]] [[(∞,1)Cat]].
## Definitions
There are a number of different ways to make the idea of an $(\infty,1)$-category precise, including [[quasi-categories]], [[simplicial set|simplicially]] [[enriched categories]], [[topologically enriched categories]], [[Segal categories]], [[complete Segal spaces]], and $A_\infty$-[[A-infinity categories|categories]] (most of which can be done either simplicially or topologically). Additionally, any notion of [[∞-category]] can be specialized to a notion of $(\infty,1)$-category by simply requiring all $n$-cells for $n\gt 1$ to be invertible.
Unlike the case for general notions of $n$-category, almost all the definitions of $(\infty,1)$-category are known to form [[model categories]] that are [[Quillen equivalence|Quillen equivalent]]. See also [[n-category]] for a summary of the state of the art about definitions of $n$-category and comparisons between them.
### Quasi-categories
We start with the definition of "$(\infty,1)$-category" that was promoted by [[Andre Joyal]] as a good model for the theory. This goes back to Boardman-Vogt in the 1970s and was further developed, by [[Jean-Marc Cordier]] and [[Tim Porter]] in the early 1980s.
This is a [[geometric definition of higher category]] which conceives an $(\infty,1)$-category as a [[simplicial set]] with extra [[stuff, structure, property|property]]. It is a straightforward generalization of the definition of [[∞-groupoid]] as a [[Kan complex]], and, in fact, one alternative term used early on was 'weak Kan complex'; see below.
Recall that a [[Kan complex]] is a [[simplicial set]] in which every [[horn]] $\Lambda^k[n]$, $0 \leq k \leq n$ has a _filler_. This condition may be read in words as: every collection of adjacent $n$-cells has a composite $n$-cell, even if the orientations of the cells don't match. This implicitly encodes the _invertibility_ of every cell: if the orientation does not match, we can invert the cell and then compose.
From this perspective one observes, by looking closely at the combinatorics, that the invertibility of the 1-cells in the simplicial set is enforced particularly by the condition that the [[horn|outer horns]] $\Lambda^0[n]$ and $\Lambda^n[n]$ have fillers.
Therefore in a [[simplicial set]] in which _only the inner horns_ $\Lambda^k[n]$ for $0 \lt k \lt n$ have fillers all cells are required to have a kind of inverse, except the 1-cells. (They may have inverses, too, but are not required to).
This is evidently a realization of the idea of an [[(n,r)-category]] with $n = \infty$ and $r = 1$.
Such a simplicial set with fillers for all inner horns
* Boardman and Vogt called a _weak Kan complex_ ;
* [[Andre Joyal]] called a [[quasi-category]];
* [[Jacob Lurie]] called an $\infty$-category.
Here we follow Joyal and say [[quasi-category]] when we mean concretely the simplicial sets with extra property. We use the more general term
"$(\infty,1)$-category" for this or any of its equivalent models, discussed below, in order to distinguish from the term [[∞-category]] or [[∞-category]] that is more traditionally understood to generically mean an $\infty$-category with no conditions on invertibility
(in terms of [[(n,r)-category]]: an $(\infty,\infty)$-category).
With [[quasi-category|quasi-categories]] being just [[simplicial sets]] with extra property, there are evident and simple definitions of
* the [[(infinity,1)-category of (infinity,1)-functors|quasi-category of (∞,1)-functors]] between two [[quasi-categories]] $C$ and $D$;
* the quasi-category of all [[∞-groupoids]];
* the [[(infinity,1)-category of (infinity,1)-categories|quasi-category of all quasi-categories]].
Similarly, [[Andre Joyal]] and [[Jacob Lurie]] have shown that all other constructions in [[category theory]] have good generalizations to quasi-categories, which usually have conceptually simple formulations: see [[Higher Topos Theory]] for more.
### $Top$-, $Kan$- and simplicially enriched categories
Despite the conceptual simplicity of [[quasi-category|quasi-categories]],
for computations and in particular for obtaining examples,
it is often useful to pass to a slightly different model.
Recall that we said at the beginning that an $(\infty,1)$-category is supposed to be like an [[enriched category]] which is enriched over the category of [[∞-groupoids]].
This turns out to make sense literally
if one takes care to remember that $\infty$-groupoids
themselves form a higher category.
As discussed at [[homotopy hypothesis]] there is a Quillen equivalence of the [[model category|model categories]] of
* the [[model structure on topological spaces|standard model structure]] on the [[nice category of spaces|nice category]] of compactly generated weakly Hausdorff [[topological spaces]];
* the [[model structure on simplicial sets|standard model structure]] on the category of [[Kan complex]]es.
In fact, this is also equivalent to
* the [[model structure on simplicial sets|standard model structure]] on the category [[SimpSet|SSet]] of [[simplicial sets]].
If we take the notion of [[Kan complex]] to be the most manifest incarnation of the idea "[[∞-groupoid]]", then under these equivalences one may think of
* a [[simplicial set]] as representing the [[Kan complex]] which is obtained from it by "freely throwing in the missing inverses" of cells (technically: as representing its fibrant replacement);
* a [[topological space]] $X$ as representing the [[Kan complex]] $\Pi(X)$, whose
* 0-cells are the points of $X$;
* 1-cells are the paths in $X$;
* 2-cells are the triangles in $X$;
* etc.
With this interpretation understood (i.e. with these model structures understood), [[SimpSet|SSet]]-enriched categories do model $(\infty,1)$-categories.
For more see
* [[relation between quasi-categories and simplicial categories]]
### Homotopical categories
A [[homotopical category]] is a category $C$
equipped with a class $W$ of [[category with weak equivalences|weak equivalences]].
Every homotopical category $(C,W)$ has a _quasi-localisation_ $C[W(-1)]$
which is a [[quasi-category]]. The simplicial set $C[W(-1)]$ is
obtained from the [[nerve]] of $C$ by freely gluing a [[homotopy]] inverse to each [[morphism]] in $W$, and then, by adding simplices to turn it into a quasi-category (this last step is called a fibrant completion).
The [[quasi-category]] $C[W(-1)]$ is equivalent to the [[Dwyer-Kan localisation]] of $C$ with respect to $W$, via the equivalence between [[quasi-category|quasi-categories]] and [[simplicially enriched category|simplicial categories]] mentioned above.
Conversely, every quasi-category is equivalent to
the quasi-localisation of a homotopical category.
This gives a representation of all $(\infty,1)$-categories
in terms of [[homotopical category|homotopical categories]].
It follows that many aspects of the theory of $(\infty,1)$-categories
can be expressed in terms of [[category theory]].
When the homotopical category (C,W) is obtained from a Quillen
model structure (by forgetting the cofibrations and the fibrations)
the quasi-category C[W^(-1)] has finite limits and colimits.
Conversely, I conjecture that every quasi-category
with finite limits and colimits is equivalent to the quasi-localisation
of a model category. In fact, every locally presentable quasi-category
is a quasi-localisation of a combinatorial model by a result of Lurie.
More can be said: the underlying category can taken to be
a category of presheaves by a result of Daniel Dugger.
http://arxiv.org/abs/math/0007070
### Model categories
A specific notion of [[homotopical category]] is that of a [[model category]]. $(\infty,1)$-categories obtained as the Dwyer-Kan simplicial localizations of model categories have for instance finite $(\infty,1)$-[[limit]]s and $(\infty,1)$-[[colimit]]s. The [[presentable (∞,1)-category|locally presentable (∞,1)-categories]] are precisely those presented this way by [[combinatorial model category|combinatorial model categeories]].
At the very beginning, a [[model category]] was understood as a "model for the category [[Top]] of topological spaces," or more precisely [[homotopy types]]: some [[category]] with extra [[stuff, structure, property|structure and properties]] which allows one to perform all operations familiar of the [[homotopy theory]] of [[topological spaces]].
As mentioned above, from the point of view of [[(∞,1)-categories]], [[Top]] may naturally be regarded an as [[(∞,1)-category]] and is in fact the archetypical example, analogous to how [[Set]] is the archetypical example of an ordinary [[category]].
This indicates that, more generally, a [[model category]] should actually be a means to model (i.e. encode) in 1-categorical terms an $(\infty,1)$-category, and of course this is true since indeed any [[category with weak equivalences]] presents an $(\infty,1)$-category via Dwyer-Kan simplicial localization. In the case of a model category, however, or at least a [[simplicial model category]], this $(\infty,1)$-category has a different, simpler construction.
* A [[simplicial model category]] $\mathbf{A}$ is, in particular, a [[simplicially enriched category]].
* the full [[SSet]]-[[subcategory]] $\mathbf{A}^\circ$ on the fibrant-cofibrant objects of $\mathbf{A}$ happens to be [[Kan complex]]-[[enriched category|enriched]];
* the [[homotopy coherent nerve]] $N(\mathbf{A}^\circ)$ of $\mathbf{A}^\circ$ is the [[quasi-category]] _presented_ by $A$.
Up to equivalence, this gives the same $(\infty,1)$-category as the Dwyer-Kan hammock localization. With the relation between [[simplicially enriched categories]] and [[quasi-category|quasi-categories]] via [[homotopy coherent nerve]] understood, we shall here often not distinguish between $\mathbf{A}^\circ$ and $N(\mathbf{A}^\circ)$ as the $(\infty,1)$-category [[presentable (infinity,1)-category|presented]] by a [[model category]] $A$.
### Segal categories and complete Segal spaces
Other models for $(\infty,1)$-categories are
* [[Segal categories]];
* [[complete Segal spaces]].
Segal categories can be thought of as categories which are
_weakly_ enriched in topological spaces/simplicial sets/Kan complexes,
where the definition of "weak" makes use of the notion of [[homotopy]]
and [[homotopy limit]] in [[Top]] or [[SimpSet|SSet]].
Complete Segal spaces are like [[internal categories in an (∞,1)-category]].
This construction principle in particular lends itself
to iteration and hence to an inductive definition of
[[(∞,n)-category]] via [[Segal n-categories]] and [[n-fold complete Segal spaces]].
### $A_\infty$-categories
An $A_\infty$-[[A-infinity category|category]] can also be thought of as a category "weakly enriched" in spaces (i.e. $\infty$-groupoids), except that in contrast to the Segal approaches the "weakness" is specified [[algebraic definition of higher category|algebraically]] and parametrized by an [[operad]]. This approach can be generalized to the [[Trimble n-category|Trimble]] definition of $n$-category or $(\infty,n)$-category.
## Properties
### $(\infty,1)$-Category theory
A crucial point about the notion of _$(\infty,1)$-category_ is that it supports all the standard constructions and theorems of [[category theory]], if only the consistent replacements are made ([[isomorphism]] becomes [[equivalence]], etc.).
See _[[(∞,1)-category theory]]_.
### The collection of all $(\infty,1)$-categories
The collection of all $(\infty,1)$-categories
forms an [[(∞,2)-category]] called [[(∞,1)Cat]].
Often it is useful to regard that as a (large) $(\infty,1)$-category itself, by discarding the non-invertible [[natural transformations]].
### Model category presentations
There is a wealth of different presentations of $(\infty,1)$-categories.
See _[[table - models for (∞,1)-categories]]_.
## A model-independent approach
In practice, it can be useful to be able to treat all "presentations of $(\infty,1)$-categories" on the same equal footing (e.g. relative categories and topologically-enriched categories). While truly model-independent foundations of $(\infty,1)$-category theory do not (yet) exist, this can be accomplished _within_ any model of $(\infty,1)$-categories, which we proceed to describe. As quasicategories are by far the most well-developed, we use them as an ambient framework. We also take care to make as few choices (even "contractible" ones) as possible. However, we do not explicitly mention set-theoretic issues, though these are easily handled using Grothendieck universes.
1. Consider the $Kan$-enriched category $\underline{QCat}$ of quasicategories; for quasicategories $C$ and $D$, the Kan complex of morphisms between them is $\underline{hom}_{\underline{QCat}} = \iota(\underline{hom}_{sSet}(C,D))$, the largest Kan complex contained in their internal hom simplicial set.
1. Define a _relative quasicategory_ to be a quasicategory equipped with a wide sub-quasicategory of "weak equivalences" containing all equivalences. For relative quasicategories $(C,W_C)$ and $(D,W_D)$, write $\underline{hom}_{\underline{RelQCat}}((C,W_C),(D,W_D)) \subset \underline{hom}_{\underline{QCat}}(C,D)$ for the sub-Kan complex consisting of those maps which take $W_C$ into $W_D$. Note that using this definition, this is actually the inclusion of a disjoint union of connected components among Kan complexes (in the strictest possible sense).
1. There is an evident inclusion $min : \underline{QCat} \to \underline{RelQCat}$, which takes a quasicategory $C$ to the relative quasicategory $(C,C^\simeq)$.
1. Although a Quillen equivalence $M_1 \rightleftarrows M_2$ between model categories determines an equivalence of homotopy categories, note that neither adjoint functor need preserve weak equivalences. On the other hand, the restrictions $M_1^c \hookrightarrow M_1 \rightarrow M_2$ and $M_1 \leftarrow M_2 \hookleftarrow M_2^f$ (to the cofibrant objects of $M_1$ and the fibrant objects of $M_2$) do preserve weak equivalences, and these determine a hexagonal diagram of weak equivalences between relative categories (in the Barwick--Kan model structure), as in [MazelGee16, Figure 1](http://nyjm.albany.edu/j/2016/22-4.html).
1. Using the previous observation, expand the diagram in the introduction of [BarwickSchommerPries](http://arxiv.org/abs/1112.0040) (relating a great many Quillen equivalent model categories presenting "the homotopy theory of $(\infty,1)$-categories") into a diagram of weak equivalences between relative categories. As relative categories are particular examples of relative quasicategories, this defines a functor $F : K \to \underline{RelQCat}$ among fibrant objects of $(Cat_{sSet})_{Bergner}$.
1. Now, apply the right Quillen equivalence $N^{hc} : (Cat_{sSet})_{Bergner} \to sSet_{Joyal}$ (the homotopy-coherent nerve) to this cospan $\underline{QCat} \xrightarrow{min} \underline{RelQCat} \leftarrow K$.
1. The morphism $N^{hc}(min)$ of quasicategories admits a contractible Kan complex worth of quasicategorical left adjoints, any of which presents the _localization_ of relative quasicategories. Choose one, and denote this quasicategorical adjunction by $L : N^{hc} (\underline{RelQCat}) \rightleftarrows N^{hc} ( \underline{QCat}) : min$.
1. It follows from the main theorem of [Toen](http://arxiv.org/abs/math/0409598) that the composite map $L \circ N^{hc}(F) : N(K) \cong N^{hc}(K) \to N^{hc}(\underline{RelQCat}) \to N^{hc}(\underline{QCat})$ is "essentially contractible" in the quasicategorical sense. More precisely, for any cofibration into an acyclic object $i : N(K) \to K' \approx pt$ in $sSet_{Joyal}$, there exists a contractible Kan complex worth of extensions of $L \circ N^{hc}(F)$ over $i$.
1. Define $(The(\infty,1)Cats) \subset N^{hc}(\underline{QCat})$ to be the maximal sub-Kan complex generated by the image of $L \circ N^{hc}(F)$. We write $Cat_{(\infty,1)} \in (The(\infty,1)Cats)$ for _any_ vertex, and propose that to work "model-independently" is to work _within_ $Cat_{(\infty,1)}$.
This sequence of maneuvers balances twin aims. On the one hand, Toen's theorem asserts that after choosing a basepoint, this Kan complex is a model for $B(\mathbb{Z}/2)$. Thus, any sort of object which might be considered as "a presentation of an $(\infty,1)$-category" canonically determines an object of $Cat_{(\infty,1)}$ (where "canonical" must still be taken in the quasicategorical sense). On the other hand, it is completely independent of which vertex of $(The(\infty,1)Cats)$ we choose.
[This diagram](http://ncatlab.org/nlab/files/relcats-modelcats-qcats-inftycats.jpg), taking place in $N^{hc}(\underline{QCat})$, elaborates on certain salient aspects of the passage from models of $(\infty,1)$-categories to a model-independent approach. (For a small amount of explanation of this diagram, see [here](https://nforum.ncatlab.org/discussion/7029/a-diagram-relating-different-models-of-inftycategories/#Item_0).)
## Related concepts
* [[table - models for (∞,1)-categories]]
* [[0-category]], [[(0,1)-category]]
* [[category]]
* [[2-category]], [[(2,1)-category]]
* [[3-category]]
* [[n-category]]
* [[(∞,0)-category]]
* [[(n,1)-category]]
* **(∞,1)-category**, [[internal (∞,1)-category]], [[∞-groupoid]]
* [[locally cartesian closed (∞,1)-category]]
* [[(∞,1)-topos]]
* [[semiadditive (∞,1)-category]], [[additive (∞,1)-category]]
* [[stable (∞,1)-category]]
* [[monoidal (∞,1)-category]]
* [[locally presentable (∞,1)-category]], [[accessible (∞,1)-category]], [[compactly generated (∞,1)-category]]
* [[disjunctive (∞,1)-category]]
* [[model (∞,1)-category]]
* [[(∞,2)-category]]
* [[(∞,n)-category]]
* [[(n,r)-category]]
## References
### General
For several years [[Andre Joyal]] -- who was one of the first to promote the idea that for studying [[higher category theory]] it is good to first study $(\infty,1)$-categories in terms of [[quasi-category|quasi-categories]] -- has been preparing a textbook on the subject. This still doesn't quite exist, but an extensive write-up of lecture notes does:
* {#Joyal08} [[André Joyal]], *[[The Theory of Quasi-Categories and its Applications]]*, lectures at: _[Advanced Course on Simplicial Methods in Higher Categories](https://lists.lehigh.edu/pipermail/algtop-l/2007q4/000017.html)_, Quadern **45** 2, Centre de Recerca Matemàtica, Barcelona 2008 ( [[JoyalTheoryOfQuasiCategories.pdf:file]])
Further notes (where the term "[[logos]]" is used instead of [[quasi-category]]):
* {#Joyal08} [[André Joyal]], _Notes on Logoi_, 2008 ([pdf](http://www.math.uchicago.edu/~may/IMA/JOYAL/Joyal.pdf), [[JoyalOnLogoi2008.pdf:file]])
Meanwhile [[Jacob Lurie]], building on Joyal's work, has considerably pushed the theory further. A comprehensive discussion of the theory of $(\infty,1)$-categories in terms of the models [[quasi-category]] and [[simplicially enriched category]] is
* [[Jacob Lurie]], _[[Higher Topos Theory]]_ .
An brief exposition from the point of view of [[algebraic topology]] is in
* [[Jacob Lurie]], _What is... an $\infty$-category?_, _Notices of the AMS_, September 2008 ([pdf](http://www.ams.org/notices/200808/tx080800949p.pdf))
A useful comparison of the four [[model category]] structures on
* [[quasi-categories]];
* [[simplicially enriched categories]];
* [[Segal categories]];
* [[complete Segal spaces]].
is in
* [[Julie Bergner]], _A survey of $(\infty,1)$-categories_, In: [[John Baez]], [[Peter May]] (eds.), _[[Towards Higher Categories]]_, The IMA Volumes in Mathematics and its Applications, vol 152, Springer 2007 ([arXiv:math/0610239](http://arxiv.org/abs/math/0610239), [doi:10.1007/978-1-4419-1524-5_2](https://doi.org/10.1007/978-1-4419-1524-5_2))
* [[Julia Bergner]], _Equivalence of models for equivariant $(\infty,1)$-categories_, Glasgow Mathematical Journal, Volume 59, Issue 1 (2016) ([arXiv:1408.0038](https://arxiv.org/abs/1408.0038), [doi:10.1017/S0017089516000136](https://doi.org/10.1017/S0017089516000136))
More discussion of the other two models can be found at
* [[Jacob Lurie]], _[[On the Classification of Topological Field Theories]]_
and in the references listed at _[[(∞,n)-category]]_.
The relation between [[quasi-category|quasi-categories]] and [[simplicially enriched categories]] was discussed in detail in
* [[Dan Dugger]], [[David Spivak]], _Rigidification of quasi-categories_ ([arXiv:0910.0814](http://arxiv.org/abs/0910.0814))
* [[Dan Dugger]], [[David Spivak]], _Mapping spaces in quasi-categories_, Algebraic & Geometric Topology 11 (2011) 263–325 ([arXiv:0911.0469](http://arxiv.org/abs/0911.0469), [doi:10.2140/agt.2011.11.263](http://dx.doi.org/10.2140/agt.2011.11.263))
The presentation of $(\infty,1)$-categories by [[homotopical categories]] and [[model categories]] is discussed in
* [[William Dwyer]], [[Philip Hirschhorn]], [[Daniel Kan]], [[Jeff Smith]], _[[Homotopy Limit Functors on Model Categories and Homotopical Categories]]_ , volume 113 of Mathematical Surveys and Monographs
A model by [[stratified spaces]] is in
* {#AyalaFrancisRozenblyum15} [[David Ayala]], [[John Francis]], [[Nick Rozenblyum]], _A stratified homotopy hypothesis_ ([arXiv:1502.01713](http://arxiv.org/abs/1502.01713))
A more model-independent abstract formulation is discussed in
* {#RiehlVerity16} [[Emily Riehl]], [[Dominic Verity]], _Infinity category theory from scratch_, Higher Structures **4** 1 (2020) [[arXiv:1608.05314](https://arxiv.org/abs/1608.05314), [pdf](http://www.math.jhu.edu/~eriehl/scratch.pdf), [lectures](https://www.epfl.ch/labs/hessbellwald-lab/seminar/ytm2015/)]
* {#RiehlVerity22} [[Emily Riehl]], [[Dominic Verity]], _[[Elements of ∞-Category Theory]]_, Cambridge studies in advanced mathematics **194**, Cambridge University Press (2022) $[$[doi:10.1017/9781108936880](https://doi.org/10.1017/9781108936880), ISBN:978-1-108-83798-9, [pdf](https://emilyriehl.github.io/files/elements.pdf)$]$
For discussion in [[homotopy type theory]] see _[[internal category in homotopy type theory]]_ and see
* {#RiehlShulman17} [[Emily Riehl]], [[Michael Shulman]], _A type theory for synthetic $\infty$-categories_ ([arXiv:1705.07442](https://arxiv.org/abs/1705.07442))
* {#Riehl18} [[Emily Riehl]], _The synthetic theory of ∞-categories vs the synthetic theory of ∞-categories_, talk at [Vladimir Voevodsky Memorial Conference 2018](http://www.math.ias.edu/vvmc2018) ([pdf](http://www.math.jhu.edu/~eriehl/Voevodsky.pdf))
### Surveys and lecture notes
{#LectureNotes}
An introduction to [[higher category theory]] through $(\infty,1)$-categories:
* Omar Antolín Camarena, _A whirlwind tour of the world of $(\infty,1)$-categories_, 2013 ([arXiv:1303.4669](http://arxiv.org/abs/1303.4669))
Elementary exposition with an eye towards [[homotopy type theory]]:
* {#Riehl22} [[Emily Riehl]], *$\infty$-Category theory for undergraduates*, [talk](Center+for+Quantum+and+Topological+Systems#RiehlDec2022) at *[[CQTS]]* (Dec. 2022) [[web](Center+for+Quantum+and+Topological+Systems#RiehlDec2022), video: [YT](https://www.youtube.com/watch?v=7g2rkiFsbXo)]
* {#Riehl23} [[Emily Riehl]], *[Could $\infty$-category theory be taught to undergraduates?](https://www.ams.org/journals/notices/202305/noti2692/noti2692.html?adat=May%202023&trk=2692&galt=feature&cat=feature&pdfissue=202305&pdffile=rnoti-p727.pdf)*, Notices of the AMS (May 2023) [[published pdf](https://www.ams.org/journals/notices/202305/rnoti-p727.pdf), [arxiv:2302.07855](https://arxiv.org/abs/2302.07855)]
A foundational set of lecture notes:
* [[Denis-Charles Cisinski]], _[[Higher Categories and Homotopical Algebra]]_, Cambridge University Press 2019 ([doi:10.1017/9781108588737](https://doi.org/10.1017/9781108588737), [pdf](http://www.mathematik.uni-regensburg.de/cisinski/CatLR.pdf))
A survey with an eye towards [[higher algebra]] is in
* [[Moritz Groth]], _A short course on $\infty$-categories_ ([pdf](https://arxiv.org/pdf/1007.2925))
A survey on various notions of [[homotopical categories]]:
* [[Emily Riehl]], _Homotopical categories: from model categories to (∞,1)-categories_, in: [[Andrew J. Blumberg]], [[Teena Gerhardt]], [[Michael A. Hill]] (eds,) *[[Stable categories and structured ring spectra]]*, MSRI Book Series, Cambridge University Press ([arXiv:1904.00886](https://arxiv.org/abs/1904.00886))
Also:
* [[Markus Land]], *Introduction to Infinity-Categories*, Birkhäuser 2021 ([doi:10.1007/978-3-030-61524-6](https://link.springer.com/book/10.1007/978-3-030-61524-6))
Lecture notes:
* [[Dylan Wilson]], _Lectures on higher categories_ ([pdf](https://sites.google.com/a/uw.edu/dwilson/notes))
* [[Emily Riehl]], _[[Categorical Homotopy Theory]]_
See also
* [[Zhen Lin Low]], _[[Notes on homotopical algebra]]_
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### Context
#### Higher category theory
+-- {: .hide}
[[!include higher category theory - contents]]
=--
=--
=--
#Contents#
* table of contents
{:toc}
## Idea
The collection of all [[(∞,1)-categories]] forms naturally the [[(∞,2)-category]] [[(∞,1)Cat]].
But for many purposes it is quite sufficient to regard only invertible [[natural transformations]] between [[(∞,1)-functor]], which means that one needs just the maximal [[(∞,1)-category]] inside that $(\infty,2)$-category of all $(\infty,1)$-categories.
Given that an $(\infty,1)$-category is a context for abstract [[homotopy theory]], the $(\infty,1)$-category of $(\infty,1)$-categories is also called the the _homotopy theory of homotopy theories_ ([Rezk 98](#Rezk98), [Bergner 07](#Bergner07)).
(Another, complementary, truncation is to the [[homotopy 2-category of (∞,1)-categories]].)
## Definition
### Intrinsic definition
The full [[SSet]]-[[enriched category theory|enriched]]-[[subcategory]] of [[SSet]] on those simplicial sets which are [[quasi-categories]] is -- by the properties discussed at [[(∞,1)-category of (∞,1)-functors]] -- itself a [[quasi-category]]-[[enriched category]]. This is the [[(∞,2)-category]] of [[(∞,1)-categories]].
The [[sSet]]-[[subcategory]] of that obtained by picking of each [[hom-object]] the [[core]], i.e. the maximal [[∞-groupoid]]/[[Kan complex]] yields an [[∞-groupoid]]/[[Kan complex]]-[[enriched category]]. This is the **$(\infty,1)$-category of $(\infty,1)$-categories** in its incarnation as a [[simplicially enriched category]]. Forming its [[homotopy coherent nerve]] produces the **quasi-category of quasi-categories** .
### Models
The [[Andre Joyal|Joyal]]-[[model structure for quasi-categories]] is an $sSet_{Joyal}$-[[enriched model category]] and hence its full [[SSet]]-[[subcategory]] on cofibrant-fibrant objects is the $(\infty,2)$-category of $(\infty,1)$-categories.
An $SSet_{Quillen}$-[[enriched model category]] (i.e. enriched over the ordinary [[model structure on simplicial sets]]) whose full subcategory of fibrant-cofibrant objects is the $(\infty,1)$-category $(\infty,1)Cat$ is the [[model structure on marked simplicial over-sets|model structure on marked simplicial sets]] (over the terminal set). Its underlying plain model category is [[Quillen equivalence|Quillen equivalent]] to the Joyal-model structure, but it is indeed $sSet_{Quillen}$-enriched.
Other model structures that present the $(\infty,1)$-category of all $(\infty,1)$-categories are
* [[model structure on categories with weak equivalences]]
* [[model structure on simplicially enriched categories]]
* [[model structure for complete Segal spaces]].
### The nerve into simplicial spaces
The nerve functor
$$
N : (\infty,1)Cat_1 \to PSh(\Delta, \infty Gpd) : C \mapsto n \mapsto Core(C^{[n]})
$$
is fully faithful. Thus, the $(\infty,1)$-category of $(\infty,1)$-categories can be identified with the $(\infty,1)$-category of [[category object in an (infinity,1)-category#GrpdInCatIsChoiceOfGroupoidObjects|internal categories in $\infty Gpd$]]
This is closely related to the complete Segal space model.
$N$ is, in fact, the embedding of a [[reflective sub-(infinity,1)-category]]. The $(\infty,1)$-categories can be identified with the subcategory of $PSh(\Delta, \infty Gpd)$ of local objects with respect to the spine inclusions $Sp^n \subseteq \Delta^n$ and with the map $J \to 1$, where $J$ is the indiscrete simplicial space on two discrete objects.
Alternatively, the map $J \to 1$ can be replaced with the projection from the simplicial discrete space formed from the union of two 2-simplices expressing the idea of a morphism with a left and right inverse $fg \simeq 1$ and $gh \simeq 1$.
## Applications
* of particular interest is the $(\infty,1)$-subcategory $(\infty,1)PresCat_1 \hookrightarrow (\infty,1)Cat_1$ of [[presentable (infinity,1)-category|presentable (∞,1)-categories]].
## Related concepts
* [[homotopy 2-category of (∞,1)-categories]]
## References
In terms of [[complete Segal spaces]]:
* {#Rezk98} [[Charles Rezk]], _A model for the homotopy theory of homotopy theory_, Trans. Amer. Math. Soc. 353 (2001), 973-1007 ([arXiv:math/9811037](https://arxiv.org/abs/math/9811037), [doi:10.1090/S0002-9947-00-02653-2](https://doi.org/10.1090/S0002-9947-00-02653-2))
* {#Bergner07} [[Julia Bergner]], _Three models for the homotopy theory of homotopy theories_, Topology Volume 46, Issue 4, September 2007, Pages 397-436 ([arXiv:math/0504334](https://arxiv.org/abs/math/0504334), [doi:10.1016/j.top.2007.03.002](https://doi.org/10.1016/j.top.2007.03.002))
In terms of [[quasi-categories]]:
* [[Jacob Lurie]], Chapter 3 of: _[[Higher Topos Theory]]_ (2009)
[[!redirects (∞,1)-category of (∞,1)-categories]]
[[!redirects homotopy theory of homotopy theories]]
[[!redirects homotopy theory of homotopy theory]]
[[!redirects homotopy theories of homotopy theories]]
[[!redirects homotopy theories of homotopy theory]]
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###Context###
#### $(\infty,1)$-Category theory
+--{: .hide}
[[!include quasi-category theory contents]]
=--
=--
=--
#Contents#
* table of contents
{:toc}
## Idea
The generalization of the notion of [[functor category]] from [[category theory]] to [[(∞,1)-category|(∞,1)]]-[[higher category theory]].
## Definition
Let $C$ and $D$ be [[(∞,1)-categories]], taken in their incarnation as [[quasi-category|quasi-categories]]. Then
$$
Func(C,D) := sSet(C,D)
$$
is the [[simplicial set]] of morphisms of simplicial sets between $C$ and $D$ (in the standard [[sSet]]-[[enriched category|enrichment]] of $SSet$):
$$
sSet(C,D) := [C,D] := ([n] \mapsto Hom_{sSet}(\Delta[n]\times C,D))
\,.
$$
The objects in $Fun(C,D)$ are the [[(∞,1)-functors]] from $C$ to $D$, the morphisms are the corresponding [[natural transformations]] or [[homotopy|homotopies]], etc.
+-- {: .num_prop }
###### Proposition
The simplicial set $Fun(C,D)$ is indeed a [[quasi-category]].
In fact, for $C$ and $D$ any simplicial sets, $Fun(C,D)$ is a [[quasi-category]] if $D$ is a [[quasi-category]].
=--
+-- {: .proof}
###### Proof
Using that [[sSet]] is a [[closed monoidal category]] the [[horn]] filling conditions
$$
\array{
\Lambda[n]_i &\to& [C,D]
\\
\downarrow & \nearrow
\\
\Delta[n]
}
$$
are equivalent to
$$
\array{
C \times \Lambda[n]_i &\to& D
\\
\downarrow & \nearrow
\\
C \times \Delta[n]
}
\,.
$$
Here the vertical map is [[fibrations of quasi-categories|inner anodyne]] for inner horn inclusions $\Lambda[n]_i \hookrightarrow \Delta[n]$, and hence the lift exists whenever $D$ has all inner horn fillers, hence when $D$ is a [[quasi-category]].
=--
For the definition of $(\infty,1)$-functors in other models for $(\infty,1)$-categories see [[(∞,1)-functor]].
## Properties
### Model category presentation
{#ModelCategoryPresentation}
The projective and injective [[global model structure on functors]] (as well as the [[Reedy model structure]] if $C$ is a [[Reedy category]]) [[presentable (infinity,1)-category|presents]] $(\infty,1)$-categories of $(\infty,1)$-functors, at least when there exists [[combinatorial simplicial model category|combinatorial simplicial]]-[[structure]] model on the [[codomain]] [[model category]].
Let
* $C$ be a [[small category|small]] [[sSet]]-[[enriched category]];
* $A$ a [[combinatorial simplicial model category]] and
$A^\circ$ its [[full subcategory|full]] [[sSet-enriched category|sSet-]][[subcategory]] of [[bifibrant objects]];
* $[C,A]$ the [[sSet]]-[[enriched functor category]] equipped with the projective[^1] [[global model structure on functors]], and
$[C,A]^\circ$ its full [[sSet]]-[[subcategory]] on [[bifibrant objects]].
* $N \colon sSet\text{-}Cat \to sSet$ the [[homotopy coherent nerve]].
[^1]: [Lurie's Remark 4.2.4.5](#Lurie) claims that Prop. \ref{PresentationByModelStructuresOnFunctors} also works also for the injective structure, but this is not so clear, see also [MO:q/440965](https://mathoverflow.net/q/440965/381).
[[right adjoints preserve limits|Since]]$\;N$ is a [[right adjoint]] it [[preserved limit|preserves]] [[products]] so that we obtain a morphism
\[
N(C) \times N\big([C,A]\big)
\xrightarrow{\; \sim \;}
N\big(C \times [C,A]\big)
\xrightarrow{\; N(ev) \;}
N(A)
\]
induced from the [[internal hom]]-[[adjunct]] of $Id \colon [C,A] \to [C,A]$.
Noticing that the [[bifibrant objects]] of $[C,A]$ are [[enriched functors]] that, in particular, take values in [[bifibrant objects]] of $A$, this restricts to a morphism of the form
$$
N(C) \times N\big([C,A]^\circ\big)
\xrightarrow{\;\; N(ev) \;\;}
N\big(A^\circ\big)
\,,
$$
which, by the [[internal hom]]-[[adjunction]], corresponds to a morphism
$$
N\big([C,A]^\circ\big)
\xrightarrow{\phantom{--}}
sSet\big(
N(C)
,\,
N(A^\circ)
\big)
\,.
$$
Here $A^\circ$ is [[Kan complex]]-[[enriched category|enriched]], by the [[axioms]] of an [[classical model structure on simplicial sets|$sSet_{Quillen}$-]][[enriched model category]], and so $N(A^\circ)$ is a [[quasi-category]]. Therefore we may write this as
$$
\cdots
\;=\;
Func\big(
N(C)
,\,
N(A^\circ)
\big)
\,.
$$
\begin{proposition}
\label{PresentationByModelStructuresOnFunctors}
This canonical morphism
\[
\label{TheNerveComparisonEquivalence}
N\big([C,A]^\circ\big)
\xrightarrow{\phantom{---}}
Func\big(
N(C)
,\,
N(A^\circ)
\big)
\]
is an [[equivalence of (infinity,1)-categories|equivalence of $\infty$-categories]] in that it is a [[weak equivalence]] in the [[model structure for quasi-categories]].
\end{proposition}
This is [Lurie (2009), Prop. 4.2.4.4](#Lurie).
\begin{proof}
The strategy is to show that the objects on both sides are both [[exponential objects]] in the [[homotopy category]] of [[Joyal model structure|$sSet_{Joyal}$]], which, by the [uniqueness of adjoints](adjoint+functor#UniquenessOfAdjoints), implies that they are [[isomorphism|isomorphic]] in the homotopy category, which finally is equivalent to the statement to be proven.
That $Func\big(N(C), N(A^\circ)\big) \simeq \big(N(A^\circ)\big)^{N(C)}$
is an [[exponential object]] in the [[homotopy category]] is pretty immediate.
That the left hand is an isomorphic exponential follows from [Lurie 09, corollary A.3.4.12](#Lurie), which asserts that for $C$ and $D$ [[sSet-enriched categories]] with $C$ [[cofibrant object|cofibrant]] and $A$ as above, we have that [[composition]] with the [[evaluation map]] induces a [[bijection]]
$$
Hom_{Ho(sSet Cat)}\big(D, [C,A]^\circ\big)
\xrightarrow{\simeq}
Hom_{Ho(sSet Cat)}\big(C \times D, A^\circ\big)
\,.
$$
Since $Ho(sSet Cat_{Bergner}) \simeq Ho(sSet_{Joyal})$ this
identifies also $N\big([C,A]^\circ\big)$ with the exponential object in question.
\end{proof}
> The following proof is fresh, still needs double-checking.
\begin{corollary}
\label{CompatibilityOfModelPresentationWithPrecomposition}
In the above situation, consider an [[sSet]]-[[enriched functor]] $f \,\colon\, C' \longrightarrow C$ between any [[small category|small]] [[sSet-enriched categories]]. Then under the identification of Prop. \ref{PresentationByModelStructuresOnFunctors} the two $\infty$-functors given by
1. [[homotopy coherent nerve]] of the [[derived functor]] $\mathbb{R}f^\ast$ of precomposition with $f$
1. the precomposition with the [[homotopy coherent nerve]] of $f$
are related by a [[natural equivalence]] of $\infty$-functors:
\begin{tikzcd}[sep=30pt]
N\big(
[C,A]^\circ
\big)
\ar[dd, "{ N\big( \mathbb{R}f^\ast \big) }"{description}]
\ar[rr, "{ \sim }"]
&&
\mathrm{Func}\big(
N(C)
,\,
N(A^\circ)
\big)
\ar[dd, "{ N(f)^\ast }"{description}]
\\
\\
N\big(
[C',A]^\circ
\big)
\ar[uurr, Rightarrow, shorten=50pt, "{ \sim }"{sloped}]
\ar[rr, "{ \sim }"]
&&
\mathrm{Func}\big(
N(C')
,\,
N(A^\circ)
\big)
\end{tikzcd}
\end{corollary}
\begin{proof}
\label{ProofOfCompatibilityOfModelPresentationWithPrecomposition}
Consider the following diagram of [[simplicial sets]] (the outer ones being [[quasi-categories]]):
\begin{tikzcd}[
row sep=30pt,
column sep=20pt
]
N\big(
[C,A]^\circ
\big)
\ar[dr, hook]
\ar[rrrr]
\ar[ddddd, "{ N(\mathbb{R}f^\ast) }"{description} ]
&[-10pt]
&&
&[-35pt]
\mathrm{Func}\big(
N(C)
,\,
N(A^\circ)
\big)
\ar[dl, hook]
\ar[ddddd, "{ N(f)^\ast }"{description}]
\\
&
N\big(
[C,A]
\big)
\ar[rr]
\ar[dd, "{ N(f^\ast) }"{description} ]
&&
\mathrm{Func}\big(
N(C)
,\,
N(A)
\big)
\ar[dd, "{ N(f)^\ast }"{description}]
\\
\\
&
N\big(
[C',A]
\big)
\ar[rr]
\ar[d, "{ N(Q) }"{description}]
\ar[dr, "{\mathrm{id}}"{description}, "{\ }"{name=t, swap}]
&&
\mathrm{Func}\big(
N(C')
,\,
N(A)
\big)
\\
{}
&
N\big(
[C',A]
\big)
\ar[r, "{ \mathrm{id} }"{description} ]
\ar[to=t, Rightarrow]
&
N\big(
[C',A]
\big)
\ar[ur]
&&
\\[-10pt]
N\big(
[C',A]^\circ
\big)
\ar[ur, hook]
\ar[rrrr]
&&&&
\mathrm{Func}\big(
N(C')
,\,
N(A^\circ)
\big)
\ar[uul, hook]
\end{tikzcd}
Here $Q$ denotes any [[functorial factorization|functorial]] [[cofibrant replacement]] (which exists, by [this Example](functorial+factorization#InCombinatorialModelCategories), since $[C',A]$ is a [[combinatorial model category]] by the above discussion) and the double arrow denotes (the image under the hc-nerve of) the [[natural transformation]] with components the corresponding resolution equivalences $Q(\text{-}) \xrightarrow{\;\sim\;} (\text{-})$ (which are components of a [[natural transformation]], by the nature of [[functorial factorization]]).
The left square commutes by the construction of right derived functors of [[right Quillen functors]] (eg. [this Prop.](Introduction+to+Homotopy+Theory#ComputationOfLeftRightDerivedFunctorsViaResolutions)) and the middle square is the [[naturality square]] of the comparison map discussed [above](#ModelCategoryPresentation). The remaining outer squares just exhibit the restriction to [[bifibrant objects]], as discussed above.
The total diagram is of the claimed from. It just remains to see that the 2-morphism filling it is really an equivalence, but this follows by Prop. \ref{EquivalencesDetectedOnObjects}.
\end{proof}
\linebreak
### Limits and colimits {#Limits}
For $C$ an ordinary [[category]] that admits small [[limit]]s and [[colimit]]s, and for $K$ a [[small category]], the [[functor category]] $Func(D,C)$ has all small limits and colimits, and these are computed objectwise. See [[limits and colimits by example]]. The analogous statement is true for $(\infty,1)$-categories of $(\infty,1)$-functors
\begin{proposition}
Let $K$ and $C$ be [[quasi-categories]], such that $C$ has all [[limit in a quasi-category|colimits]] indexed by $K$.
Let $D$ be a small quasi-category. Then
* The $(\infty,1)$-category $Func(D,C)$ has all $K$-indexed colimits;
* A morphism $K^\triangleright \to Func(D,C)$ is a colimiting cocone precisely if for each object $d \in D$ the induced morphism $K^\triangleright \to C$ is a colimiting cocone.
\end{proposition}
This is ([Lurie, corollary 5.1.2.3](#Lurie)).
### Equivalences
\begin{proposition}
\label{EquivalencesDetectedOnObjects}
**(equivalences of $\infty$-functors detected on objects)**
\linebreak
A morphism $\alpha$ in $Func(D,C)$ (that is, a [[natural transformation]]) is an [[equivalence in an (infinity,1)-category|equivalence]] if and only if each component $\alpha_d$ is an equivalence in $C$.
\end{proposition}
This is due to [Joyal (2008), Chapter 5, Theorem C ([p. 125](https://ncatlab.org/nlab/files/JoyalTheoryOfQuasiCategories.pdf#page=125))](#Joyal08).
## Examples
* Between ordinary categories, it reproduces the ordinary [[category of functors]].
* Since the standard [[model structure on simplicial sets]] presents [[∞Grpd]]
$$
(sSet_{Quillen})^\circ \simeq \infty Grpd
$$
the [[model structure on simplicial presheaves]] (more precisely and more generally the [[model structure on sSet-enriched presheaves]]) on the [[opposite (∞,1)-category]] $C^{op}$ [[presentable (infinity,1)-category|presents]] the [[(∞,1)-category of (∞,1)-presheaves]] on $C$:
$$
N([C^{op},sSet_{Quillen}]^\circ) \simeq Func(C^{op},\infty Grpd)
= PSh_{(\infty,1)}(C)
\,.
$$
## Related concepts
* [[functor category]]
* [[2-functor 2-category]]
## References
The intrinsic definition is in section 1.2.7 of
* {#Lurie} [[Jacob Lurie]], _[[Higher Topos Theory]]_
The discussion of [[model category]] models is in A.3.4.
The theorem about equivalences (Prop. \ref{EquivalencesDetectedOnObjects}) is due to:
* {#Joyal08} [[André Joyal]], *[[The Theory of Quasi-Categories and its Applications]]*, lectures at *[Advanced Course on Simplicial Methods in Higher Categories](https://lists.lehigh.edu/pipermail/algtop-l/2007q4/000017.html)*, CRM (2008) [[pdf](http://mat.uab.cat/~kock/crm/hocat/advanced-course/Quadern45-2.pdf), [[JoyalTheoryOfQuasiCategories.pdf:file]]]
[[!redirects (∞,1)-category of (∞,1)-functors]]
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[[!redirects (infinity,1)-functor category]]
[[!redirects (infinity,1)-functor categories]]
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|
(infinity,1)-category of (infinity,1)-presheaves | https://ncatlab.org/nlab/source/%28infinity%2C1%29-category+of+%28infinity%2C1%29-presheaves |
+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### $(\infty,1)$-Category theory
+--{: .hide}
[[!include quasi-category theory contents]]
=--
#### $(\infty,1)$-Topos Theory
+--{: .hide}
[[!include (infinity,1)-topos - contents]]
=--
=--
=--
#Contents#
* table of contents
{:toc}
## Definition ##
For [[∞Grpd]] the [[(∞,1)-category]] of [[∞-groupoids]], and for $S$ a [[(∞,1)-category]] (or in fact any [[simplicial set]]), an **$(\infty,1)$-presheaf** on $S$ is an $(\infty,1)$-functor
$$
F : S^{op} \to \infty Grpd
\,.
$$
The **$(\infty,1)$-category of $(\infty,1)$-presheaves** is the [[(∞,1)-category of (∞,1)-functors]]
$$
PSh_{(\infty,1)}(S) := Func(S^{op}, \infinity Grpd)
\,.
$$
## Properties
### Models {#Models}
A [[model category|model]] for an $(\infty,1)$-presheaf categories is the [[model structure on simplicial presheaves]]. See also the discussion at [[models for ∞-stack (∞,1)-toposes]].
+-- {: .un_prop}
###### Proposition
For $C$ a [[simplicially enriched category]] with [[Kan complex]]es as hom-objects, write $[C^{op}, sSet_{Quillen}]_{proj}$ and $[C^{op}, sSet_{Quillen}]_{inj}$ for the projective or injective, respectively, global [[model structure on simplicial presheaves]]. Write $(-)^\circ$ for the full [[sSet]]-[[enriched category|enriched]] [[subcategory]] on fibrant-cofibrant objects, and $N(-)$ for the [[homotopy coherent nerve]] that sends a Kan-complex enriched category to a [[quasi-category]].
Then there is an [[equivalence of quasi-categories]]
$$
PSh(N(C))
\simeq
N ([C^{op}, sSet_{Quillen}]_{proj})^\circ
\,.
$$
Similarly for the injective model structure.
=--
+-- {: .proof}
###### Proof
This is a special case of the more general statement that the [[model structure on functors]] models an [[(∞,1)-category of (∞,1)-functors]]. See there for more details.
=--
Notice that the result in particular means that any $(\infty,1)$-presheaf -- an "$\infty$-[[pseudofunctor]]" -- may be _straightened_ or _rectified_ to a genuine [[sSet]]-[[enriched functor]], that respects horizontal compositions strictly.
### Limits and colimits
In an ordinary [[category of presheaves]], [[limit]]s and colimits are computed objectwise, as described at [[limits and colimits by example]]. The analogous statement is true for [[limit in a quasi-category|(∞,1)-limits]] and colimits in an $(\infty,1)$-category of $(\infty,1)$-presheaves.
This is a special case of the general existence of limits and colimits in an [[(∞,1)-category of (∞,1)-functors]]. See there for more details.
+-- {: .un_cor}
###### Corollary
For $C$ a small $(\infty,1)$-category, the $(\infty,1)$-category $PSh(C)$ admits all small limits and colimits.
=--
See around [[Higher Topos Theory|HTT, cor. 5.1.2.4]].
### As the free completion under colimits {#FreeColimCompletion}
An ordinary [[category of presheaves]] on a small category $C$ is the [[free cocompletion]] of $C$, the free completion under forming colimits.
The analogous result holds for $(\infty,1)$-category of $(\infty,1)$-presheaves.
+-- {: .un_lemma}
###### Lemma
Let $C$ be a small [[quasi-category]] and $j \colon S \to PSh(C)$ the [[(∞,1)-Yoneda embedding]].
The identity [[(∞,1)-functor]] $Id \,\colon\, PSh(C) \to PSh(C)$ is the left [[(∞,1)-Kan extension]] of $j$ along itself.
=--
This is [[Higher Topos Theory|HTT, lemma 5.1.5.3]].
For $D$ a [[quasi-category]] with all small [[limit in a quasi-category|colimits]], write $Func^L(PSh(C),D) \subset Func(PSh(C),D)$ for the full [[sub-quasi-category]] of the [[(∞,1)-category of (∞,1)-functors]] on those that [[preserved colimit|preserve]] small [[limit in a quasi-category|colimits]].
+-- {: .un_lemma}
###### Lemma
Pre-composition with the Yoneda embedding $j \colon C \to PSh(C)$ induces an [[equivalence of quasi-categories]]
$$
Func^L(PSh(C),D) \to Func(C,D)
\,.
$$
=--
This is [[Higher Topos Theory|HTT, theorem 5.1.5.6]].
In terms of the model given by the [[model structure on simplicial presheaves]], this is statement made in [Dugger (2001)](#Dugger2001), which gives that article its name.
+-- {: .un_def}
###### Definition
Let $A$ and $B$ be [[model categories]], $D$ a plain [[category]] and
$$
\array{
D &\overset{r}{\longrightarrow}& A
\\
& \mathllap{{}_{\gamma}}\searrow
\\
&& B
}
$$
two plain [[functors]]. Say that a **model-category theoretic factorization** of $\gamma$ through $A$ is
1. a [[Quillen adjunction]] $(L \dashv R) : A \stackrel{\overset{L}{\to}}{\underset{R}{\leftarrow}} B$
1. a [[natural transformation|natural]] weak equivalence $\eta : L \circ r \to \gamma$
$$
\array{
D &&\stackrel{r}{\to}&& A
\\
& \mathllap{{}_\gamma}\searrow
&{}^\eta\swArrow
& \swarrow_{\mathrlap{L}}
\\
&& B
}
\,.
$$
Let the [[category]] of such factorizations have morphisms $\big((L \dashv R), \eta \big) \to \big((L' \dashv R'\big), \eta' )$ given by [[natural transformation]]s $L \to L'$ such that for all all objects $d \in D$ the diagrams
$$
\array{
L\circ r(d) &&\longrightarrow&& L'\circ r(d)
\\
& {}_{\eta_{d}}\searrow && \swarrow_{\eta'_{d}}
\\
&&
\gamma()
}
$$
commutes.
=--
Notice that the [[(∞,1)-category]] presented by a [[model category]] -- at least by a [[combinatorial model category]] -- has all [[limit in a quasi-category|(∞,1)-categorical colimits]], and that the Quillen left adjoint functor $L$ presents, via its [[derived functor]], a left [[adjoint (∞,1)-functor]] that preserves $(\infty,1)$-categorical colimits. So the notion of factorization as above is really about factorizations through colimit-preserving $(\infty,1)$-functors into $(\infty,1)$-categories that have all colimits.
+-- {: .un_theorem}
###### Theorem
**(model category presentation of free $(\infty,1)$-cocompletion)**
For $C$ a [[small category]], the projective global [[model structure on simplicial presheaves]] $[C^{op}, sSet]_{proj}$ on $C$ is universal with respect to such factorizations of functors out of $C$:
every functor $C \to B$ to any [[model category]] $B$ has a factorization through $[C^{op}, sSet]_{proj}$ as above, and the category of such factorizations is [[contractible]].
=--
This is [Dugger (2001), theorem 1.1](#Dugger2001),
where the proof appears on page 30.
+-- {: .proof}
###### Proof
To produce the factorization $[C^{op},sSet] \to B$ given the functor $\gamma$, first
notice that the ordinary [[Yoneda extension]] $[C^{op},Set] \to B$ would be given by the left [[Kan extension]] given by the [[coend]] formula
$$
F \mapsto \int^{c \in C} \gamma(c) \cdot F(c)
\,,
$$
where the dot in the integrand is the [[copower|tensoring]] of cocomplete category $B$ over [[Set]]. To refine this to a [[Quillen adjunction|left Quillen functor]] $L : [C^{op},sSet] \to B$, _choose_ a [[cosimplicial object|cosimplicial]] [[resolution]]
$$
\Gamma \;\colon\; C \to [\Delta,B]
$$
of $\gamma$. Then set
$$
L \;\colon\;
F \mapsto
\int^{c \in C}
\int^{[n] \in \Delta}
\Gamma^n(c) \cdot F_n(c)
\,.
$$
The [[right adjoint]] $R \colon B \to [C^{op},sSet]$ of this functor is given by
$$
R(X) \colon c \mapsto Hom_B(\Gamma^\bullet(c), X)
\,.
$$
For $(L \dashv R) \colon [C^{op}, sSet]_{proj} \stackrel{\to}{\leftarrow} B$ to be a [[Quillen adjunction]], it is sufficient to check that $R$ preserves [[fibrations]] and [[acyclic fibrations]]. By definition of the projective model structure this means that for every (acyclic) fibration $b_1 \to b_2$ in $B$ we have for every object $c \in C$ that that
$$
Hom_C\big(\Gamma^\bullet(c), b_1 \to b_2\big)
$$
is an (acyclic) fibration of simplicial sets. But this is one of the standard properties of [[cosimplicial object|cosimplicial [[resolutions]].
Finally, to find the natural weak equivalence $\eta \colon L \circ j \simeq \gamma$, write $j : C \to [C^{op},sSet]$ for the [[Yoneda embedding]] and notice that by [[Yoneda reduction]] it follows that for $x \in C$ we have
$$
L(j(x))
\;=\;
\int^{c \in C} \int^{[n] \in \Delta}
\Gamma^n(c) \cdot C(c,x)
\;=\;
\Gamma^0(x)
$$
(where equality signs denote [[isomorphisms]]).
By the very definition of cosimplicial resolutions, there is a natural weak equivalence $\Gamma(x) \stackrel{\simeq}{\to}$. We can take this to be the component of $\eta$.
=--
+-- {: .un_cor}
###### Corollary
The [[(∞,1)-Yoneda embedding]] $j \colon C \to PSh(C)$ generates $PSh(C)$ under small colimits:
a full [[sub-quasi-category|(∞,1)-subcategory]] of $PSh(C)$ that contains all representables and is closed under forming $(\infty,1)$-colimits is already equivalent to $PSh(C)$.
=--
+-- {: .proof}
###### Proof
This is [[Higher Topos Theory|HTT, corollary 5.1.5.8]].
=--
### Functorality
{#Functoriality}
The two descriptions of $PSh(S)$ as the functor category $Func(S^{op}, \infty Grpd)$ and as the free cocompletion $S \to PSh(S)$ ([above](#FreeColimCompletion)) both extend to contravariant and covariant [[(infinity,1)-functors|functors]] on [[(infinity,1)Cat|$(\infty, 1)Cat$]].
These two constructions are related.
$Func(-, \infty Grpd) \colon (\infty, 1)Cat^{op} \to (\infty, 1)\widehat{Cat}$ takes values in the subcategory of [[presentable (∞,1)-categories]] and functors [[preserved colimit|preserving]] [[small limit|small]] [[(infinity,1)-limit|limits]] and [[(infinity,1)-colimit|colimits]], and thus having [[left adjoint|left]] and [[right adjoint|right]] [[adjoint (infinity,1)-functor|adjoints]].
Let $Pr^L(\infty,1)Cat$ be the (∞,1)category of [[presentable (∞,1)-categories]] and functors that are left adjoints,
and similarly for $Pr^R(\infty,1)Cat$.
+-- {: .un_theorem}
###### Theorem
The [[adjoint (infinity,1)-functor#category_of_adjunctions|local left adjoint]]
to $Func(-, \infty Grpd) \colon (\infty, 1)Cat^{op} \to Pr^R(\infty,1)Cat$
is naturally equivalent to the free cocompletion functor $P \colon (\infty, 1)Cat \to Pr^L(\infty,1)Cat$
by a natural equivalence whose components at a small (∞,1)-category $S$ are homotopic to the [[identity functor]] on $PSh(S)$.
=--
+-- {: .proof}
###### Proof
Consider the (∞,1)-category of possibly large (∞,1)-categories that have small colimits, and functors preserving small colimits. By [[Higher Topos Theory|HTT, 5.3.6.10]], the inclusion into the full (∞,1)-category of (∞,1)-categories has a left adjoint $P$, which is characterized by the natural equivalence
$$
Func^L\big(P(S), -\big) \simeq Func(S, -)
$$
given by pre-composition with the [[(infinity,1)-Yoneda embedding|Yoneda embedding]].
Let $Func(-, \infty Grpd)^{ladj}$ denote the [local left adjoint](adjoint+infinity1-functor#LadjAndRadj) to $Func(-, \infty Grpd)$.
Then for $C \in Pr^L(\infty,1)Cat$
and $S \in (\infty,1)Cat$, there are natural equivalences
$$
\begin{aligned}
Func^L\big(Func(S^{op}, \infty Grpd)^{ladj}, C\big)
&\simeq Func^R\big(C^{radj}, Func(S^{op}, \infty Grpd)\big)^{op}
\\ &\simeq Func\big(S^{op}, Func^R(C^{radj}, \infty Grpd)\big)^{op}
\\&\simeq Func\big(S, Func^L(\infty Grpd, C)\big)
\\&\simeq Func(S, C)
\end{aligned}
$$
The second equivalence holds because, for presentable $C$, the property of being in $Func^R$ is characterized by being accessible and preserving small limits, which can be determined pointwise.
The first and third equivalences are general properties of [local left adjoints](adjoint+infinity1-functor#LadjAndRadj) and opposite functor categories.
The last equivalence is because $Func^L(\infty Grpd, C) \simeq Func(1, C)$.
Since both constructions corepresent the same functor $Func(S, -)$, the yoneda lemma ensures there is a natural equivalence $ P(S) \simeq Func(S^{op}, \infty Grpd)^{ladj} $.
Now consider the chain of equivalences
$$
Func(S, C)
\leftarrow Func^L\big(Func(S^{op}, \infty Grpd)^{ladj}, C\big)
\to Func^L\big(P(S), C\big)
\to Func(S, C)
\mathrlap{\,,}
$$
where the first map is inverse to composition with the Yoneda embedding, the second map is the equivalence constructed previously, and the third map is composition with the Yoneda embedding.
Be careful to note that the first equivalence has not yet been shown to be natural in $S$.
The overall composite is essentially uniquely determined by an automorphism of $\alpha_S$ of $S$. It remains to be shown that $\alpha_S$ is an identity.
When $S = \Delta^n$, $S$ has no nontrivial automorphisms, so
$\alpha_{\Delta^n}$ is the identity.
For a functor $\phi : \Delta^n \to S$. The naturality of the above equivalence gives a commutative square
\begin{tikzcd}
P(\Delta^n) \arrow[r] \arrow[d] &
\mathrm{Func}(\Delta^{op}, \infty \mathrm{Grpd})^{ladj} \arrow[d]
\\ P(S) \arrow[r] &
\mathrm{Func}(S^{op}, \infty \mathrm{Grpd})^{ladj}
\end{tikzcd}
[[Higher Topos Theory|HTT, 5.2.6.3]] asserts $P(\phi)$ is left adjoint to $\Func(\phi, \infty Grpd)$, so the left and right arrows are homotopic.
We've already seen the top arrow is homotopic to the identity, and so we infer $\alpha_S \phi \simeq \phi$. Since the $\Delta^n$ generate $(\infty,1)Cat$, $\alpha_S$ is thus homotopic to the identity.
=--
Note that a natural automorphism of either functor extending $PSh$ would induce a natural automorphism of the identity functor on $(\infty,1)Cat$. By [[Higher Topos Theory|HTT, 5.2.9.1]], the space of such automorphisms is contractible.
### Relation to slicing
{#WithOvercategories}
The following analog of the corresponding result for
1-[[categories of presheaves]] holds for $(\infty,1)$-presheaves. See [[functors and comma categories]].
+-- {: .num_prop #SlicingCommutesWithFormingPresheaves}
###### Proposition
**(slicing commutes with passing to presheaves)**
Let $\mathcal{C}$ be a [[small (∞,1)-category]] and
$p \colon \mathcal{K} \to \mathcal{C}$ a [[diagram]].
Write $\mathcal{C}_{/p}$ and $PSh_\infty(\mathcal{C})/_{y p}$
for the corresponding [[over quasi-category|over categories]], where
$y \colon \mathcal{C} \to PSh_\infty(\mathcal{C})$ is the [[(∞,1)-Yoneda embedding]].
Then we have an [[equivalence of quasi-categories|equivalence of (∞,1)-categories]]
$$
PSh_\infty(\mathcal{C}_{/p})
\stackrel{\simeq}{\to}
PSh_\infty(\mathcal{C})_{/y p}
\,.
$$
=--
This appears as [[Higher Topos Theory|HTT, 5.1.6.12]].
## $(\infty,1)$-subcategories of $(\infty)$-presheaf categories
### Locally presentable $(\infty,1)$-categories
A [[reflective (∞,1)-subcategory]] of an $(\infty,1)$-category of $(\infty,1)$-presheaves is called a [[presentable (∞,1)-category]].
### $(\infty,1)$-Sheaf $(\infty,1)$-categories
If that [[left adjoint|left]] [[adjoint (∞,1)-functor]] to the embedding of the [[reflective (∞,1)-subcategory]] furthermore preserves finite [[limit]]s, then the subcategory is an [[(∞,1)-category of (∞,1)-sheaves]]: an [[(∞,1)-topos]]
## Related concepts
[[!include locally presentable categories - table]]
## References ##
* {#Dugger01} [[Dan Dugger]], *[[Universal homotopy theories]]*, Advances in Mathematics **164** (2001) 144-176 [[arXiv:math/0007070](https://arxiv.org/abs/math/0007070), [doi:10.1006/aima.2001.2014](https://doi.org/10.1006/aima.2001.2014)]
* {#Lurie09} [[Jacob Lurie]], section 5.1 in: _[[Higher Topos Theory]]_, Annals of Mathematics Studies **170**, Princeton University Press (2009) [[pup:8957](https://press.princeton.edu/titles/8957.html)]
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[[!redirects infinity-categories of infinity-presheaves]]
[[!redirects infinity1-category of infinity1-presheaves]]
[[!redirects infinity1-categories of infinity1-presheaves]]
|
(infinity,1)-category of (infinity,1)-sheaves | https://ncatlab.org/nlab/source/%28infinity%2C1%29-category+of+%28infinity%2C1%29-sheaves |
+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### $(\infty,1)$-Topos Theory
+--{: .hide}
[[!include (infinity,1)-topos - contents]]
=--
#### Locality and descent
+--{: .hide}
[[!include descent and locality - contents]]
=--
=--
=--
#Contents#
* table of contents
{:toc}
## Idea
The notion of $(\infty,1)$-category of $(\infty,1)$-sheaves is the generalization of the notion of [[category of sheaves]] from [[category theory]] to the [[higher category theory]] of [[(∞,1)-categories]].
## Definition
{#Definition}
+-- {: .num_defn #CategoryOfSheaves}
###### Definition
An **$(\infty,1)$-category of $(\infty,1)$-sheaves** is a [[reflective sub-(∞,1)-category]]
$$
Sh(C) \stackrel{\overset{L}{\leftarrow}}{\hookrightarrow}
PSh(C)
$$
of an [[(∞,1)-category of (∞,1)-presheaves]] such that the following equivalent conditions hold
* $L$ is a [[topological localization]];
* there is the structure of an [[(∞,1)-site]] on $C$ such that the objects of $Sh(C)$ are precisely those [[(∞,1)-presheaves]] $A$ that are [[local objects]] with respect to the _covering_ [[monomorphism in an (∞,1)-category|monomorphisms]] $p : U \to j(c)$ in $PSh(C)$ in that
\[
\label{DescentCondition}
A(c) \simeq PSh(j(c),A) \stackrel{PSh(p,A)}{\to} PSh(U,A)
\]
is an [[(∞,1)-equivalence]] in [[∞Grpd]].
=--
This is [[Higher Topos Theory|HTT, def. 6.2.2.6]].
An $(\infty,1)$-category of $(\infty,1)$-sheaves is an **[[(∞,1)-topos]]**.
+-- {: .num_remark}
###### Remark
Equivalence (eq:DescentCondition) is the _[[descent]]_ condition and the presheaves satisfying it are the **[[(∞,1)-sheaves]]** .
Typically $U$ here is the [[Cech nerve]]
$$
C(\{U_i\}) = \lim_{\to_{[n]}} U_{i_0, \cdots U_{i_n}}
$$
of a [[covering]] family $\{U_i \to c\}$
(where the colimit is the [[limit in a quasi-category|(∞,1)-categorical colimit]] or [[homotopy colimit]]) so that the above [[descent]] condition becomes
$$
A(c) \simeq PSh(\lim_\to U_\cdots, A)
\simeq
\lim_{\leftarrow} A(U_\cdots)
=
\lim_{\leftarrow}
\left(
\cdots
\stackrel{\to}{\stackrel{\to}{\to}}
\prod_{i,j} A(U_i) \times_{A(c)} A(U_j)
\stackrel{\to}{\to}\prod_i A(U_i)
\right)
\,.
$$
=--
+-- {: .num_remark}
###### Remark
Sometimes [[(∞,1)-sheaves]] are called **[[∞-stacks]]**, though sometimes the latter term is reserved for [[hypercomplete]] $(\infty,1)$-sheaves and at other times again it refers to [[(∞,2)-sheaves]].
The [[(n,1)-category|(n,1)-categorical]] counting is:
* [[sheaf]] = 0-stack = 0-[[truncated]] $(\infty,1)$-sheaf
* $(2,1)$-sheaf = [[stack]] = 1-truncated $(\infty,1)$-sheaf
* $(3,1)$-sheaf = 2-stack = 2-truncated $(\infty,1)$-sheaf
* etc.
* $(\infty,1)$-sheaf = [[∞-stack]] (or = [[hypercomplete]] $(\infty,1)$-sheaf).
=--
## Properties
### Localizations and Grothendieck topology {#LocAndTopology}
We reproduce the proof that the two characterization in def. \ref{CategoryOfSheaves} above are indeed equivalent.
+-- {: .num_prop}
###### Proposition
For $C$ an [[(∞,1)-site]], the full [[sub-(∞,1)-category]] of $PSh(C)$ on [[local objects]] with respect to the covering monomorphisms in $PSh(C)$ is indeed a [[topological localization]], and hence $Sh(C)$ is indeed an exact [[reflective sub-(∞,1)-category]] of $PSh(C)$ and hence an [[(∞,1)-topos]].
=--
This is [[Higher Topos Theory|HTT, Prop. 6.2.2.7]]
+-- {: .proof}
###### Proof
We must prove that the [[(∞,1)-sheafification]] functor $L \colon PSh(C)\to Sh(C)$ preserves [[finite (∞,1)-limits]]. To do so we give an explicit construction of $L$. Given a presheaf $F\in PSh(C)$, define a new presheaf $F^+$ by the formula
$$F^+(c)={\lim_{\rightarrow}}_U {\lim_{\leftarrow}}_{u\in U} F(u)$$
where the colimit is taken over all covering sieves $U$ of $c$; this is called the _[[plus construction]]_. It defines a functor $PSh(C)\to PSh(C)$ and there is an obvious morphism $F\to F^+$ natural in $F$.
It is clear that the construction $F\mapsto F^+$ preserves
[[finite (∞,1)-limits]], since [[filtered (∞,1)-colimits]] do, and it is easy to see that the map $F\to F^+$ becomes an [[equivalence in an (∞,1)-category|equivalence]] in $Sh(C)$. Given an [[ordinal]] $\alpha$, let $F^{(\alpha)}$ be the $\alpha$-iteration of the [[plus construction]] applied to the presheaf $F$. Then the functor $F\mapsto F^{(\alpha)}$ preserves finite limits and the canonical map $F\to F^{(\alpha)}$ becomes an equivalence in $Sh(C)$. In particular, if $F^{(\alpha)}$ is a sheaf, then $F^{(\alpha)}\simeq L(F)$. Thus, it suffices to show that there exists an ordinal $\alpha$ such that, for every $F\in PSh(C)$, $F^{(\alpha)}$ is a sheaf.
Fix $c\in C$ and a covering sieve $U$ of $C$. Given a presheaf $G\in PSh(C/c)$, we define an auxiliary presheaf $Match(U,G)\in PSh(C/c)$ by the formula
$$Match(U,G)(f: d\to c)={\lim_{\leftarrow}}_{u\in f^\ast U}G(u).$$
Restriction maps induce a morphism $\theta_G: G\to Match(U,G)$. Since we clearly have $G(u)\stackrel{\sim}{\to} Match(U,G)(u)$ for $u\in U$, the functor $Match(U,-)$ is _idempotent_ in the sense that $Match(U,\theta_G)$ and $\theta_{Match(U,G)}$ are (equivalent) equivalences.
By definition, $F\in PSh(C)$ is a sheaf if and only if $F(c)\stackrel{\sim}{\to} Match(U,F|_{C/c})(c)$ for every $c\in C$ and every covering sieve $U$ of $c$. Our goal is therefore to find an ordinal $\alpha$ (depending only on the (∞,1)-site $C$) such that, for every $F\in PSh(C)$, the map
$$F^{(\alpha)}(c) \to \Match(U,F^{(\alpha)}|_{C/c})(c)$$
is an equivalence.
The morphism $G\to G^+$ in $PSh(C/c)$ factors as
$$G\to Match(U,G)\to G^+.$$
Applying $Match(U,-)$ to this factorization, we get a commutative diagram
$$
\array{
G &\to& Match(U,G) &\to& G^+
\\
\downarrow^{\mathrlap{\theta_G}} && \downarrow^{\mathrlap{\theta_{Match(U,G)}}} && \downarrow^{\mathrlap{\theta_{G^+}}}
\\
Match(U,G) &\to& Match(U,Match(U,G)) &\to& Match(U,G^+)
}
$$
in which the map $\theta_{Match(U,G)}$ is an equivalence since $Match(U,-)$ is idempotent. By cofinality, the colimit of the maps $\theta_{G^{(n)}}$ as $n\to\infty$ is an equivalence. Applying this to $G=F|_{C/c}$, we get
$$ F^{(\omega)}(c)\stackrel{\sim}{\to} {\lim_{\rightarrow}}_{n\to\infty} Match(U,F^{(n)}|_{C/c})(c).$$
This _almost_ means that $F^{(\omega)}$ is a sheaf. The problem is that the filtered colimit on the right-hand side need not commute with the limit appearing in the definition of $Match(U,-)$, that is, the canonical map
$$ {\lim_{\rightarrow}}_{\alpha \lt \omega} Match(U,F^{(\alpha)}|_{C/c})(c) \to \Match(U,F^{(\omega)}|_{C/c})(c) $$
need not be an equivalence. To solve this problem, we choose a cardinal $\kappa$ such that for every $c\in C$ and every covering sieve $U$ of $c$, the functor $Match(U,(-)|_{C/c})(c):Psh(C)\to \infty Grpd$ preserves $\kappa$-filtered colimits. This is possible because $C$ is small and each of these functors, being the composition of the restriction functor $PSh(C)\to PSh(U)$ and the limit functor $PSh(U)\to\infty Grpd$, has a [[left adjoint|left]] [[adjoint (∞,1)-functor]] and is therefore accessible (see [[HTT|HTT Prop. 5.4.7.7]]). Then the above map with $\omega$ replaced by $\kappa$ is an equivalence. For every ordinal $\alpha\lt\kappa$, applying the above to $F^{(\alpha)}$ shows that
$$ F^{(\alpha+\omega)}(c)\stackrel{\sim}{\to} {\lim_{\rightarrow}}_{n\to\infty} Match(U,F^{(\alpha+n)}|_{C/c})(c),$$
Since $\kappa$ is a limit ordinal, we deduce that $F^{(\kappa)}$ is a sheaf by cofinality.
=--
And conversely:
+-- {: .num_prop}
###### Proposition
**(equivalence of site structures and categories of sheaves)**
For $C$ a [[small (∞,1)-category]], there is a bijective correspondence between structure of an [[(∞,1)-site]] on $C$ and equivalence classes of [[topological localization]]s of $PSh(C)$.
=--
This is [[Higher Topos Theory|HTT, prop. 6.2.2.9]].
+-- {: .num_lemma}
###### Lemma
For $C$ a small [[(∞,1)-site]] and $Sh(C) \stackrel{\overset{L}{\leftarrow}}{\hookrightarrow} PSh(C)$ the corressponding reflective inclusion of [[(∞,1)-sheaves]] into [[(∞,1)-presheaves]] on $C$ we have that the image under $L$ of a sub-$(\infty,1)$-functor $p : U \to j(c)$ of a [[representable functor|representable]] $j(c)$ is covering precisely if $L(p)$ is an equivalence.
=--
This is [[Higher Topos Theory|HTT, lemma 6.2.2.8]].
+-- {: .proof}
###### Proof of the Lemma
Since $Sh(C)$ is the reflectuive localization of $PSh(C)$ at covering monomorphisms, it is clear that if $p : U \to j(c)$ is covering, then $L(p)$ is an equivalence.
To see the converse, form the 0-truncation of $L i$ and conclude as for ordinary sheaves on the [[homotopy category of an (infinity,1)-category|homotopy catgegory]] of $C$.
...
=--
+-- {: .proof}
###### Proof of the Proposition
We have seen in (...) that for every structure of an $(\infty,1)$-site on $C$ we obtain a topological localization of the presheaf category, and that this is an injective map from site structures to equivalence classes of sheaf categories. It remains to show that it is also a surjective map, i.e. that every [[topological localization]] of $PSh(C)$ comes from the structure of an [[(∞,1)-site]] on $C$.
So consider $S \subset Mor(PSh(C))$ a strongly saturated class of morphisms which s topological (closed under pullbacks). Write $S_0 \subset S$ for the subcalss of those that are [[monomorphism in an (infinity,1)-category|monomorphisms]] of the form $U \to j(c)$.
Observe that then $S$ is indeed generated by (is the smallest strongly saturated class containing) $S_0$: since by the [[co-Yoneda lemma]] every object $X \in PSh(C)$ is a colimit $x \simeq {\lim_\to}_k j(\Xi_k)$ over representables. It follows that every monomorphism $f : Y \to X$ is a colimit (in $Func(\Delta[1],PSh(C))$) of those of the form $U \to j(c)$: for consider the pullback diagram
$$
\array{
f^* ({\lim_\to}_k \Xi_k) &\to& Y
\\
\downarrow^{\mathrlap{\simeq f}} && \downarrow^{\mathrlap{f}}
\\
{\lim_\to}_k \Xi_k &\stackrel{\simeq}{\to}& X
}
\;\;\;\;\;
\simeq
\;\;\;\;\;
\array{
({\lim_\to}_k f^* \Xi_k) &\to& Y
\\
\downarrow^{\mathrlap{\simeq f}} && \downarrow^{\mathrlap{f}}
\\
{\lim_\to}_k \Xi_k &\stackrel{\simeq}{\to}& X
}
$$
where the equivalence is due to the fact that we have [[universal colimits]] in $PSh(C)$. This realizes $f$ as a colimit over morphisms of the form $f^* j(\Xi_k) \to j(\Xi_k)$ that are each a pullback of a monomorphism. Since monomorphisms are stable under pullback (see [[monomorphism in an (∞,1)-category]] for details), all these component morphisms are themselves monomorphisms.
So every monomorphism in $S$ is generated from $S_0$, but by the assumption that $S$ is topological, it is itself entirely generated from monomorphisms, hence is generated from $S_0$.
So far this establishes that evry topological localization of $PSh(C)$ is a localization at a collection of sieves/ subfunctors $U \to j(c)$ of representables. It remains to show that this collection of subfunctors is indeed an Grothendieck topology and hence exhibits on $C$ the structure of an [[(∞,1)-site]]. We check the necessary three axioms:
1. _equivalences cover_ -- The equivalences $j(c) \stackrel{\simeq}{\to} j(c)$ belong to $S$ and are monomorphisms, hence belong to $S_0$.
1. _pullback of a cover is covering_ - Since monomorphisms are stable under pullback, we haave for every $p : U \to j(c)$ in $S$ and every $j(f) : j(d) \to j(c)$ that also the pullback $f^* p$
$$
\array{
f^* U &\to& U
\\
\downarrow^{\mathrlap{f^* p}} && \downarrow^{\mathrlap{p}}
\\
j(d) &\stackrel{f}{\to}& j(c)
}
$$
is a monomorphism and in $S$, hence in $S_0$.
1. _if restriction of a sieve to a cover is covering, then the sieve is covering_ -- Let $p : U \to j(c)$ be an arbitrary monomorphism and $f : X \to j(d)$ in $S_0$. Write $X \simeq {\lim_\to}_k \Xi_k$ and consider the pullback
$$
\array{
{\lim_\to}_k p^* \Xi_k &\stackrel{p^* f}{\to}& U
\\
\downarrow^{{\lim_\to}_k f_k^* p} && \downarrow^{\mathrlap{p}}
\\
{\lim_\to}_k \Xi_k &\stackrel{f}{\to}& j(c)
}
\,,
$$
where again we made use of the [[universal colimits]] in $PSh(C)$. Now notice that
1. $f$ is in $S$ by assumption;
1. $p^* f$ is by pullback stability of $S$;
1. each of the $f_k p$ is in $S$ by assumption, hence ${\lim_k f_k^* p}$
is by the fact that $S$ is strongly saturated.
1. so by commutativity $p \circ p^*f$ is in $S$.
1. finally by 2-out-of-3 this means that $p$ is in $S$.
=--
### Over paracompact topological spaces {#OverParacompactSpaces}
We discuss how $(\infty,1)$-sheaves over a [[paracompact topological space]] are equivalent to topological spaces [[overcategory|over]] $X$. This is the analogue of the 1-categorical statement that [[sheaves]] on $X$ are equivalent to [[etale space]]s over $X$: an etale space over $X$ is one whose [[fiber]]s are [[discrete topological space]], hence 0-[[truncated]] spaces. The [[n-category]] analogy has [[homotopy n-type]]s as fibers.
+-- {: .num_defn}
###### Definition
For $Y \to X$ a [[morphism]] in [[Top]], and $U \in Op(X)$ an [[open subset]] of $X$, write
$$
Sing_X(Y,U) := Hom_X(U \times \Delta^\bullet, X)
$$
for the [[simplicial set]] (in fact a [[Kan complex]]) of [[continuous map]]s
$$
\array{
U \times \Delta^k && \to && Y
\\
& \searrow && \swarrow
\\
&& X
}
$$
from $U$ times the topological $k$-[[simplex]] $\Delta^k$ into $Y$, that are [[section]]s of $Y \to X$.
=--
This is a relative version of the [[singular simplicial complex]] functor.
+-- {: .num_prop #OverTopModelStructure}
###### Proposition
Let $(X, \mathcal{B})$ be a [[topological space]] equipped with a [[base for the topology]] $\mathcal{B}$.
There is a [[model category]] structure on the [[over category]] $Top/X$ with weak equivalences and fibration precisely those morphisms $Y \to Z$ over $X$ such that for each $U \in \mathcal{B}$ the induced morphism $Sing_X(Y,U) \to Sing_X(Z,U)$ is a weak equivalence or fibration, respectively, in the standard [[model structure on simplicial sets]].
=--
This is [[Higher Topos Theory|HTT, prop 7.1.2.1]].
Write $(Top/X)^\circ$ for the [[(∞,1)-category]] [[locally presentable (∞,1)-category|presented]] by this model structure.
+-- {: .num_prop}
###### Proposition
Let $X$ be a [[paracompact topological space]] and write as usual $Sh_{(\infty,1)}(X) := Sh_{(\infty,1)}(Op(X))$ for the $(\infty,1)$-category of $(\infty,1)$-sheaves on the [[category of open subsets]] of $X$; equipped with the canonical structure of a [[site]].
Let $\mathcal{B}$ be the set of **$F_\sigma$-open subsets** of $X$. This are those [[open subset]]s that are countable unions of [[closed subset]]s, equivalently the 0-sets of [[continuous function]]s $X \to [0,1]$.
Let $Top/X^\circ$ be the corresponding $(\infty,1)$-categoty according to the [above proposition](#OverTopModelStructure). Then $Sing_X(-,-)$ constitutes an [[equivalence of (∞,1)-categories]]
$$
Top/X^\circ \simeq Sh_{(\infty,1)}(X)
\,.
$$
=--
This is [[Higher Topos Theory|HTT, corollary 7.1.4.4]].
### Difference to more general $(\infty,1)$-toposes {#DiffToOthers}
The [[(∞,1)-topos]]es that are $(\infty,1)$-categories of sheaves, i.e. that arise by [[topological localization]] from an [[(∞,1)-category of (∞,1)-presheaves]], enjoy a number of special properties over other classes of $(\infty,1)$-toposes, such as notably [[hypercomplete (∞,1)-topos]]es.
The following lists these properties. ([[Higher Topos Theory|HTT, section 6.5.4]].)
#### Universal property
The construction of [[(∞,1)-sheaf]] [[(∞,1)-topos]]es on a given [[locale]] is singled out over the construction of other kinds of $(\infty,1)$-toposes (such as [[hypercomplete (∞,1)-topos]]es) by the following universal property:
forming $(\infty,1)$-sheaves is, roughly, [[right adjoint]] to the functor $\tau_{\leq -1}$ that sends each $(\infty,1)$-topos to its underlying [[locale]] of [[subobject]]s of the [[terminal object]].
See [[Higher Topos Theory|HTT, item 1) of section 6.5.4]].
For $X,Y$ two $(\infty,1)$-toposes, write $Geom(X,Y) \subset Func(X,Y)$ for the full [[sub-(∞,1)-category]] of the [[(∞,1)-category of (∞,1)-functors]] on those that are [[geometric morphism]]s.
+-- {: .num_lemma}
###### Lemma
For $C$ an [[essentially small (∞,1)-category|small]] [[(n,1)-category]] with [[finite (∞,1)-limits]] and equipped with the structure of an [[(∞,1)-site]] and for $Y$ an [[(∞,1)-topos]], the [[truncated|truncation functor]]
$$
\tau_{\leq n-1} :
Geom(Y, Sh(C)) \to Geom(\tau_{\leq n-1} Y, \tau_{\leq n-1} Sh(C))
$$
is an [[equivalence in a quasi-category|equivalence]] (of [[(∞,1)-categories]]).
=--
This is [[Higher Topos Theory|HTT, lemma 6.4.5.6]].
See also [[n-localic (∞,1)-topos]].
#### Compact generation
+-- {: .num_prop}
###### Proposition
Let $X$ be a [[coherent topological space]] and let $Op(X)$ be its [[category of open subsets]] with the standard structure of an [[(∞,1)-site]].
Then $Sh_{(\infty,1)}(X) := Sh_{(\infty,1)}(Op(X))$ is _compactly generated_ in that it is generated by [[filtered colimit]]s of [[compact object]]s.
Moreover, the compact objects of $Sh_{(\infty,1)}(X)$ are those that are [[stalk]]wise compact objects in [[∞Grpd]] and [[locally constant ∞-stack|locally constant]] along a suitable [[stratification]] of $X$.
=--
This is [[Higher Topos Theory|HTT, prop. 6.5.4.4]].
This statement is false for the [[hypercompletion]] of $Sh_{(\infty,1)}(X)$, in general.
#### Nonabelian cohomology {#NonabelianCohomology}
For $X$ a [[topological space]], let
$$
(LConst \dashv \Gamma) : Sh_{(\infty,1)}(X)
\stackrel{\overset{LConst}{\leftarrow}}{\underset{\Gamma}{\to}}
$$
be the [[global section]]s terminal [[geometric morphism]].
For $A \in \infty Grpd$, the ([[nonabelian cohomology|nonabelian]]) [[cohomology]] of $X$ with coefficients in $A$ is usually defined in [[∞Grpd]] as
$$
H(X,A) := \pi_0 Func(Sing X, A)
\,,
$$
where $Sing X$ is the [[fundamental ∞-groupoid]] of $X$. On the other hand, if we send $A$ into $Sh_{(\infty,1)}(X)$ via $LConst$, the there is the _intrinsic_ [[cohomology]] of the $(\infty,1)$-topos $Sh_{(\infty,1)}(X)$
$$
H'(X,A) := \pi_0 Sh_{(\infty,1)}(X)(X, LConst A)
\,.
$$
Noticing that $X$ is in fact the [[terminal object]] of $Sh_{(\infty,1)}(X)$ and that $Sh_{(\infty,1)}(X)(X,-)$ is in fact that [[global section]]s functor, this is equivalently
$$
\cdots \simeq \pi_0 \Gamma LConst A
\,.
$$
+-- {: .num_theorem }
###### Theorem
If $X$ is a [[paracompact space]], then these two definitions of [[nonabelian cohomology]] of $X$ with [[constant ∞-stack|constant coefficients]] $A \in \infty Grpd$ agree:
$$
H(X,A) := \pi_0 \infty Grpd(Sing X,A) \simeq Sh_{(\infty,1)}(X)(X,LConst A)
\,.
$$
=--
This is [[Higher Topos Theory|HTT, theorem 7.1.0.1]].
## Models
The topological localizations of an [[(∞,1)-category of (∞,1)-presheaves]] are [[presentable (∞,1)-category|presented]] by the [[Bousfield localization of model categories|left Bousfield localization]] of the global [[model structure on simplicial presheaves]] at the set of [[Cech cover]]s.
The [[hypercompletion|hypercomplete]] $(\infty,1)$-sheaf toposes are [[presentable (infinity,1)-category|presented]] by the local Joyal-Jardine [[model structure on simplicial presheaves]].
Detailed discussion of this [[model category]] presentation is at
* [[models for infinity-stack (infinity,1)-toposes|models for (infinity,1)-category of (infinity,1)-sheaves]] .
## Related concepts
* [[geometric homotopy type theory]]
## References
The study of simplicial presheaves apparently goes back to
* K. Brown, [[BrownAHT|Abstract Homotopy Theory and Generalized Sheaf Cohomology]]
which considers locally [[Kan complex|Kan]] [[simplicial presheaf|simplicial presheaves]] as a [[category of fibrant objects]].
This was later conceived in terms of a [[model structure on simplicial presheaves]] and on simplicial sheaves by Joyal and Jardine. Toën summarizes the situation and emphasizes the interpretation in terms of [[∞-stacks]] living in $(\infty,1)$-categories for instance in
B. Toën, _Higher and derived stacks: a global overview_ ([arXiv](http://arxiv.org/abs/math/0604504)) .
This concerns mostly [[hypercomplete]] $(\infty,1)$-sheaves, though.
The full picture in terms of Grothendieck-[[(∞,1)-topos]]es of [[(∞,1)-sheaves]] is the topic of
* [[Jacob Lurie]], _[[Higher Topos Theory]]_ .
* localization $(\infty,1)$-functors ($(\infty,1)$-sheafification for the present purpose) are discussed in section 5.2.7;
* local objects ($(\infty,1)$-sheaves for the present purpose) and [[local isomorphism]]s are discussed in section 5.5.4;
* the definition of $(\infty,1)$-topoi of $(\infty,1)$-sheaves is then definition 6.1.0.4 in section 6.1;
* the characterization of $(\infty,1)$-sheaves in terms of [[descent]] is in section 6.1.3
* the relation between the [[model structure on simplicial presheaves|Brown?Joyal?Jardine model]] and the general story is discussed at length in section 6.5.4
An overview is in
* [[Denis-Charles Cisinski]], _Catégories supérieures et théorie des topos_, Séminaire Bourbaki, 21.3.2015, [pdf](http://www.math.univ-toulouse.fr/~dcisinsk/1097.pdf).
[[!redirects (infinity,1)-category of (infinity,1)-sheaves]]
[[!redirects (∞,1)-category of (∞,1)-sheaves]]
[[!redirects (infinity,1)-categories of (infinity,1)-sheaves]]
[[!redirects (∞,1)-categories of (∞,1)-sheaves]]
[[!redirects (∞,1)-sheaf (∞,1)-topos]]
[[!redirects (∞,1)-sheaf (∞,1)-toposes]]
[[!redirects (∞,1)-sheaf (∞,1)-topoi]]
[[!redirects ∞-stack (∞,1)-topos]]
[[!redirects ∞-stack (∞,1)-toposes]]
[[!redirects ∞-stack (∞,1)-topoi]]
[[!redirects infinity-stack (infinity,1)-topos]]
[[!redirects infinity-stack (infinity,1)-toposes]]
[[!redirects infinity-stack (infinity,1)-topoi]]
[[!redirects (∞,1)-sheaf (∞,1)-category]]
[[!redirects (∞,1)-sheaf (∞,1)-categories]]
[[!redirects (infinity,1)-sheaf (infinity,1)-category]]
[[!redirects (infinity,1)-sheaf (infinity,1)-categories]]
[[!redirects (∞,1)-topos of (∞,1)-sheaves]]
[[!redirects (∞,1)-toposes of (∞,1)-sheaves]]
[[!redirects (infinity,1)-topos of (infinity,1)-sheaves]]
[[!redirects (infinity,1)-toposes of (infinity,1)-sheaves]]
[[!redirects (∞,1)-sheaf ∞-topos]]
[[!redirects (∞,1)-sheaf ∞-toposes]]
|
(infinity,1)-category of cartesian sections | https://ncatlab.org/nlab/source/%28infinity%2C1%29-category+of+cartesian+sections |
# Idea #
Let $p : E \to X$ be an [[(∞,1)-functor]] of [[(infinity,1)-category|(∞,1)-categories]]. A _cartesian section_ of $p$ is a [[section]] $\sigma : X \to E$ that sends all 1-morphisms in $X$ to [[Cartesian morphism]]s in $E$.
#Definition#
...
#Remarks#
If $p : E \to X$ is a [[Cartesian fibration]] classified by an [[(∞,1)-functor]] $F : X \to (\infty,1)Cat^{op}$ then $\Gamma_X^{cart}(E)$ is equivalent to the [[limit in a quasi-category|limit]] of $F$
$$
\Gamma_X^{cart}(E) \simeq lim F
\,.
$$
See the discussion at [[limit in a quasi-category]] for details.
#References#
In corollary 3.3.3.2 of
* [[Jacob Lurie]], [[Higher Topos Theory]]
the collection of cartesian sections of $p : E \to X$ appears as $Maps_X^\flat(X^#, E^{cart})$.
Here
* the [[simplicial set]] $Maps_X^\flat(\cdots)$ is the simplicial set underlying the [[internal hom]] of [[marked simplicial set]]s over $X$ (beginning of section 3.1.3);
* $X^#$ is the [[simplicial set]] $X$ with _all_ cells marked (beginning of section 3.1)
* and $E^{cart}$ is $E$ with precisely all [[Cartesian morphism]]s marked (def. 3.1.1.9).
[[!redirects (∞,1)-category of cartesian sections]]
[[!redirects (infinity,1)-category of cartesian section]]
|
(infinity,1)-category of chain complexes | https://ncatlab.org/nlab/source/%28infinity%2C1%29-category+of+chain+complexes |
+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Homological algebra
+--{: .hide}
[[!include homological algebra - contents]]
=--
#### $(\infty,1)$-Category theory
+--{: .hide}
[[!include quasi-category theory contents]]
=--
=--
=--
#Contents#
* table of contents
{:toc}
## Idea
An [[(∞,1)-category]] presented by a [[model structure on chain complexes]].
Its [[homotopy category of an (∞,1)-category]] is the [[derived category]] of the underlying [[abelian category]].
## Related concepts
* [[stable (∞,1)-category]], [[triangulated category]]
* [[truncation of a chain complex]], [[suspension of a chain complex]]
[[!redirects (∞,1)-category of chain complexes]]
[[!redirects (∞,1)-categories of chain complexes]]
[[!redirects (infinity,1)-categories of chain complexes]]
[[!redirects (∞,1)-category of unbounded chain complexes]]
[[!redirects (infinity,1)-category of unbounded chain complexes]]
[[!redirects (∞,1)-categories of unbounded chain complexes]]
[[!redirects (infinity,1)-categories of unbounded chain complexes]]
[[!redirects stable (∞,1)-category of chain complexes]]
[[!redirects stable (∞,1)-categories of chain complexes]]
[[!redirects stable (infinity,1)-category of chain complexes]]
[[!redirects stable (infinity,1)-categories of chain complexes]]
|
(infinity,1)-category theory | https://ncatlab.org/nlab/source/%28infinity%2C1%29-category+theory |
+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### $(\infty,1)$-Category theory
+--{: .hide}
[[!include quasi-category theory contents]]
=--
#### Higher category theory
+--{: .hide}
[[!include higher category theory - contents]]
=--
=--
=--
#Contents#
* automatic table of contents goes here
{:toc}
## Idea
In the pattern of [[(n,r)-categories]] the notion of [[(∞,1)-category]] is special in that it is precisely the structure that
* admits [[homotopy theory]] in that it encodes the notion of higher coherent equivalence;
* admits [[category theory]] in terms of [[universal construction]]s: [[limit]]s, [[adjunction]]s, [[Grothendieck construction]]s, etc.
This entry surveys the _category theory of $(\infty,1)$-categories_ .
## Basic notions {#BasicCatNotions}
* [[nLab:homotopy category of an (∞,1)-category]]
* presentation by a [[nLab:category with weak equivalences]]:
[[nLab:Dwyer-Kan localization]]
* [[nLab:(∞,1)-functor]]
* [[nLab:(∞,1)-category of (∞,1)-categories]]
* the archetypical $(\infty,1)$-category: [[nLab:∞Grpd]]
* [[(∞,1)-category of (∞,1)-presheaves]]
* [[opposite quasi-category]]
* [[nLab:terminal object in a quasi-category]]
## Universal constructions {#UniversalConstructions}
The [[universal construction]]s of [[category theory]] generalize, with unique existence of universal morphisms replaced by the requirement of a _[[contractible space]]_ of universal morphisms.
* [[limit in a quasi-category]]
* [[adjoint (∞,1)-functor]]
* [[(∞,1)-Grothendieck construction]]
## Related concepts
* [[(∞,1)-category of (∞,1)-categories]]
* [[homotopy 2-category of (∞,1)-categories]]
* [[synthetic (infinity,1)-category theory]]
[[!include table of category theories]]
## References
See the references at _[[(infinity,1)-category|$(\infty,1)$-category]]_.
[[!redirects (∞,1)-category theory]] |
(infinity,1)-functor | https://ncatlab.org/nlab/source/%28infinity%2C1%29-functor |
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#### $(\infty,1)$-Category theory
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[[!include quasi-category theory contents]]
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#Contents#
* table of contents
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## Idea
An _$(\infty,1)$-functor_ is a [[homomorphism]] between [[(∞,1)-categories]]. It generalizes
* the notion of [[functor]] between [[categories]];
* the notion of [[pseudofunctor]] from a [[category]] to a [[(2,1)-category]];
* the notion of [[2-functor]] between [[(2,1)-categories]];
* the notion of [[n-functor]] between [[(n,r)-category|(n,1)-categories]].
An $(\infty,1)$-functor is _functorial_ (respects [[composition]]) only up to [[coherent]] higher [[homotopies]]. It may be thought of as a **[[homotopy coherent functor]]** or **strongly homotopy functor**.
The collection of all $(\infty,1)$-functors between two $(\infty,1)$-categories form an [[(∞,1)-category of (∞,1)-functors]].
## Definition
The details of the definition depend on the model chosen for
[[(∞,1)-categories]].
1. [[quasi-category]]
1. [[simplicially enriched category]]
1. [[Segal category]]
1. [[complete Segal space]]
### In terms of quasi-categories
+-- {: .un_defn}
###### Definition
For $C$ and $D$ [[quasi-category|quasi-categories]], an **$(\infty,1)$-functor** $F : C \to D$ is simply a morphism of the underlying [[simplicial sets]].
A **natural transformation** $\eta : F \to G$ between two such $(\infty,1)$-functors is a simplicial [[homotopy]]
$$
\array{
C
\\
{}^{\mathllap{i_0}}\downarrow & \searrow^{\mathrlap{F}}
\\
C \times \Delta[1] &\stackrel{\eta}{\to}& D
\\
{}^{\mathllap{i_1}}\uparrow & \nearrow_{G}
\\
C
}
\,.
$$
A **[[modification]]** $\rho$ between natural transformations is an order 2 simplicial homotopy
$$
\rho : C \times \Delta[2] \to D
\,.
$$
Generally a **$k$-[[transfor]]** $\phi$ of $(\infty,1)$-functors is a simplicial homotopy of order $k$ between the corresponding quasi-categories
$$
\phi : C \times \Delta[k] \to D
\,.
$$
In total, the [[(∞,1)-category of (∞,1)-functors]] between given quasi-categories $C$ and $D$ is the simplicial function complex
$$
(\infty,1)Cat(C,D) :=
sSet(C,D)
:=
\int^{k \in \Delta} \Delta[k] \cdot Hom_{sSet}(C \times \Delta[k], D)
$$
as computed by the canonical [[sSet]]-[[enriched category|enrichment]] of $sSet$ itself.
=--
This serves to define the [[(∞,1)-category of (∞,1)-functors]].
## Examples
### $\infty$-Pseudo-functors / homotopy presheaves
Let $C$ be an ordinary [[category]]. The above definition in particular serves to generalize the notion of a [[pseudofunctor]] (functor up to homotopy)
$$
F : C^{op} \to Grpd
$$
with values in the [[2-category]] [[Grpd]] as it appears in the theory of [[stack]]s/[[2-sheaves]]:
let $KanCplx \subset sSet$ be the [[full subcategory]] of [[sSet]] on the [[Kan complex]]es. This is naturally a [[simplicially enriched category]]. Write $N(\mathbf{KanCplx})$ for the [[homotopy coherent nerve]] of this simplicially enriched category. This is the [[quasi-category]]-incarnaton of [[∞Grpd]].
Write $N(C^{op})$ for the ordinary [[nerve]] of the ordinary category $C^{op}$ (passing to the [[opposite category]] is just a convention here, with no effect on the substance of the statement). Then an **$\infty$-pseudofunctor** or **[[(∞,1)-presheaf]]** or **homotopy presheaf** on $C$ is a morphism of simplicial sets
$$
F : N(C^{op}) \to N(\mathbf{KanCplx})
\,.
$$
One sees easily in low degrees that this does look like the a [[pseudofunctor]] there:
1. the 1-cells of $N(C)$ are just the [[morphism]]s in $C$, so that on 1-cells we have that $F$ is an assignment
$$
F : (x \stackrel{f}{\leftarrow} y) \mapsto (F(x) \stackrel{F(f)}{\to} F(y)
$$
of morphisms in $C$ to morphisms in $KanCplx$, as befits a functor;
1. the 2-cells of $N(C)$ are pairs of composable morphisms, so that on 2-cells we have that $F$ is an assignment
$$
F :
\left(
\array{
&& y
\\
& {}^{\mathllap{g}}\swarrow
& & \nwarrow^{\mathrlap{f}}
\\
x &&\stackrel{g \circ f}{\leftarrow}&& z
}
\right)
\;\;
\mapsto
\;\;
\left(
\array{
&& F(y)
\\
& {}^{\mathllap{F(g)}}\nearrow
& \Downarrow^{\mathrlap{F(f,g)}} &
\searrow^{\mathrlap{F(f)}}
\\
F(x) &&\stackrel{F(g \circ f)}{\rightarrow}&& F(z)
}
\right)
$$
which means that $F$ does not necessarily respect the [[composition]] of morphisms, but instead does introduce [[homotopies]] $F(f,g)$ for very pairs of composable morphisms, which measure how $F(g)\circ F(f)$ differs from $F(g \circ f)$. These are precisely the homotopies that one sees also in an ordinary [[pseudofunctor]]. But for our $(\infty,1)$-functor there are now also higher and higher homotopies:
1. the 3-cells of $N(C)$ are triples of composable morphisms $(f,g,h)$ in $C$. They are sent by $F$ to a tetrahedron that consists of a homotopy-of-homotopies from the $F(f,g) \cdot F( h , g\circ f )$ to $F(g, h) \cdot F(f , h \circ g)$;
1. and so on.
For more see [[(∞,1)-presheaf]].
## Properties
It turns out that every $(\infty,1)$-functor $C \to \infty Grpd$ can be **rectified** to an _ordinary ([[sSet]]-[[enriched functor|enriched]]) functor_ with values in [[Kan complex]]es.
+-- {: .un_theorem}
###### Theorem
For $C = N(\mathbf{C})$ a quasi-category given as the [[homotopy coherent nerve]] of a Kan-complex enriched category $\mathbf{C}$ (which may for instance be just an ordinary 1-category), write
$$
[\mathbf{C}^{op}, \mathbf{sSet}]
$$
for the [[sSet]]-[[enriched category]] of _ordinary_ ($sSet$-[[enriched functor|enriched]]) functors (respecting composition strictly).
Then: every $(\infty,1)$-functor $N(\mathbf{C}^{op}) \to \infty Grpd$ is equivalent to a strictly composition respecting functor of this sort. Precisely: write $[\mathbf{C}^{op}, \mathbf{sSet}]^\circ$ for the full $\mathbf{sSet}$-enriched subcategory on those strict functors that are fibrant and cofibrant in a [[model structure on simplicial presheaves]] on $\mathbf{C}$. Then we have an [[equivalence of quasi-categories|equivalence]] of [[(∞,1)-categories]]
$$
Hom_{(\infty,1)Cat}(N(\mathbf{C}^{op}), \infty Grpd)
\simeq
N([\mathbf{C}^{op}, \mathbf{sSet}]^\circ)
\,.
$$
=--
More on this is at [[(∞,1)-category of (∞,1)-presheaves]].
## Related concepts
* [[function]]
* [[functor]]
* [[2-functor]] / [[pseudofunctor]] / [[(2,1)-functor]]
* [[monoidal functor]]
* [[n-functor]]
* **(∞,1)-functor**
* [[monoidal (∞,1)-functor]]
* [[adjoint (∞,1)-functor]], [[(∞,1)-monad]]
* [[(∞,n)-functor]]
[[!include properties of functors -- contents]]
## References
section 1.2.7 in
* [[Jacob Lurie]], _[[Higher Topos Theory]]_
discusses morphisms of [[quasi-category|quasi-categories]].
[[!redirects (infinity,1)-functors]]
[[!redirects (∞,1)-functor]]
[[!redirects (∞,1)-functors]]
|
(infinity,1)-geometric morphism | https://ncatlab.org/nlab/source/%28infinity%2C1%29-geometric+morphism |
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#### $(\infty,1)$-Topos Theory
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[[!include (infinity,1)-topos - contents]]
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#Contents#
* table of contents
{:toc}
## Idea
By [[categorification]] of the notion of [[geometric morphism]], an _$(\infty,1)$-geometric morphism_ is a pair of [[adjoint (∞,1)-functor]]s between [[(∞,1)-topos]]es where the leftadjoint is left-exact.
## Definition
For $\mathbf{H}$ and $\mathbf{K}$ two [[(∞,1)-topos]]es, a **$(\infty,1)$-geometric morphism** $f : \mathbf{H} \to \mathbf{K}$ is
* an [[(∞,1)-functor]] $f_* : \mathbf{H} \to \mathbf{K}$
(called the [[direct image]] of the geometric morphism)
* which has a left [[adjoint (∞,1)-functor]] $\mathbf{H} \leftarrow \mathbf{K} : f^*$;
(called the [[inverse image]] of the geometric morphism)
* such that $f^*$ is a left [[exact (∞,1)-functor]].
The (non-full) [[sub-(∞,1)-category]] of [[(∞,1)Cat]] on [[(∞,1)-toposes]] and $(\infty,1)$-geometric morphisms between them is [[(∞,1)Toposes]].
## Examples
* [[terminal geometric morphism]]
* [[locally infinity-connected (infinity,1)-topos|locally $\infty$-connected $\infty$-topos]]
* [[cohesive (infinity,1)-topos|cohesive $\infty$-topos]]
## Related concepts
* [[geometric morphism]]
* [[terminal geometric morphism]]
## References
section 6.3.1 in
* [[Jacob Lurie]], _[[Higher Topos Theory]]_
[[!redirects (∞,1)-geometric morphism]]
[[!redirects (∞,1)-geometric morphisms]]
[[!redirects (infinity,1)-geometric morphisms]]
|
(infinity,1)-Grothendieck construction | https://ncatlab.org/nlab/source/%28infinity%2C1%29-Grothendieck+construction |
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#### $(\infty,1)$-Category theory
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[[!include quasi-category theory contents]]
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#Contents#
* table of contents
{:toc}
## Idea
The **$(\infty,1)$-Grothendieck construction** is the generalization of the [[Grothendieck construction]] from [[category theory]] to [[(∞,1)-category theory]].
Recall that the [[1-category]]-theoretic [[Grothendieck construction]] establishes [[equivalence of categories|equivalences of categories]],
$$
Fib(C) \simeq 2Func(C^{op}, Cat)
{\phantom{AAAA}}
and
{\phantom{AAAA}}
Fib_{Grpd}(C) \simeq 2Func(C^{op}, Grpd)
$$
between (a) [[fibered category|fibered categories]] (or just [[categories fibered in groupoids]]) and (b) [[pseudofunctors]] to [[Cat]] (or just to [[Grpd]]).
Analogously,
the full Grothendieck construction for [[(∞,1)-categories]] constitutes an [[equivalence of (∞,1)-categories]]
$$
CartFib(C) \,\simeq\, \infty Func\big( C^{op}, (\infty,1)Cat \big)
$$
between (a) [[Cartesian fibrations]] [[fibrations of quasi-categories|of quasi-categories]] and (b) [[indexed (∞,1)-categories]], that is, [[(∞,1)-functors]] to [[(∞,1)Cat]];
while its restriction to the Grothendieck construction for [[∞-groupoids]] constitutes an [[equivalence of (∞,1)-categories]]
$$
RFib(C) \,\simeq\, \infty Func\big(C^{op}, \infty Grpd\big)
$$
between [[right fibrations]] [[fibrations of quasi-categories|of quasi-categories]] and [[(∞,1)-functors]] to [[∞Grpd]].
This correspondence may be [[model category|modeled]]:
* in the case of $\infty$-groupoids by a [[Quillen equivalence]] between the [[model structure for right fibrations]] and the projective [[global model structure on simplicial presheaves]],
* in the case of $(\infty,1)$-categories by a Quillen equivalence between the [[model structure for Cartesian fibrations]] and the [[global model structure on functors]] with values in the [[model structure on marked simplicial over-sets|model structure on marked simplicial sets]].
## For fibrations in $\infty$-groupoids
The generalization of a [[category fibered in groupoids]] to [[quasi-category]] theory is a [[right fibration|right]] [[fibrations of quasi-categories|fibration of quasi-categories]].
\begin{prop}\label{Infinity0GrothendieckConstruction}
**($(\infty,0)$-Grothendieck construction)**
Let $\mathcal{C}$ be a small [[(∞,1)-category]]. The operation of [[homotopy pullback]] of the [[universal fibration of infinity-groupoids|universal fibration of $\infty$-groupoids]] constitutes an [[equivalence of (∞,1)-categories]]
$$
Func\big(\mathcal{C}^{op}, \infty Grpd\big)
\;\simeq\;
RFib(C)
$$
* from the [[(∞,1)-category of (∞,1)-functors]] from the [[opposite category]] $\mathcal{C}^{op}$ to [[∞Grpd]], i.e. the [[(∞,1)-category of (∞,1)-presheaves]] on $C$;
* to the [[right fibration|(∞,1)-category of right fibrations]] $RFib(C)$ -- incarnated as the full [[SSet]]-[[subcategory]] of the [[overcategory]] $SSet/C$ on [[right fibrations]];
\end{prop}
Cisinski19
In the next section we discuss how this statement is presented in terms of [[model categories]].
\begin{prop}\label{GrpdFibsOverGrpds}
**($(\infty,0)$-fibrations over an $\infty$-groupoid)**
\linebreak
If $C$ itself be an [[infinity-groupoid|$\infty$-groupoid]], then $RFib(C) \simeq \infty Grpd/C$ is [[equivalence of (infinity,1)-categories|equivalently]] the [[slice (infinity,1)-category|slice $\infty$-category]] of [[∞Grpd]] over $C$, and hence the above Prop. \ref{Infinity0GrothendieckConstruction} reduces to
$$
\infty Grpd/C \;\simeq\; Func\big(C^{op}, \infty Grpd\big)
\,.
$$
\end{prop}
\begin{proof}
By the fact that there is the [[classical model structure on simplicial sets]] we have that every morphism of $\infty$-groupoids $X \to C$ factors as
$$
\array{
X &&\stackrel{\simeq}{\to}&& \hat X
\\
& \searrow && \swarrow_{\mathrlap{fib}}
\\
&& C
}
\,,
$$
where the top morphism is an equivalence and the right morphism a [[Kan fibration]]. Moreover, as discussed at [[right fibration]], over an $\infty$-groupoid the notions of left/right fibrations and Kan fibrations coincide. This shows that the full [[sub-(∞,1)-category]] of $\infty Grpd/X$ on the right fibrations is equivalent to all of $\infty Grpd/X$.
\end{proof}
### Model category presentation {#GrpdModelCatVersion}
We discuss a [[presentable (∞,1)-category|presentation]] of the $(\infty,0)$-Grothendieck construction by a [[simplicial Quillen adjunction]] between [[simplicial model categories]]. ([[Higher Topos Theory|HTT, section 2.2.1]]).
+-- {: .un_def #UnmarkedStraighteningFunctor}
###### Definition
**(extracting a simplicial presheaf from a fibration)**
Let
* $S$ be a [[simplicial set]], $\tau_hc(S)$ the corresponding [[SSet-category]] (under the [[left adjoint]] $\tau_{hc} : SSet \to SSet Cat$ of the [[homotopy coherent nerve]], denoted $\mathfrak{C}$ in [[Higher Topos Theory|HTT]]);
* $C$ an [[SSet-category]];
* $\phi : \tau_{hc}(S) \to C$ a morphism of [[SSet-categories]].
> In particular we will be interested in the case that $\phi$ is the identity, or at least an equivalence, identifying $C$ with $\tau_{hc}(S)$.
For any object $(p : X\to S)$ in $sSet/S$ consider the [[sSet-category]] $K(\phi,p)$ obtained as the (ordinary) [[pushout]] in [[SSet Cat]]
$$
\array{
\tau_{hc}(X) &\stackrel{}{\to}& \tau_{hc}(X^{\triangleright})
\\
{}^{\mathllap{\phi(p)}}\downarrow && \downarrow
\\
C &\to& K(\phi,p)
}
\,,
$$
where $X^{\triangleright} = X \star \{v\}$ is the [[join of simplicial sets]] of $X$ with a single vertex $v$.
Using this construction, define a functor, the **[[straightening functor]]**,
$$
St_\phi : sSet/S \to [C^{op}, sSet]
$$
from the [[overcategory]] of [[sSet]] over $S$ to the [[enriched functor category]] of [[sSet]]-[[enriched functor]]s from $C^{op}$ to $sSet$ by defining it on objects $(p : X \to S)$ to act as
$$
St_\phi(X) := K(\phi,p)(-,v) : C^{op} \to SSet
\,.
$$
=--
+-- {: .un_example}
###### Example
The straightening functor effectively computes the fibers of a [[Cartesian fibration]] $(p : X \to C)$ over every point $x \in C$. As an illustration for how this is expressed in terms of morphisms in that pushout, consider the simple situation where
* $C = *$ only has a single point;
* $X = \left\{ a \to b \;\;\; c\right\}$ is a category with three objects, two of them connected by a morphism
* $p \colon X \to C$ is the only possible functor, sending everything to the point.
Then
* $X^{\triangleright} =
\left\{
\array{
a &\to& b && c
\\
& \searrow \Leftarrow& \downarrow & \swarrow
\\
&& v
}
\right\}
$
and
* $X^{\triangleright} \coprod_{X} C =
\left\{
\array{
&& \bullet
\\
& \swarrow & \downarrow & \searrow
\\
\downarrow& \Leftarrow & \downarrow
\\
& \searrow & \downarrow & \swarrow
\\
&& v
}
\right\}
$
Therefore the category of morphisms in this pushout from $*$ to $v$ is indeed again the category $\{a \to b \;\;\; c\}$.
More on this is at [[Grothendieck construction]] in the section of adjoints to the Grothendieck construction.
=--
+-- {: .un_prop}
###### Proposition
With the definitions [as above](#UnmarkedStraighteningFunctor), let $\pi : C \to C'$ be an [[sSet]]-[[enriched functor]] between [[sSet-categories]]. Write
$$
\pi_! : [C^{op}, sSet] \to [{C'}^{op}, sSet]
$$
for the left [[sSet]]-[[Kan extension]] along $\pi$.
There is a [[natural isomorphism]] of the straightening functor for the composite $\pi \circ \phi$ and the original straightening functor for $\phi$ followed by left [[Kan extension]] along $\pi$:
$$
St_{\pi \circ \phi} \simeq \pi_! \circ St_\phi
\,.
$$
=--
This is [[Higher Topos Theory|HTT, prop. 2.2.1.1.]]. The following proof has kindly been spelled out by [[Harry Gindi]].
+-- {: .proof}
###### Proof
We unwind what the [[sSet-categories]] with a single object adjoined to them look like:
let
$$
F : C^{op} \to sSet
$$
be an [[sSet]]-[[enriched functor]]. Define from this a new [[sSet-category]] $C_F$ by setting
* $Obj(C_F) = Obj(C) \coprod \{\nu\}$
* $C_F(c,d) = \left\{
\array{
C(c,d) & for c,d \in Obj(C)
\\
F(c) & for c \in Obj(c) and d = \nu
\\
\emptyset & for c = \nu and d \in Obj(C)
\\
* & for c = d = \nu
}
\right.$
The composition operation is that induced from the composition in $C$ and the enriched functoriality of $F$.
Observe that the [[sSet-category]]
$K(\phi,p)$ appearing in the [definition of the straightening functor](#UnmarkedStraighteningFunctor) is
$$
K(\phi,p) \simeq C_{St_\phi(X)}
$$
(because $K(\phi,p)$ is $C$ with a single object $\nu$ and some morphisms to $\nu$ adjoined, such that there are no non-degenerate morphisms originating at $\nu$, we have that $K(\phi,p)$ is of form $C_F$ for some $F$; and $St_\phi(X)$ is that $F$ by definition).
To prove the proposition, we need to compute the pushout
$$
\array{
\tau_{hc}(X) &\to& \tau_{hc}(X^{\triangleright})
\\
\downarrow && \downarrow
\\
C &\to& K(\phi,p) = C_{St_\phi(X)}
\\
{}^{\mathllap{\pi}}\downarrow && \downarrow
\\
C' &\to& Q
}
$$
and show that indeed $Q \simeq C'_{\pi_! St_\phi(X)}$.
Using the pasting law for [[pushout]]s (see [[pullback]]) we just have to compute the lower square pushout. Here the statement is a special case of the following statement: for every [[sSet-category]] of the form $C_F$, the pushout of the canonical inclusion $C\to C_F$ along any $sSet$-functor $\pi \colon C \to C'$ is $C'_{\pi_! F}$.
This follows by inspection of what a cocone
$$
\array{
C &\stackrel{\iota}{\to}& C_F
\\
{}^{\mathllap{\pi}}\downarrow && \downarrow^{\mathrlap{d}}
\\
C' &\underset{r}{\to}& Q
}
$$
is like: by the nature of $C_F$ the functor $d$ is characterized by a functor $d|_C : C \to Q$, an object $d(\nu) \in Q$ together with a [[natural transformation]]
$$
F(c) \to Q(d|_C(c), d(\nu))
$$
being the component $F_{c,\nu} : C_F(c,\nu) \to Q(d(c), d(\nu))$ of the $sSet$-functor.
We may write this natural transformation as
$$
F \to (d|_C)^* Q(-,d(\nu)) = \iota^* d^* \nu Q(-,d(\nu))
\,,
$$
where $d^*$ etc. means precomposition with the functor $d$.
By commutativity of the diagram this is
$$
\cdots \simeq \pi^* r^* Q(-,d(\nu))
\,.
$$
Now by the definition of left [[Kan extension]] $\pi_!$ as the [[left adjoint]] to prescomposition with a functor, this is bijectively a transformation
$$
\eta : \pi_! F \to r^* Q(-,d(\nu))
\,.
$$
Using this we see that we may find a universal cocone by setting
$Q := C'_{\pi_! F}$ with $r : C' \to Q$ the canonical inclusion and
$C_{F} \to C'_{\pi_! F}$ given by $\pi$ on the restriction to $C$ and by the [[unit of an adjunction|unit]] $F \to \pi^* \pi_! F$ on $C_F(c,\nu)$. For this the [[adjunct]] transformation $\eta$ is the identity, which makes
this universal among all cocones.
=--
+-- {: .un_prop}
###### Proposition
This functor has a [[right adjoint]]
$$
Un_\phi : [C^{op}, sSet] \to sSet/S
\,,
$$
that takes a [[simplicial presheaf]] on $C$ to a simplicial set over $S$ -- this is the **unstraightening functor**.
=--
+-- {: .proof}
###### Proof
One checks that $St_\phi$ preserves [[colimit]]s. The claim then
follows with the [[adjoint functor theorem]].
=--
\begin{prop}
**(presentation of the $(\infty,0)$-Grothendieck construction)**
\linebreak
The straightening and the unstraightening functor constitute a [[Quillen adjunction]]
$$
(St_\phi \dashv Un_\phi)
\,\colon\,
sSet/S
\underoverset
{\underset{St_\phi}{\longrightarrow}}
{\overset{Un_{\phi}}{\longleftarrow}}
{}
[C^{op}, sSet]
$$
between the [[model structure for right fibrations]] and the global projective [[model structure on simplicial presheaves]] on $S$.
If $\phi$ is a weak equivalence in the [[model structure on simplicial categories]] then this Quillen adjunction is a [[Quillen equivalence]].
\end{prop}
This is [[Higher Topos Theory|HTT, theorem 2.2.1.2]].
This models the Grothendieck construction for [[∞-groupoids]] in the following way:
* the [[presentable (∞,1)-category|(∞,1)-category presented by]] $sSet/S$ is $RFib(S)$
([[Higher Topos Theory|HTT, lemma 2.2.3.9]])
* the [[presentable (∞,1)-category|(∞,1)-category presented by]]
the global [[model structure on simplicial presheaves]]
$[C^{op}, SSet]$ is [[(∞,1)-category of (∞,1)-presheaves]]
$PSh_{(\infty,1)}(N_{hc}(C))$
Hence the unstraightening functor is what models the [[Grothendieck construction]] proper, in the sense of a construction that generalizes the construction of a [[fibered category]] from a [[pseudofunctor]].
## For general fibered $(\infty,1)$-categories
The generalization of a [[fibered category]] to [[quasi-category]] theory is a [[Cartesian fibration|Cartesian]] [[fibrations of quasi-categories|fibration of quasi-categories]].
+-- {: .un_theorem}
###### Theorem
**($(\infty,1)$-Grothendieck construction)**
Let $C$ be a small [[(∞,1)-category]]. There is an equivalence
$$
Cart(C) \simeq Func(C^{op}, (\infty,1) Cat)
$$
* on the left we have the $(\infty,1)$-category of [[Cartesian fibration]]s over $C$ with small fibers and cartesian functors between them -- incarnated as a [[sSet]]-[[subcategory]]
of the [[overcategory]] $sSet/C$ on [[Cartesian fibration]]s;
* and on the right the [[(∞,1)-category of (∞,1)-functors]] from $C^{op}$ to the [[(∞,1)-category of (∞,1)-categories]].
Furthermore, this is equivalence is natural for $C$, where $Cart(-)$ acts by taking pullbacks and $Func(-^{op}, (\infty,1) Cat)$ acts by composition.
=--
A reference for the naturality in small $C$ is corollary A.32 of [Gepner-Haugseng-Nikolaus 15](#GepnerHaugsengNikolaus15). The dual statement is made in remark 1.13 of [Mazel-Gee](#MazelGee).
In the next section we discuss how this statement is presented in terms of [[model categories]].
### Model category presentation {#ModCatCart}
Regard the [[(∞,1)-category]] $C$ in its incarnation as a [[simplicially enriched category]].
Let $S$ be a [[simplicial set]], $\tau_{hc}(S)$ the corresponding [[simplicially enriched category]] (where $\tau_{hc}$ is the [[left adjoint]] of the [[homotopy coherent nerve]]) and let $\phi : \tau_{hc}(S) \to C$ be an [[sSet]]-[[enriched functor]].
+-- {: .un_def}
###### Definition
**(extracting a marked simplicial presheaf from a marked fibration)**
(HTT, section 3.2.1)
The **straightening functor**
$$
St_\phi : sSet^+/S \to [C^{op}, sSet^+]
$$
from [[marked simplicial set]]s over $S$ to marked [[simplicial presheaves]] on $C^{op}$ is on the underlying simplicial sets (forgetting the marking) the same straightening functor as [above](#GrpdModelCatVersion).
On the markings the functor acts as follows.
Each edge $f: d \rightarrow e$ of $X \in sSet/S$ gives rise to an edge $\tilde f \in St_\phi (X)(d) = K(\phi,p)(d,v)$: the [[join of simplicial sets|join]] 2-simplex $f \star v$ of $X^{\triangleright}$
$$
\array{
d && \stackrel{f}{\to} && e
\\
& {}_{\mathllap{\tilde d}}\searrow
& \stackrel{\tilde f}{\Rightarrow}
& \swarrow_{\mathrlap{\tilde e}}
\\
&& v
}
$$
with image $\tilde f : \tilde d \to f^* \tilde e$
in the pushout $K(\phi,p)(d,v)=St_\phi X(d)$.
We define the straightening functor to assign that
marking of edges which is the minimal one such that all
such morphisms $\tilde f$ are marked in
$St_\phi X(d)$, for all marked $f : d \to e$
in $X$:
this means that this marking is being completed under the constraint
that $St_\phi(X)$ be [[sSet]]-[[enriched functor|enriched functorial]].
For that, [[marked simplicial set|recall]] that the hom simplicial sets of $sSet^+$ are the spaces $Map^\sharp(X,Y)$, which consist of those simplices of the [[internal hom]] $Map(X,Y) := Y^X$ whose edges are all marked:
$$
Map(X,Y)_n = Hom_{sSet^+}(X \times \Delta[n]^#, Y)
\,.
$$
So we need to find a marking on the $St_\phi(X)(-)$ such that for all
$g : \Delta[1] \to C(c,d)$ the composite
$$
\Delta[1]
\stackrel{g}{\to}
C(c,d)
\stackrel{St_\phi(X)(c,d)}{\to}
Map(St_\phi(X)(d), St_\phi(X)(c))
$$
is a marked edge of the mapping complex. By the internal hom-adjunction this edge corresponds to a morphism
$$
St_\phi(X)(g)
:
St_\phi(X)(d) \times \Delta[1] \rightarrow St_\phi(X)(c)
$$
and to be marked needs to carry edges of the form $\tilde f \times \{0 \to 1\}$ i.e. 1-cells $(\tilde f , Id) : \Delta[1] \to St_\phi(X)(d) \times \Delta[1]$ to marked edges
$$
g^* \tilde f : \Delta[1] \stackrel{(\tilde f,Id)}{\to}
St_\phi(X)(d)\times \Delta[1] \stackrel{St_\phi(X)(g)}{\to}
St_{\phi}(X)(c)
$$
in $St_\phi(X)(c)$. So we need to ensure that the edges of this form are marked:
we define that the straightening functor marks an edge in $St_\phi(X)(c)$ iff it is of this form $g^* \tilde f$, for $f : d \to e$ a marked edge of $X$ and $g \in C(c,d)_1$.
As in the unmarked cae, the straightening functor has an [[sSet]]-[[right adjoint]], the **unstraightening functor**
$$
n_\phi : [C^{op}, sSet^+] \to sSet^+/S
\,.
$$
=--
This functor $Un_\phi$ exhibits the $(\infty,1)$-Grothendieck-construction proper, in that it constructs a [[Cartesian fibration]] from a given $(\infty,1)$-functor:
+-- {: .un_theorem}
###### Theorem
**(presentation of $(\infty,1)$-Grothendieck construction)**
This induces a [[Quillen adjunction]]
$$
(St_\phi \dashv Un_\phi) : SSet^+/S
\stackrel{\overset{Un_{\phi}}{\leftarrow}}{\underset{St_\phi}{\to}}
[C^{op}, SSet^+]
$$
between the [[model structure for Cartesian fibrations]] and the projective [[global model structure on functors]] with values in the [[model structure on marked simplicial over-sets|model structure on marked simplicial sets]].
If $\phi$ is an equivalence in the [[model structure on simplicial categories]] then this Quillen adjunction is a [[Quillen equivalence]].
=--
+-- {: .proof}
###### Proof
This is [[Higher Topos Theory|HTT, theorem 3.2.0.1]].
=--
#### Over an ordinary category {#OverCat}
In the case that $C$ happens to be an ordinary [[category]], the $(\infty,1)$-Grothendieck construction can be simplified and given more explicitly.
This is [[Higher Topos Theory|HTT, section 3.2.5]].
+-- {: .un_def}
###### Definition
**(relative nerve functor)**
Let $C$ be a small category and let $f : C \to sSet$ be a functor. The simplicial set $N_f(C)$, the **relative nerve** of $C$ under $f$ is defined as follows:
an $n$-cell of $N_f(C)$ is
1. a functor $\sigma : [n] \to C$;
1. for every $[k] \subset [n]$ a morphism $\tau(k) : \Delta[k] \to f(\sigma(k))$;
1. such that for all $[j] \subset [k] \subset [n]$ the diagram
$$
\array{
\Delta[j] &\stackrel{\tau(j)}{\to}& f(\sigma(j))
\\
\downarrow && \downarrow^{\mathrlap{f(\sigma(j\to k))}}
\\
\Delta[k] &\stackrel{\tau(k)}{\to}& f(\sigma(k))
}
$$
commutes.
There is a canonical morphism
$$
N_f(C) \to N(C)
$$
to the ordinary [[nerve]] of $C$, obtained by forgetting the $\tau$s.
=--
This is [[Higher Topos Theory|HTT, def. 3.2.5.2]].
+-- {: .un_remark}
###### Remark
When $f$ is constant on the point, then $N_f(C) \to N(C)$ is an isomorphism of simplicial sets, so $N_f(C)$ this is the ordinary [[nerve]] of $C$.
The [[fiber]] of $N_f(C) \to N(C)$ over an object $c \in C$ is given by taking $\sigma$ to be constant on $C$. Then all the $\tau$s are fixed by the maximal $\tau(n) : \Delta[n] \to f(c)$. So the fiber of $N_f(C)$ over $c$ is $f(c)$.
=--
+-- {: .un_def}
###### Definition
**(marked relative nerve functor)**
Let $C$ be a small [[category]]. Define a functor
$$
sSet^+/N(C) \leftarrow [C, sSet^+] : N^+
$$
by
$$
(C \stackrel{F}{\to} sSet^+)
\mapsto
(N_f(C), E_F)
\,,
$$
where $f : C^{op} \stackrel{F}{\to} sSet^+ \to sSet$ is $F$ with the marking forgotten, where $N_f(C)$ is the relative nerve of $C$ under $f$, and where the marking $E_F$ is given by ...
=--
This is [[Higher Topos Theory|HTT, def. 3.2.5.12]].
This functor has a [[left adjoint]] $\mathcal{F}^+$. The value of $\mathcal{F}^+(F)$ on some $c \in C$ is equivalent to the value of $St(F)$.
This is [[Higher Topos Theory|HTT, Lemma 3.2.5.17]].
+-- {: .un_prop}
###### Proposition
**($(\infty,1)$-Grothendieck construction over a category)**
The adjunction
$$
(\mathcal{F}^+ \dashv N^+)
:
sSet^+_{/N(C)} \stackrel{\overset{\mathcal{F}^+}{\to}}{\underset{N^+}{\leftarrow}}
[C,sSet^+]
\,.
$$
is a [[Quillen equivalence]] between the [[model structure for Cartesian fibrations|model structure for coCartesian fibrations]] and the projective [[global model structure on functors]].
=--
+-- {: .proof}
###### Proof
This is [[Higher Topos Theory|HTT, prop. 3.2.5.18]].
=--
### Relation beween the model structures
\begin{prop}
Let $S$ be a [[simplicial set]].
There is a sequence of [[Quillen adjunction]]s
$$
(sSet/S)_{Joyal}
\stackrel{\overset{}{\to}}{\overset{}{\leftarrow}}
sSet^+/S
\stackrel{\overset{}{\to}}{\overset{}{\leftarrow}}
(sSet^+/S)^{loc}
\stackrel{\overset{}{\to}}{\overset{}{\leftarrow}}
(sSet/S)_{rfib}
\stackrel{\overset{}{\to}}{\overset{}{\leftarrow}}
(sSet/S)_{Quillen}
\,.
$$
Where from left to right we have
1. the [[model structure on an overcategory]] for the Joyal [[model structure for quasi-categories]];
1. the [[model structure for Cartesian fibrations]];
1. some localizaton of the model structure for Cartesian fibrations;
1. the [[model structure for right fibrations]]
1. the [[model structure on an overcategory]] for the Quillen [[model structure on simplicial sets]];
The first and third Quillen adjunction here is a [[Quillen equivalence]] if $S$ is a [[Kan complex]].
\end{prop}
([[Higher Topos Theory|HTT, section 3.1.5]])
## Properties
### As an (op)lax $\infty$-colimit
The $(\infty,1)$-Grothendieck construction on an $\infty$-functor is equivalently its [[lax (infinity,1)-colimit]] ([Gepner-Haugseng-Nikolaus 15](#GepnerHaugsengNikolaus15)).
See also at _[[Grothendieck construction]]_ _[as a lax colimit](Grothendieck%20construction#AsALaxColimit)_.
## Examples
### Cartesian fibrations over the point
For the base category $S$ being the point $S = {*}$, the $(\infty,1)$-Grothendieck construction naturally becomes essentially trivial. However, its model by the Quillen functor
$$
St_\phi : sSet/* \simeq sSet \to [*,sSet] \simeq sSet
$$
is not entirely trivial and in fact produces a Quillen auto-equivalence of $sSet_{Quillen}$ with itself that plays a central role in the proof of the corresponding Quillen equivalence over general $S$.
**Definition**
Let $Q : \Delta \to sSet$ be the [[cosimplicial object|cosimplicial]] [[simplicial set]] given by
$$
Q[n] := |J^n|(x,v)
\,,
$$
where
$$
J^n = C^{\triangleleft}(\Delta[n] \to \{x\})
\,.
$$
Then: [[nerve and realization]] associated to $Q$ yield a [[Quillen equivalence]] of $sSet_{Quillen}$ with itself.
[[Higher Topos Theory|HTT, section 2.2.2]].
...
### Cartesian fibrations over the interval {#FibsOverInterval}
A [[Cartesian fibration]] $p : K \to \Delta[1]$ over the 1-[[simplex]] corresponds to a morphism
$\Delta[1]^{op} \to $ [[(∞,1)Cat]], hence to an [[(∞,1)-functor]] $F : D \to C$.
By the above procedure we can express $F$ as the image of $p$ under the straightening functor. The characterization via lax colimits leads to describing $K$ as the mapping cylinder $(\Delta^1 \times B) \amalg_{\Delta^{\{0\}} \times B} A$.
However, there is a more immediate way to extract this functor, which we now describe. This construction also provides additional strictness properties in the quasicategory model.
First recall the situation for the ordinary [[Grothendieck construction]]: given a [[Grothendieck fibration]] $K \to \{0 \to 1\}$, we obtain a functor $f : K_1 \to K_0$ between the fibers, by _choosing_ for each object $d \in K_1$ a [[Cartesian morphism]] $e_d \to d$. Then the universal property of Cartesian morphism yields for every morphism $d_1 \to d_2$ in $K_1$ the unique left vertical filler in
$$
\array{
e_{d_1} &\to& d_1
\\
\downarrow && \downarrow
\\
e_{d_2} &\to& d_2
}
\,.
$$
And again by universality, this assignment is functorial: $K_1 \to K_0$.
Diagrammatically, the choice of Cartesian morphisms here is a lift $e$ in the diagram
$$
\array{
K_1 &\hookrightarrow& K
\\
\downarrow &\nearrow_e& \downarrow
\\
K_1 \times \{0 \to 1\} &\to& \{0 \to 1\}
}
\,.
$$
This diagrammatic way of encoding the functor associated to a Grothendieck fibration over the interval generalizes straightforwardly to the [[quasi-category]] context.
+-- {: .un_defn}
###### Definition
Given a [[Cartesian fibration]] $p : K \to \Delta[1]$ with fibers the [[quasi-categories]] $C := K_{0}$ and $D := K_{1}$, an **$(\infty,1)$-functor associated to the Cartesian fibration** $p$ is a functor $f : D \to C$ such that there exists a commuting diagram in [[sSet]]
$$
\array{
D \times \Delta[1] &&\stackrel{F}{\to}&& K
\\
& \searrow && \swarrow_{\mathrlap{p}}
\\
&& \Delta[1]
}
$$
such that
* $F|_{1} = Id_D$;
* $F|_{0} = f$;
* and for all $d \in D$, $F(\{d\}\times \{0 \to 1\})$ is a [[Cartesian morphism]] in $K$.
More generally, if we also specify possibly nontrivial [[equivalence of quasi-categories|equivalences of quasi-categories]] $h_0 : C \stackrel{\simeq}{\to} K_{0}$ and $h_1 : D \stackrel{\simeq}{\to} K_{1}$, then a functor is associated to $K$ and this choice of equivalences if the first twoo conditions above are generalized to
* $F|_{1} = h_1$;
* $F|_{0} = h_0 \circ f$;
=--
This is [[Higher Topos Theory|HTT, def. 5.2.1.1]].
+-- {: .un_prop}
###### Proposition
For $p : K \to \Delta[1]$ a Cartesian fibration, the associated functor exists and is unique up to equivalence in the [[(∞,1)-category of (∞,1)-functors]] $Func(K_{0}, K_{1})$.
=--
+-- {: .proof}
###### Proof
This is [[Higher Topos Theory|HTT, prop 5.2.1.5]].
Set $C := K_{0}$ and $D := K_{1}$.
With the notation described at [[model structure for Cartesian fibrations]], consider the commuting diagram
$$
\array{
D^\flat \times \{1\} &\hookrightarrow& K^{\sharp}
\\
\downarrow && \downarrow^{\mathrlap{p}}
\\
D^{\flat} \times \Delta[1]^{#} &\to&
\Delta[1]^#
}
$$
in the category $sSet^+$ of marked simplicial sets.
Here the left vertical morphism is [marked anodyne](http://ncatlab.org/nlab/show/model+structure+for+Cartesian+fibrations#MarkedAnodyne): it is the [[smash product]] of the marked cofibration (monomorphism) $Id : D^\flat \to D^\flat$ with the marked anodyne morphism $\Delta[1]^# \to \Delta[0]$. By the stability properties discussed at [Marked anodyne morphisms](http://ncatlab.org/nlab/show/model+structure+for+Cartesian+fibrations#MarkedAnodyne), this implies that the morphism itself is marked anodyne.
As discussed there, this means that a lift $d : D^\flat \times \Delta[1]^# \to K^{\sharp}$ against the Cartesian fibration in
$$
\array{
D^\flat \times \{1\} &\hookrightarrow& K^{\sharp}
\\
\downarrow &\nearrow_{s}& \downarrow^{\mathrlap{p}}
\\
D^{\flat} \times \Delta[1]^{#} &\to&
\Delta[1]^#
}
$$
exists. This exhibits an associated functor $f := s_0$.
Suppose now that another associated functor $f'$ is given. It will correspondingly come with its diagram
$$
\array{
D^\flat \times \{1\} &\hookrightarrow& K^{\sharp}
\\
\downarrow &\nearrow_{s'}& \downarrow^{\mathrlap{p}}
\\
D^{\flat} \times \Delta[1]^{#} &\to&
\Delta[1]^#
}
\,.
$$
Together this may be arranged to a diagram
$$
\array{
D^\flat \times \Lambda[2]_2
&\stackrel{(s,s')}{\to}& K^{\sharp}
\\
\downarrow &\nearrow_{q}& \downarrow^{\mathrlap{p}}
\\
D^{\flat} \times \Delta[2]^{#} &\to&
\Delta[1]^#
}
\,,
$$
where the top horizontal morphism picks the 2-[[horn]] in $K$ whose two edges are labeled by $s$ and $s'$, respectively.
Now, the left vertical morphism is still marked anodyne, and hence the lift $k$ exists, as indicated. Being a morphism of marked simplicial sets, it must map for each $d \in D$ the edge $\{d\}\times \{0\to 1\}$ to a [[Cartesian morphism]] in $K$, and due to the commutativity of the diagram this morphism must be in $K_0$, sitting over $\{0\}$. But as discussed there, a [[Cartesian morphism]] over a point is an equivalence. This means that the restriction
$$
k|_{D \times \{0 \to 1\}} \to K_0
$$
is an invertible [[natural transformation]] between $f$ and $f'$, hence these are equivalent in the functor category.
=--
Conversely, every functor $f : D \to C$ gives rise to a Cartesian fibration that it is associated to, in the above sense.
+-- {: .un_prop}
###### Proposition
Every $(\infty,1)$-functor $f : D \to C$ is associated to some Cartesian fibration $p : K \to \Delta[1]$, and this is unique up to equivalence.
=--
+-- {: .proof}
###### Proof
This is [[Higher Topos Theory|HTT, prop 5.2.1.3]].
The idea is that we obtain $K$ from first forming the cylinder $D \times \Delta[1]$ and the identifying the left boundary of that with $C$, using $f$.
Formally this means that we form the [[pushout]]
$$
N := (D^\natural \times \Delta[1]^#) \coprod_{D^\natural \times \{0\}^#} C^\natural
$$
in $sSet^+$, where $C^\natural$ and $D^\natural$ are $C$ and $D$ with precisely the [[equivalence in a quasi-category|equivalences]] marked. This comes canonically with a morphism
$$
N \to \Delta[1]^{\sharp}
$$
and does have the property that $N_0 = C$, $N_1 = D$ and that $f$ is associated to it in that the restriction of the canonical morphism $D \times \Delta[1] \to K$ to the 0-fiber is $f$. But it may fail to be a Cartesian fibration.
To fix this, use the [[small object argument]] to factor $N \to \Delta[1]$ as
$$
N \to K \to \Delta[1]^#
\,,
$$
where the first morphism is [marked anodyne](http://ncatlab.org/nlab/show/model+structure+for+Cartesian+fibrations#MarkedAnodyne) and the second has the [[right lifting property]] with respect to all marked anodyne morphisms and is hence (since every morphism in $\Delta[1]^#$ is marked) a Cartesian fibration.
It then remains to check that $f$ is still associated to this $K \to \Delta[1]^#$. This is done by observing that in the small object argument $K$ is built succesively from [[pushout]]s of the form
$$
\array{
A &\to& N_\alpha
\\
\downarrow && \downarrow & \searrow
\\
B &\to& N_{\alpha+1} &\to& \Delta[1]
}
\,,
$$
where the morphisms on the left are the generators of marked anodyne morphisms (see [here](http://ncatlab.org/nlab/show/model+structure+for+Cartesian+fibrations#MarkedAnodyne)). from this one checks that if the fiber $N_\alpha \times_{\Delta[1]} \{0\}$ is equivalent to $C$, then so is $N_{\alpha +1} \times_{\Delta[1]} \{0\}$ and similarly for $D$. By induction, it follows that $f$ is indeed associated to $K \to \Delta[1]$.
To see that the $K$ obtained this way is unique up to equivalence, consider...
=--
### Cartesian fibrations over simplices {#CartOverSimplex}
... for the moment see [[Higher Topos Theory|HTT, section 3.2.2]] ...
### The universal Cartesian fibration
for the moment see
* [[universal fibration of (∞,1)-categories]].
## Related concepts
* [[operadic (∞,1)-Grothendieck construction]]
* [[Grothendieck construction for model categories]]
* [[Joyal locus]], [[tangent (infinity,1)-topos]]
## References
Textbook accounts:
* {#Lurie09} [[Jacob Lurie]], Section 3.2 of: _[[Higher Topos Theory]]_, Annals of Mathematics Studies **170**, Princeton University Press 2009 ([pup:8957](https://press.princeton.edu/titles/8957.html), [pdf](https://www.math.ias.edu/~lurie/papers/HTT.pdf))
* {#Cisinski19} [[Denis-Charles Cisinski]], Section 5.2 of: _[[Higher Categories and Homotopical Algebra]]_, Cambridge University Press 2019 ([doi:10.1017/9781108588737](https://doi.org/10.1017/9781108588737), [pdf](http://www.mathematik.uni-regensburg.de/cisinski/CatLR.pdf))
> (for $\infty$-groupoids)
More on model-category theoretic construction of the $\infty$-Grothendieck construction with values in $\infty$-groupoids is in
* {#HeutsMoerdijk13} [[Gijs Heuts]], [[Ieke Moerdijk]], _Left fibrations and homotopy colimits_ ([arXiv:1308.0704](http://arxiv.org/abs/1308.0704))
* [[Gijs Heuts]], [[Ieke Moerdijk]], *Left fibrations and homotopy colimits II* [[arXiv:1602.01274](https://arxiv.org/abs/1602.01274)]
Discussion in terms of [[lax (infinity,1)-colimits]] is in
* {#GepnerHaugsengNikolaus15} [[David Gepner]], [[Rune Haugseng]], [[Thomas Nikolaus]], _Lax colimits and free fibrations in $\infty$-categories_ ([arXiv:1501.02161](http://arxiv.org/abs/1501.02161))
Section 1 of this paper reviews properties of the Grothendieck construction for quasicategories:
* {#MazelGee} [[Aaron Mazel-Gee]], _All about the Grothendieck construction_ ([arXiv:1510.03525](https://arxiv.org/abs/1510.03525))
Another review is
* Anders Jess Pedersen, Magnus Baunsgaard Kristensen, _Straightening and Unstraightening_ [(Dropbox)](https://www.dropbox.com/s/alys3wg59rir3gt/Grothendieck_construction.pdf?dl=0)
Discussion of the $\infty$-Grothendieck construction for [[(infinity,1)-functors|$\infty$-functors]] to [[(infinity,1)-toposes|$\infty$-toposes]] from [[inverse images]] of [[infinity-groupoid|$\infty$-groupoids]]:
* {#BeardsleyPéroux22} [[Jonathan Beardsley]], [[Maximilien Péroux]], Lemma 3.10 in: *Koszul Duality in Higher Topoi*, Homol. Homot. Appl. [(2022)](https://www.intlpress.com/site/pub/pages/journals/items/hha/_home/acceptedpapers/index.php), [[arXiv:1909.11724](https://arxiv.org/abs/1909.11724)]
Discussion entirely within the context of [[quasi-categories]]:
* [[Denis-Charles Cisinski]], [[Hoang Kim Nguyen]], *The universal coCartesian fibration* [[arXiv:2210.08945](https://arxiv.org/abs/2210.08945)]
which lends itself to understanding the [[universal coCartesian fibration]] as [[categorical semantics]] for the [[type universe]] in [[directed homotopy type theory]]:
* {#CisinskiEtAl23} [[Denis-Charles Cisinski]], [[Hoang Kim Nguyen]], Tashi Walde: *Univalent Directed Type Theory*, lecture series in the *[CMU Homotopy Type Theory Seminar](https://www.cmu.edu/dietrich/philosophy/hott/seminars/index.html)* (13, 20, 27 Mar 2023) [[web](https://www.cmu.edu/dietrich/philosophy/hott/seminars/index.html#230313), video 1:[YT](https://www.youtube.com/watch?v=5YOltuTcBK8), 2:[YT](https://www.youtube.com/watch?v=xWmELBvHMPo), 3:[YT](https://www.youtube.com/watch?v=P0Cfb4eUJo4); slides 0:[pdf](https://www.cmu.edu/dietrich/philosophy/hott/seminars/slides/cisinski-nguyen-walde-intro_talk1.pdf), 1:[pdf](https://www.cmu.edu/dietrich/philosophy/hott/seminars/slides/cisinski-nguyen-walde-talk1.pdf), 2:[pdf](https://www.cmu.edu/dietrich/philosophy/hott/seminars/slides/cisinski-nguyen-walde-talk2.pdf), 3:[pdf](https://www.cmu.edu/dietrich/philosophy/hott/seminars/slides/cisinski-nguyen-walde-talk3.pdf)]
(see [video 3 at 1:16:58](https://youtu.be/P0Cfb4eUJo4?t=4618) and [slides 3.33](https://www.cmu.edu/dietrich/philosophy/hott/seminars/slides/cisinski-nguyen-walde-talk3.pdf#page=33))
[[!redirects (∞,1)-Grothendieck construction]]
[[!redirects infinity1-Grothendieck construction]]
|
(infinity,1)-Kan extension | https://ncatlab.org/nlab/source/%28infinity%2C1%29-Kan+extension |
> under construction
+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### $(\infty,1)$-Category theory
+--{: .hide}
[[!include quasi-category theory contents]]
=--
#### Limits and colimits
+--{: .hide}
[[!include infinity-limits - contents]]
=--
=--
=--
#Contents#
* table of contents
{:toc}
## Idea
The notion of $(\infty,1)$-Kan extension is the generalization of the notion of [[Kan extension]] from [[category theory]] to [[(∞,1)-category theory]].
## Definition
### General
Independent of any models or concrete realizations chosen, the notion of $(\infty,1)$-Kan extension is intrinsically determined from just the notions of
* [[(∞,1)-functor]],
* [[(∞,1)-category of (∞,1)-functors]]
* [[adjoint (∞,1)-functor]].
In terms of these, for $f : C \to C'$ any [[(∞,1)-functor]] and any [[(∞,1)-category]] $A$, there is an induced $(\infty,1)$-functor $f^* : Func_{(\infty,1)}(C',A) \to Func_{(\infty,1)}(C,A)$.
The **left $(\infty,1)$-Kan extension functor** is the left [[adjoint (∞,1)-functor]] to $f^*$.
The **right $(\infty,1)$-Kan extension functor** is the right [[adjoint (∞,1)-functor]] to $f^*$.
Given different incarnations of or models for the notion of [[(∞,1)-category]], there are accordingly different incarnations and models of this general abstract prescription.
#### In terms of quasi-categories
([LurieHTT, def. 4.3.2.2, 4.3.3.2](#LurieHTT))
#### In terms of Kan-complex enriched categories
see [[homotopy Kan extension]]
#### In terms of simplicial model categories
see [[homotopy Kan extension]]
## Properties
### Pointwise (strong)
{#Pointwise}
$\infty$-Kan extensions as above are [pointwise/strong](Kan+extension#Pointwise). That is in fact the very content of ([LurieHTT, def. 4.3.2.2, 4.3.3.2](#LurieHTT)).
### As adjoints to pullbacks
left/right $\infty$-Kan extension is left/right [[adjoint (∞,1)-functor]] to restriction. ([LurieHTT, prop. 4.3.3.7](#LurieHTT))
## Related concepts
* [[Kan extension]]
* **$(\infty,1)$-Kan extension**
## References
A general concept of $(\infty,1)$-Kan extensions in terms of quasi-categories are discussed in section 4.3 of
* {#LurieHTT} [[Jacob Lurie]], _[[Higher Topos Theory]]_ .
For [[simplicially enriched categories]] and [[model categories]] a discussion is in section A.3.3 there.
Coinciding left/right ([[ambidextrous adjunction|ambidextrous]]) $\infty$-Kan extensions along maps of [[∞-groupoids]] are discussed in
* {#HopkinsLurie14} [[Michael Hopkins]], [[Jacob Lurie]], section 4 of _[[Ambidexterity in K(n)-Local Stable Homotopy Theory]]_
Pointwise [[homotopy Kan extensions]] are discussed in
* {#Radulescu-Banu06} Andrei Radulescu-Banu, _Cofibrations in Homotopy Theory_ ([arXiv:0610009](http://arxiv.org/abs/math/0610009))
* {#Cisinski09} [[Denis-Charles Cisinski]], _Locally constant functors_, Math. Proc. Camb. Phil. Soc. (2009), 147, 593 ([pdf](http://www.math.univ-toulouse.fr/~dcisinsk/lcmodcat3.pdf))
* {#Gonzales11} Beatriz Rodriguez Gonzalez, section 4 of _Realizable homotopy colimits_ ([arXiv:1104.0646](http://arxiv.org/abs/1104.0646))
See also
* [[Samuel Isaacson]], _A note on unenriched homotopy coends_ ([pdf](http://www.math.uwo.ca/~sisaacso/PDFs/coends.pdf))
[[!redirects (∞,1)-Kan extension]]
[[!redirects (∞,1)-Kan extensions]]
[[!redirects (infinity,1)-Kan extensions]]
[[!redirects pointwise (∞,1)-Kan extension]]
[[!redirects pointwise (∞,1)-Kan extensions]]
[[!redirects pointwise (infinity,1)-Kan extension]]
[[!redirects pointwise (infinity,1)-Kan extensions]] |
(infinity,1)-module | https://ncatlab.org/nlab/source/%28infinity%2C1%29-module |
+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Higher algebra
+--{: .hide}
[[!include higher algebra - contents]]
=--
#### Higher linear algebra
+-- {: .hide}
[[!include homotopy - contents]]
=--
=--
=--
#Contents#
* table of contents
{:toc}
## Idea
The notion of _$(\infty,1)$-module_ over an [[monoid object in an (∞,1)-category]] (for instance an [[A-∞ ring]] or [[E-∞ ring]]) is the generalization to ([[stable homotopy theory|stable]])[[homotopy theory]] of the notion of [[module]] over a [[ring]].
## Definition
See at _[[module over an algebra over an (∞,1)-operad]]_.
## Examples
* in the [[stable (∞,1)-category of spectra]]: [[module spectrum]] over an [[ring spectrum]]
## Properties
### Compact modules
+-- {: .num_prop}
###### Propositon
Let $R$ be an [[A-∞ ring]]. The (∞,1)-category of ∞-modules $R Mod$ is a [[compactly generated (∞,1)-category]] and the [[compact object in an (∞,1)-category|compact objects]] coincide with the [[perfect modules]]
=--
([[Higher Algebra|HA, prop. 8.2.5.2]])
### Relation to fiberwise stabilization
By the discussion an [[tangent (∞,1)-category]] we may realize $E_\infty$-modules over $R$ as objects in the [[stabilization]] of the [[over-(∞,1)-category]] over $R$:
+-- {: .num_prop}
###### Proposition
Let $E_\infty := Alg^{Comm}(\infty Grpd)$ be the [[(∞,1)-category]] of [[E-∞ ring]]s and let $R \in E_\infty$. Then the [[stabilization]] of the [[over-(∞,1)-category]] over $A$
$$
Stab(E_\infty/R) \simeq A Mod(Spec)
$$
is equivalentl to the category of $R$-module spectra.
=--
This is ([Lurie, cor. 1.5.15](#Lurie)).
### Stable Dold-Kan correspondence
{#StableDoldKanCorrespondence}
For $R$ an ordinary [[ring]], write $H R$ for the corresponding [[Eilenberg-MacLane spectrum]].
+-- {: .num_theorem #StableDoldKan}
###### Theorem
For $R$ any [[ring]] (or [[ringoid]], even) there is a [[Quillen equivalence]]
$$
H R Mod \simeq Ch_\bullet(R Mod)
$$
between model structure on $H R$-module spectra and the [[model structure on chain complexes]] (unbounded) of ordinary $R$-[[module]]s.
This presents a corresponding [[equivalence of (∞,1)-categories]]. If $R$ is a commutative ring, then this is an equivalence of [[symmetric monoidal (∞,1)-categories]].
=--
This equivalence on the level of [[homotopy categories]] is due to ([Robinson](#Robinson)). The refinement to a Quillen equivalence is ([SchwedeShipley, theorem 5.1.6](#SchwedeShipley)). See also the discussion at _[[stable model categories]]_. A direct description as an equivalence of $(\infty,1)$-categories appears as ([Lurie, theorem 7.1.2.13](#Lurie)).
+-- {: .num_remark}
###### Remark
This is a stable version of the [[Dold-Kan correspondence]].
=--
See at __[[algebra spectrum]]_ for the corresponding statement for $H R$-algebra spectra and [[dg-algebras]].
+-- {: .num_example}
###### Example
For $X$ a [[topological space]] and $R$ a ring, let $C_\bullet(X, R)$ be the standard [[chain complex]] for [[singular homology]] $H_\bullet(X, R)$ of $X$ with coefficients in $R$.
Under the stable Dold-Kan correspondence, prop. \ref{StableDoldKan}, this ought to be identified with the [[smash product]] $(\Sigma^\infty_+ X) \wedge H R$ of the [[suspension spectrum]] of $X$ with the [[Eilenberg-MacLane spectrum]]. Notice that by the general theory of [[generalized homology]] the [[homotopy groups]] of the latter are again singular homology
$$
\pi_\bullet( (\Sigma^\infty_+ X) \wedge H R) \simeq H_\bullet(X, R)
\,.
$$
While the correspondence $(\Sigma^\infty_+ X) \wedge H R \sim C_\bullet(X,R)$ under the above equivalence is suggestive, maybe nobody has really checked it in detail. It is sort of stated as true for instance on <a href="http://arxiv.org/PS_cache/arxiv/pdf/0906/0906.5198v1.pdf#page=15">p. 15 </a> of ([BCT](#BCT)).
=--
## Related concepts
* [[action]], [[∞-action]]
* [[module]], **$(\infty,1)$-module**
* [[(∞,1)-category of (∞,1)-modules]]
* [[tensor product of ∞-modules]]
* [[(∞,1)-module bundle]]
* [[perfect module]]
* [[2-module]]
* [[bimodule]]
[[(∞,1)-bimodule]]
* [[(∞,n)-module]]
* [[representation]], [[∞-representation]]
## References
Discussion in terms of [[module objects]] in [[symmetric smash product of spectra|symmetric monoidal]] [[model categories of spectra]] includes
* [[Anthony Elmendorf]], [[Igor Kriz]], [[Peter May]], section 5 of _[[Modern foundations for stable homotopy theory]]_ ([pdf](http://hopf.math.purdue.edu/Elmendorf-Kriz-May/modern_foundations.pdf))
Modules over algebras over an arbitrary [[(∞,1)-operad]] are discussed in section 3.3 of
* [[Jacob Lurie]], _[[Higher Algebra]]_
Modules specifically over [[A-∞ algebras]] are discussed in section 4.2 there.
Further discussion of [[(infinity,n)-bimodules]] is in
* {#Haugseng14} [[Rune Haugseng]], _The higher Morita category of $E_n$-algebras_ ([arXiv:1412.8459](http://arxiv.org/abs/1412.8459))
The equivalence between the [[homotopy categories]] of $H R$-module spectra and $Ch_\bullet(R Mod)$ is due to
*{#Robinson} Alan Robinson, _The extraordinary derived category_ , Math. Z. 196 (2) (1987) 231–238.
The refinement of this statement to a [[Quillen equivalence]] is due to
* {#SchwedeShipley} [[Stefan Schwede]], [[Brooke Shipley]], _Stable model categories are categories of modules_ , Topology 42 (2003), 103-153 ([pdf](http://www.math.uic.edu/~bshipley/classTopFinal.pdf))
Applications to [[string topology]] are discussed in
* {#BCT} [[Andrew Blumberg]], [[Ralph Cohen]], [[Constantin Teleman]], _Open-closed field theories, string topology and Hochschild homology_ ([arXiv:0906.5198](http://arxiv.org/abs/0906.5198))
See the section on string topology at [[sigma model]] for more on this.
[[!redirects ∞-module]]
[[!redirects infinity-modules]]
[[!redirects ∞-modules]]
[[!redirects infinity-module]]
[[!redirects (∞,1)-module]]
[[!redirects (∞,1)-modules]]
[[!redirects (infinity,1)-modules]]
|
(infinity,1)-module bundle | https://ncatlab.org/nlab/source/%28infinity%2C1%29-module+bundle | [[!redirects (infinity,1)-vector bundle]]
+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Bundles
+-- {: .hide}
[[!include bundles - contents]]
=--
#### Cohomology
+--{: .hide}
[[!include cohomology - contents]]
=--
#### Stable Homotopy theory
+--{: .hide}
[[!include stable homotopy theory - contents]]
=--
#### Higher algebra
+--{: .hide}
[[!include higher algebra - contents]]
=--
=--
=--
#Contents#
* table of contents
{:toc}
## Idea
The notion of an _$(\infty,1)$-module bundle_ is a [[categorification]]/[[homotopy theory|homotopification]] of the notion of a [[module bundle]]/[[vector bundle]], where [[fields]] and [[rings]] are replaced by [[∞-rings]] and [[modules]] by [[∞-modules]]; a central notion in [[parameterized stable homotopy theory]].
Recall that for $k$ a [[field]], a [[vector space]] is a $k$-[[module]], and a [[vector bundle]] over a [[space]] $X$ is classified by a morphism
$\alpha : X \to k$[[Mod]] with $k$[[Mod]] regarded as an object in the relevant [[topos]]. For instance for _discrete_ or _flat_ vector bundles $k Mod$ is the [[category]] [[Vect]] of vector spaces.
There is the subcategory $k Line \hookrightarrow k Mod$ of 1-[[dimension]]al $k$-vector bundles, and morphisms that factor as $\alpha : X \to k Line \hookrightarrow k Mod$ are $k$-[[line bundle]]s.
In the discrete case the vector space of [[section]]s of the vector bundle classified by $\alpha$ is the [[colimit]] $\lim_\to \alpha$.
These statements [[categorification|categorify]] in a straightforward manner to the case where $k$ is generalized to a commutative [[∞-ring]]: an _[[E-∞ ring]]_ or _[[ring spectrum]]_ . Modules are replaced by [[module spectrum|module spectra]] and colimits by [[homotopy colimit]]s.
The resulting notion of $(\infty,1)$-vector bundles plays a central role in many constructions in [[orientation in generalized cohomology]], [[twisted cohomology]] and [[Thom isomorphism]]s.
Further generalization of the concept leads to [[(∞,n)-vector bundle]]s: an $(\infty,n)$-module over an [[E-∞-ring]] $K$ is an object of the [[(∞,n)-category]] $(\cdots (K Mod) Mod ) \cdots Mod$, where we are iteratively forming module $(\infty,k)$-categories over the monoidal $(\infty,k-1)$-category of $(\infty,k-1)$-modules, $n$ times.
## Discrete $(\infty,1)$-vector bundles
We discuss $(\infty,1)$-vector bundles internal to the [[(∞,1)-topos]] [[∞Grpd]] $\simeq$ [[Top]]. Since we are discussing objects with geometric interpretation, we are to think of this as the $(\infty,1)$-topos of _[[discrete ∞-groupoids]]_.
Discussion of $\infty$-vector bundles internal to _structured_ (non-discrete) $\infty$-groupoids is [below](#Structured).
### $\infty$-Modules and $\infty$-Module bundles
Assume in the following choices
* $K$ -- an [[E-∞ ring]]
* $A$ -- a $K$-[[algebra spectrum|algebra]],
hence an [[A-∞ algebra]] in [[Spec]] equipped with a $\infty$-algebra homomorphism $K \to A$.
Denote
* $A Mod$ -- the [[(∞,1)-category]] of $A$-[[module spectrum|module spectra]].
+-- {: .num_defn}
###### Definition
For $X$ a [[discrete ∞-groupoid]] (often presented as a [[topological space]]), the [[(∞,1)-category]] of **$A$-module $\infty$-bundles** over $X$ is the [[(∞,1)-functor (∞,1)-category]]
$$
A Mod(X) := Func(X, A Mod)
\,.
$$
=--
In this form this appears as ([ABG def. 3.7](#ABG)). Compare this to the analogous definition at _[[principal ∞-bundle]]_.
+-- {: .num_remark}
###### Remark
If $X$ is regarded as a [[topological space]] then the corresponding [[discrete ∞-groupoid]] is $\Pi X$, the [[fundamental ∞-groupoid]] of $X$ and the morphism encoding an $K$-module bundle over $X$ is reads
$$
\alpha : \Pi(X) \to A Mod
\,.
$$
This assignment of $A$-modules to points in $X$, of $A$-module morphism to paths in $X$ etc. may be regarded as the [[higher parallel transport]] of the (unique and flat, due to [[discrete ∞-groupoid|discreteness]]) [[connection on an ∞-bundle]] on $\alpha$.
Equivalently, this morphism may be regarded as an [[∞-representation]] of $\Pi(X)$. Notaby if $X = B G$ is the [[classifying space]] of a [[discrete group]] or [[discrete ∞-group]], a $K$-module $\infty$-bundle over $X$ is the same as an [[∞-representation]] of $G$ on $A Mod$.
=--
### $\infty$-Lines and $\infty$-line bundles
+-- {: .num_defn}
###### Definition
Write
$$
A Line \hookrightarrow A Mod
$$
for the [[full sub-(∞,1)-category]] on the _$A$-lines_ : on those $A$-modules that are equivalent to $A$ as an $A$-module. The full subcatgeory of $A Mod(X)$ on morphisms factoring through this inclusion we call the $(\infty,1)$-catgeory of **$A$-line $\infty$-bundles**.
=--
This appears as ([ABG def. 3.12](#ABG)), ([ABGHR 08, 7.5](#ABGHR08)).
+-- {: .num_defn}
###### Definition
Let $A$ be an [[A-∞ algebra|A-∞]] [[ring spectrum]].
For $\Omega^\infty A$ the underlying [[A-∞ space]] and $\pi_0 \Omega^\infty A$ the ordinary [[ring]] of connected components, writ $(\pi_0 \Omega^\infty A)^\times$ for its [[group of units]].
Then the [[∞-group of units]] of $A$ is the [[(∞,1)-pullback]] $GL_1(A)$ in
$$
\array{
GL_1(A) &\to& \Omega^\infty A
\\
\downarrow && \downarrow
\\
(\pi_0 \Omega^\infty A)^\times &\to& \pi_0 \Omega^\infty A
}
\,.
$$
=--
+-- {: .num_prop}
###### Proposition
There is an [[equivalence in an (∞,1)-category|equivalence]] of [[∞-groups]]
$$
GL_1(A) \simeq Aut_{A Line}(A)
$$
of the [[∞-group of units]] of $A$ with the [[automorphism ∞-group]] of $A$, regarded canonically as a module over itself.
Since every $A$-line is by definition equivalent to $A$ as an $A$-module, there is accordingly, an [[equivalence of (∞,1)-categories]], in fact of [[∞-groupoids]]:
$$
A Line \simeq B GL_1(A) \simeq B Aut(A)
$$
that identifies $A Line$ as the [[delooping]] [[∞-groupoid]] of either of these two [[∞-groups]].
=--
This appears in ([ABG, 3.6](#ABG)) ([p. 10](http://arxiv.org/pdf/1002.3004v2.pdf#page=10)). See also ([ABGHR 08, section 6](#ABGHR08)).
+-- {: .num_remark}
###### Remark
This means that every $A$-line $\infty$-bundle is canonically [[associated ∞-bundle|associated]] to a $GL_1(A)$-[[principal ∞-bundle]] over $X$ which is [[moduli space|modulated]] by a map $X \to B GL_1(A)$.
=--
+-- {: .num_defn}
###### Definition
A $GL_1(A)$-[[principal ∞-bundle]] on $X$ is also called a **[[twisted cohomology|twist]]** -- or better: a **[[local coefficient ∞-bundle]]** -- for
$A$-[[generalized (Eilenberg-Steenrod) cohomology|cohomology]] on $X$.
=--
For the moment see _[[twisted cohomology]]_ for more on this.
### Sections and twisted cohomology
{#SectionsAndTwistedCohomology}
+-- {: .num_defn}
###### Definition
The $A$-module of (dual) [[sections]] of an $(\infty,1)$-module bundle $f : X \to A Mod$ is the [[(∞,1)-colimit]] over this functor
$$
X^f := \lim_\to (X \stackrel{\alpha}{\to} A Mod)
\,.
$$
The corresponding _spectrum of sections_ is the $A$-dual
$$
\Gamma(f) := Mod_A(X^f, A)
\,.
$$
=--
This is ([ABG, def. 4.1](#ABG)) and ([ABG, p. 15](#ABG)), ([ABG11, remark 10.16](#ABGII)).
+-- {: .num_remark}
###### Remark
For $f$ an $A$-line bundle $\Gamma(f)$ is called in ([ABGHR 08, def. 7.27, remark 7.28](#ABGHR08)) the **[[Thom spectrum|Thom A-module]]** of $f$ and written $M f$.
=--
Because for $A = S$ the [[sphere spectrum]], $M f$ is indeed the classical [[Thom spectrum]] of the spherical fibration given by $f$:
+-- {: .num_prop}
###### Proposition
For $K = S$ the [[sphere spectrum]], $f : X \to K Line = S Line$ an $S$-line bundle -- hence a spherical fibration, and $A$ any other $\infty$-ring with canonical inclusion $S \to A$, the Thom $A$-module of the composite $X \stackrel{f}{\to} S Mod \to A Mod$ is the classical [[Thom spectrum]] of $f$ tensored with $A$:
$$
\Gamma(X \stackrel{f}{\to} S Line \to A Line \to A Mod)
\simeq
X^f \wedge_S A
\,.
$$
=--
This is ([ABGHR 08, theorem 4.5](#ABGHR08)).
### Trivializations and orientations
+-- {: .num_defn}
###### Definition
For $f : X \to A Line$ an $A$-line $\infty$-bundle, its [[∞-groupoid]] of **trivializations** is the $\infty$-groupoid of lifts
$$
\array{
&& *
\\
& \nearrow & \downarrow
\\
X &\stackrel{f}{\to}& A Line
}
\,.
$$
For $K \to A$ the canonical inclusion and $f : X \to K Line$ a $K$-line bundle, we say that an **$A$-orientation** of $f$ is a trivialization of the associated $A$-line bundle $X \stackrel{f}{\to} K Line \to A Line$.
=--
That this encodes the notion of [[orientation in generalized cohomology|orientation in A-cohomology]] is around
([ABGHR 08, 7.32](#ABGHR08)).
+-- {: .num_cor #ThomModuleInOrientedCase}
###### Corollary
Every trivialization/orientation of an $A$-line $\infty$-bundle $f : X \to A Line$ induces an equivalence
$$
\Gamma(f) \simeq (\Sigma^\infty X )\wedge A
$$
of the $A$-module of sections of $f$ / the [[Thom spectrum|Thom A-module]] of $f$ with the [[homology|generalized A-homology]]-spectrum of $X$:
$$
\pi_\bullet \Gamma(f) \simeq H_\bullet(X,A)
\,.
$$
=--
This appears as ([ABGHR 08, cor. 7.34](#ABGHR08)).
Therefore if $f$ is _not_ trivializable, we may regard its $A$-module of sections as encoding $f$-[[twisted cohomology|twisted A-cohomology]]:
+-- {: .num_defn}
###### Definition
For $f : X \to A Line$ an $A$-line $\infty$-bundle, the $f$-**[[twisted cohomology|twisted A-homology]] of $A$ is
$$
H_\bullet^f(X, A) := \pi_\bullet(\Gamma(f)) := \pi_\bullet(M f)
\,.
$$
The **$f$-[[twisted cohomology|twisted A-cohomology]] is**
$$
H^\bullet_f(X,A) := \pi_0 A Mod(M f, \Sigma^\bullet A)
\,.
$$
=--
## Structured $(\infty,1)$-vector bundles
{#Structured}
We discuss now $(\infty,1)$-vector bundles in more general [[(∞,1)-toposes]].
(...)
## Applications
* The [[string topology]] operations on a compact [[smooth manifold]] $X$ may be understood as arising from a [[sigma-model]] [[quantum field theory]] with [[target space]] $X$ whose [[background gauge field]] is a flat $A$-line $\infty$-bundle $(P,\nabla)$ which is $A$-oriented over $X$, hence trivializabe over $X$ (for instance for $A = H \mathbb{Q}$ the [[Eilenberg-MacLane spectrum]] this may be the sphereical fibration of [[Thom spectrum|Thom spaces]] induced from the [[tangent bundle]] if the manifold is [[oriented]] in the ordinary sense).
By prop. \ref{ThomModuleInOrientedCase} this implies that the space of [[state]]s of the $\sigma$-model is the $A$-homology spectrum $\Gamma(P) \simeq X \edge A$ of $X$, and that for every suitable [[surface]] $\Sigma$ with incoming and outgoing boundary components $\partial_{in} \Gamma \stackrel{in}{\to} \Gamma \stackrel{out}{\leftarrow} \partial_{out} \Gamma$ the [[mapping space]] [[span]]
$$
X^{\partial_{in} \Gamma}
\stackrel{X^{in}}{\leftarrow}
X^{\Gamma}
\stackrel{X^{out}}{\rightarrow}
X^{\partial_{out} \Gamma}
$$
acts by [[path integral as a pull-push transform]] on these spaces of states
$$
(X^{out})_* (X^{in})^! : H_\bullet(X^{\partial_{in} \Gamma},A) \to
H_\bullet(X^{\partial_{out} \Gamma}, A)
\,.
$$
## Related concepts
* [[∞-representation]]
* [[twisted cohomology]]
* [[Picard ∞-stack]]
## References
{#References}
A systematic discussion of discrete $(\infty,1)$-module bundles has a precursor in
* {#AndoHopkinsRezk10} [[Matthew Ando]], [[Michael Hopkins]], [[Charles Rezk]], _Multiplicative orientations of KO-theory and the spectrum of topological modular forms_, 2010 ([pdf](http://www.math.uiuc.edu/~mando/papers/koandtmf.pdf))
(discussing the [[string orientation of tmf]]) and is then discussed in more detail in the triple of articles
* {#ABGHR08} [[Matthew Ando]], [[Andrew Blumberg]], [[David Gepner]], [[Michael Hopkins]], [[Charles Rezk]], _Units of ring spectra and Thom spectra_ ([arXiv:0810.4535](http://arxiv.org/abs/0810.4535))
* {#ABG} [[Matthew Ando]], [[Andrew Blumberg]], [[David Gepner]], _Twists of K-theory and TMF_, in Robert S. Doran, Greg Friedman, [[Jonathan Rosenberg]], _Superstrings, Geometry, Topology, and $C^*$-algebras_, Proceedings of Symposia in Pure Mathematics [vol 81](http://www.ams.org/bookstore-getitem/item=PSPUM-81), American Mathematical Society ([arXiv:1002.3004](http://arxiv.org/abs/1002.3004))
* {#ABGII} [[Matthew Ando]], [[Andrew Blumberg]], [[David Gepner]], _Parametrized spectra, multiplicative Thom spectra, and the twisted Umkehr map_ ([arXiv:1112.2203](http://arxiv.org/abs/1112.2203))
The last of these explains the relation to
* [[Peter May|May]], Sigurdsson, _[[Parametrized Homotopy Theory]]_
A streamlined version of ([ABGHR 08](#ABGHR08)) appears as
* {#ABGHR14} [[Matthew Ando]], [[Andrew Blumberg]], [[David Gepner]], [[Michael Hopkins]], [[Charles Rezk]], _An $\infty$-categorical approach to $R$-line bundles, $R$-module Thom spectra, and twisted $R$-homology_ ([arXiv:1403.4325](http://arxiv.org/abs/1403.4325))
Lecture notes on these articles are in
* Ben Knudsen, Scott Slinker, Paul VanKoughnett, Brian Williams, and [[Dylan Wilson]], _Thom spectra reading course_ ([web](http://www.math.northwestern.edu/~bwill/thom/))
Interpretation of the [[algebraic K-theory]] $K(R)$ of a [[ring spectrum]] $R$ (see at _[[iterated algebraic K-theory]]_) as the [[Grothendieck group]] of [[(∞,1)-module bundles]] over $R$:
* [[John Lind]], _Bundles of spectra and algebraic K-theory_, Pacific J. Math. 285 (2016) 427-452 ([arXiv:1304.5676](https://arxiv.org/abs/1304.5676), [doi:10.2140/pjm.2016.285.427](https://doi.org/10.2140/pjm.2016.285.427))
[[!redirects (∞,1)-vector bundle]]
[[!redirects (infinity,1)-vector bundles]]
[[!redirects (∞,1)-vector bundles]]
[[!redirects (∞,1)-module bundle]]
[[!redirects (∞,1)-module bundles]]
[[!redirects (infinity,1)-module bundle]]
[[!redirects (infinity,1)-module bundles]]
[[!redirects (∞,1)-line]]
[[!redirects (∞,1)-lines]]
[[!redirects (infinity,1)-line]]
[[!redirects (infinity,1)-lines]]
[[!redirects (∞,1)-vector space]]
[[!redirects (∞,1)-vector spaces]]
[[!redirects (infinity,1)-vector space]]
[[!redirects (infinity,1)-vector spaces]]
[[!redirects (∞,1)-line bundle]]
[[!redirects (∞,1)-line bundles]]
[[!redirects (infinity,1)-line bundle]]
[[!redirects (infinity,1)-line bundles]]
[[!redirects ∞-line bundle]]
[[!redirects ∞-line bundles]]
[[!redirects infinity-line bundle]]
[[!redirects infinity-line bundles]] |
(infinity,1)-monad | https://ncatlab.org/nlab/source/%28infinity%2C1%29-monad |
+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Higher algebra
+--{: .hide}
[[!include higher algebra - contents]]
=--
=--
=--
#Contents#
* table of contents
{:toc}
## Idea
The notion of $(\infty,1)$-monad is the [[vertical categorification]] of that of [[monad]] from the context of [[category|categories]] to that of [[(∞,1)-category|(∞,1)-categories]].
They relate to [[(∞,1)-adjunctions]] as [[monads]] relate to [[adjunctions]].
## Properties
### Barr-Beck monadicity theorem
{#MonadicityTheorem}
+-- {: .num_prop #CanonicalMonadicAdjunction}
###### Proposition
Given an [[(∞,1)-monad]] $T$ on an [[(∞,1)-category]]
$\mathcal{C}$, there is an [[(∞,1)-adjunction]]
$$
(F \dashv U)
\;\colon\;
Alg_{\mathcal{C}}(T)
\stackrel{\overset{F}{\leftrightarrow}}{\underset{U}{\longrightarrow}}
\mathcal{C}
\,,
$$
where $Alg_{\mathcal{C}}(T)$ is the (Eilenberg-Moore) [[(∞,1)-category of algebras over an (∞,1)-monad|(∞,1)-category of algebras over the (∞,1)-monad]] and where $U$ is the [[forgetful functor]] that remembers the underlying [[object]] of $\mathcal{C}$.
=--
This appears in ([Riehl-Verity 13, def. 6.1.14](#RiehlVerity13)).
The following is the refinement to [[(∞,1)-category theory]] of the classical [[Barr-Beck monadicity theorem]] which states sufficient conditions for recognizing an [[(∞,1)-adjunction]] as being canonically [[equivalence|equivalent]] to the one in prop. \ref{CanonicalMonadicAdjunction}, hence to be a _[[monadic adjunction]]_.
+-- {: .num_theorem #InfinityBarrBeckTheorem}
###### Theorem
Let $(L \dashv R)$ a pair of [[adjoint (∞,1)-functors]] such that
1. $R$ is a [[conservative (∞,1)-functor]];
1. the [[domain]] [[(∞,1)-category]] of $R$ admits [[geometric realization]] ([[(∞,1)-colimit]]) of [[simplicial objects in an (∞,1)-category|simplicial objects]];
1. and $R$ preserves these
then for $T \coloneqq R \circ L$ the essentially unique $(\infty,1)$-monad structure on the composite endofunctor, there is an [[equivalence of (∞,1)-categories]] identifying the [[domain]] of $R$ with the [[(∞,1)-category of algebras over an (∞,1)-monad]] $Alg_{\mathcal{C}}(T)$ over $T$ and $R$ itself as the canonical [[forgetful functor]] $U$ from prop. \ref{CanonicalMonadicAdjunction}.
=--
This appears as ([[Higher Algebra|Higher Algebra, theorem 6.2.0.6, theorem 6.2.2.5]], [Riehl-Verity 13, section 7](#RiehlVerity13))
### Homotopy coherence
{#HomotopyCoherence}
+-- {: .num_remark}
###### Remark
An [[(∞,1)-adjunction]] $(L \dashv R) \colon \mathcal{C} \leftrightarrow \mathcal{D}$ is uniquely determined already by its image in the [[homotopy 2-category]] ([Riehl-Verity 13, theorem 5.4.14](#RiehlVerity13)). This is not in general true for $(\infty,1)$-monads $T \colon \mathcal{C} \to \mathcal{C}$. As these are [[monoids in an (∞,1)-category]] of [[endomorphisms]], they in general have relevant [[coherence]] data all the way up in degree. However, by the previous statement and the monadicity theorem \ref{InfinityBarrBeckTheorem}, for $(\infty,1)$-monads given via specified [[(∞,1)-adjunctions]] as $T \simeq R \circ L$ are determined by less (further) coherence data ([[Higher Algebra|Higher Algebra, remark 6.2.0.7, prop. 6.2.2.3]], [Riehl-Verity 13, page 6](#RiehlVerity13)). (Of course there is, instead, extra data/information carried by the choice of $\mathcal{D}$.) This should justify the [[simplicial model category]]-theoretic discussion in ([Hess 10](#Hess10)) in [[(∞,1)-category theory]].
=--
## Related concepts
* [[higher monadic descent]]
* [[algebraic theory]] / [[Lawvere theory]] / [[(∞,1)-algebraic theory]]
* [[monad]] / [[2-monad]]/ [[doctrine]] / **$(\infty,1)$-monad**
* [[idempotent (∞,1)-monad]]
* [[modal type theory]]
* [[operad]] / [[(∞,1)-operad]]
## References
A general treatment of $(\infty,1)$-monads in [[(∞,1)-category theory]] is in
* [[Jacob Lurie]], section 3 of _Noncommutative algebra_ ([math.CT/0702299](http://arxiv.org/abs/math/0702299))
later absorbed as
* [[Jacob Lurie]], section 6.2 of _[[Higher Algebra]]_
More explict discussion in terms of [[quasi-categories]] and [[simplicial sets]]:
* {#RiehlVerity13} {#RiehlVerity16} [[Emily Riehl]], [[Dominic Verity]], _Homotopy coherent adjunctions and the formal theory of monads_, Advances in Mathematics, Volume 286, 2 January 2016, Pages 802-888 ([arXiv:1310.8279](http://arxiv.org/abs/1310.8279), [doi:10.1016/j.aim.2015.09.011](https://doi.org/10.1016/j.aim.2015.09.011))
Some homotopy theory of ([[enriched functor|enriched]]) monads on ([[simplicial model category|simplicial]]) [[model categories]] is discussed (with an eye towards [[higher monadic descent]]) in
* {#Hess10} [[Kathryn Hess]], _A general framework for homotopic descent and codescent_ ([arXiv/1001.1556](http://arxiv.org/abs/1001.1556))
[[!redirects (∞,1)-monad]]
[[!redirects (infinity,1)-monads]]
[[!redirects (∞,1)-monads]]
[[!redirects (∞,1)-comonad]]
[[!redirects (∞,1)-comonads]]
[[!redirects (infinity,1)-comonad]]
[[!redirects (infinity,1)-comonads]] |
(infinity,1)-operad | https://ncatlab.org/nlab/source/%28infinity%2C1%29-operad |
+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Higher algebra
+--{: .hide}
[[!include higher algebra - contents]]
=--
#### $(\infty,1)$-Category theory
+--{: .hide}
[[!include quasi-category theory contents]]
=--
=--
=--
#Contents#
* table of contents
{:toc}
## Idea
The notion of _$(\infty,1)$-operad_ is to that of [[(∞,1)-category]] as [[operad]] is to [[category]].
So, roughly, an $(\infty,1)$-operad is an algebraic structure that has for each given type of input and one type of output an [[∞-groupoid]] of operations that take these inputs to that output.
There is a fairly evident notion of [[∞-algebra over an (∞,1)-operad]]s. Examples include
* [[E-∞ algebra]]s
* [[L-∞ algebra]]s;
* [[A-∞ algebra]]s.
$(\infty,1)$-Operads form an [[(∞,2)-category]] [[(∞,1)Operad]].
## Definitions
Two models for $(\infty,1)$-operads exist to date, one by [[Denis-Charles Cisinski|Cisinski]]--[[Ieke Moerdijk|Moerdijk]]--[[Ittay Weiss|Weiss]], the other by [[Jacob Lurie|Lurie]]. It is expected though not yet entirely proved that the two are equivalent ([[Higher Algebra]], draft, Remark 2.0.0.8).
The first one models $(\infty,1)$-operads as [[dendroidal set]]s in close analogy to (in fact as a generalization of how) [[simplicial set]]s model [[(∞,1)-category|(∞,1)-categories]].
The second models the [[(∞,1)-category]] version of a [[category of operators]] of an operad.
### In terms of dendroidal sets
Here [[simplicial set]]s are generalized to [[dendroidal set]]s. The theory of $(\infty,1)$-operads is then formulated in terms of dendroidal sets in close analogy to how the theory of [[(∞,1)-category|(∞,1)-categories]] is formulated in terms of [[simplicial set]]s.
There is a [[model structure on dendroidal sets]] whose fibrant objest are the **quasi-operad**s in direct analogy to the notion of [[quasi-category]].
So the [[model structure on dendroidal sets]] is a [[presentable (∞,1)-category|presention]] of the [[(∞,1)-category]] of $(\infty,1)$-operads. It is [[Quillen equivalence|Quillen equivalent]] to the standard [[model structure on operads]] enriched over [[Top]] or [[sSet]]. Therefore, conversely, the traditional homotopy-theoretic constructions on topological and chain operads (such as cofibrant [[resolution]]s in order to present homtopy algebras such as [[A-∞ algebra]]s, [[L-∞ algebra]]s, [[homotopy BV-algebra]]s and the like) are also indeed presentations of $(\infty,1)$-operads.
### In terms of $(\infty,1)$-categories of operators
{#InTermsOfInfinityCategoriesOfOperators}
Every [[operad]] $A$ encodes and is encoded by its [[category of operators]] $C_A$. In the approach to $(\infty,1)$-operators described below, the notion of category of operators is generalized to an [[(∞,1)-category]] of operators.
In this approach an $(\infty,1)$-operad $C^\otimes$ is regarded as an [[(∞,1)-category]] $C$ -- the unary part of the $(\infty,1)$-operad to be described-- with extra structure that determines [[(∞,1)-functor]]s $C^{\times n} \to C$.
This and the conditions on these are encoded in requiring that $C^\otimes$ is an $(\infty,1)$-functor $C^\otimes \to \Gamma$ over [[Segal Gamma-category|Segal's category]] $\Gamma$ of pointed finite sets, satisfying some conditions.
In particular, any [[symmetric monoidal (∞,1)-category]] yields an example of an $(\infty,1)$-operad in this sense. In fact, symmetric monoidal $(\infty,1)$-categories can be *defined* as $(\infty,1)$-operads such that the functor $C^\otimes \to \Gamma$ is a [[coCartesian fibration]]. (For the moment, see [[monoidal (infinity,1)-category]] for more comments and references on higher operads in this context.)
This is the approach described in ([LurieCommutative](#LurieCommutative))
#### Basic definitions
{#BasicDefinitions}
We are to generalize the following construction from [[categories]] to [[(∞,1)-categories]].
+-- {: .num_defn}
###### Definition
For $\mathcal{O}$ a [[symmetric multicategory]], write $\mathcal{O}^\otimes \to FinSet^{*/}$ for its [[category of operators]].
Here $\mathcal{O}^\otimes$ is the [[category]] whose
* [[objects]] are finite sequences ([[tuples]]) of objects of $\mathcal{O}$;
* [[morphisms]] $(X_1, \cdots, X_{n_1}) \to (Y_1, \cdots, Y_{n_2})$ are given by a morphism $\alpha \colon \langle n_1\rangle \to \langle n_2\rangle$ in $FinSet_*$ together with a collection of [[multimorphisms]]
$$
\left\{
\phi_j \in \mathcal{O}\left( \left\{ X_i\right\}_{i \in \alpha^{-1}\left\{j\right\}} , Y_j \right)
\right\}_{1 \leq j \leq n_2}
\,.
$$
The [[functor]] $p \colon \mathcal{O}^\otimes \to FinSet^{*/}$ is the evident [[forgetful functor]].
=--
In ([Lurie](#Lurie)) this is construction 2.1.1.7.
This motivates the following definition of the generalization of this situation to [[(∞,1)-category theory]].
+-- {: .num_defn #FinSetPointed}
###### Definition
Write $FinSet^{*/}$ for the category of [[pointed object|pointed]] [[finite set]] ([[Segal Gamma-category|Segal's Gamma-category]]).
For $n \in \mathbb{N}$ we write
$$
\langle n\rangle
\coloneqq
{*} \coprod [n]
\in
FinSet^{*/}
$$
for the [[pointed set]] with $n+1$ elements.
A morphism in $FinSet^{*/}$
* is called an **[[inert morphism]]** if it is a surjection, and an [[injection]] on those elements that are not sent to the base point. That is, the preimage of every non-base point is a singleton.
* called an **[[active morphism]]** if only the basepoint goes to the basepoint.
For $n \in \mathbb{N}$ and $1 \leq i \leq n$ write
$$
\rho^i \colon \langle n\rangle \to \langle 1\rangle
$$
for the inert morphism that sends all but the $i$th element to the basepoint.
Notice that for each $n \in \mathbb{N}$ there is a unique [[active morphism]] $\langle n\rangle \to \langle 1\rangle$.
=--
([Lurie, def. 2.1.1.8](#Lurie))
+-- {: .num_defn #InfinityOperadByCategoryOfOperators}
###### Definition
The **$(\infty,1)$-category of operators of an $(\infty,1)$-operad** is a morphism
$$
p \colon \mathcal{O}^\otimes \to FinSet^{*/}
$$
of [[quasi-categories]] such that the following conditions hold:
1. For every [[inert morphism]] in $FinSet^{*/}$ and every [[object]] over it, there is a lift to a $p$-[[coCartesian morphism]] in $\mathcal{O}^\otimes$. In particular, for $f \colon \langle n_1\rangle \to \langle n_2\rangle$ inert, there is an induced [[(∞,1)-functor]]
$$
f_!
\colon
\mathcal{O}^\otimes_{\langle n_1\rangle}
\to
\mathcal{O}^\otimes_{\langle n_2\rangle}
\,.
$$
1. The coCartesian lifts of the inert projection morphisms induce an equivalence of [[derived hom-spaces]] in $\mathcal{O}^{\otimes}$ between maps into multiple objects and the products of the maps into the separete objects:
For $f \colon \langle n_1 \rangle \to \langle n_2 \rangle$ write $\mathcal{O}^\otimes_f(-,-) \hookrightarrow \mathcal{O}^\otimes(-,-)$ for the components of the [[derived hom-space]] covering $f$, then the $(\infty,1)$-functor
$$
\mathcal{O}^\otimes_f(C_1,C_2)
\to
\underset{1 \leq k \leq n_2}{\prod}
\mathcal{O}^\otimes_{\rho^i\circ f}(C_1,(C_2)_i)
$$
induced as above is an [[equivalence of infinity-groupoids|equivalence]].
1. For every finite collection of objects $C_1, \cdots c_n \in \mathcal{O}^\otimes_{\langle 1\rangle}$ there exists a multiobject $C \in \mathcal{O}^\otimes_{\langle n\rangle}$ and a collection of $p$-[[coCartesian morphisms]] $\{C \to C_i\}$ covering $\rho^i$.
Equivalently (given the first two conditions): for all $n \in \mathbb{N}$ the $(\infty,1)$-functors $\{(\rho^i)_!\}_{1 \leq i \leq n}$ induce an [[equivalence of (∞,1)-categories]]
$$
\mathcal{O}^\otimes_{\langle n\rangle}
\to
(\mathcal{O}^\otimes_{\langle 1\rangle})^{\times^n}
$$
=--
([Lurie, def. 2.1.1.10, remark 2.1.1.14](#Lurie))
+-- {: .num_remark }
###### Remark
The conditions in def. \ref{InfinityOperadByCategoryOfOperators} mean that $p \colon \mathcal{O}^\otimes \to FinSet^{*/}$ encodes
1. an [[(∞,1)-category]] $\mathcal{O} \coloneqq \mathcal{O}^\otimes_{\langle 1\rangle}$;
1. for each $n \in \mathbb{N}$ an $n$-ary operation given by the $(\infty,1)$-functor
$$
\mathcal{O}^{n}
=
(\mathcal{O}^\otimes_{\langle 1\rangle})^{\times n}
\simeq
\mathcal{O}^{\otimes}_{\langle n\rangle}
\to
\mathcal{O}^\otimes_{\langle 1\rangle}
=
\mathcal{O}
$$
induced by the unique [[active morphism]] $\langle n\rangle \to \langle 1\rangle$
1. a coherently associative multicomposition of these operations.
=--
+-- {: .num_remark }
###### Remark
The notion of def. \ref{InfinityOperadByCategoryOfOperators} may also be called a **symmetric $(\infty,1)$-multicategory** or **colored $(\infty,1)$-operad**. The _colors_ are the [[objects]] of $\mathcal{O}$.
=--
We now turn to the definition of [[homomorphisms]] of $(\infty,1)$-operads.
+-- {: .num_defn #InertMorphismsInInfinityOperad}
###### Definition
Given an $(\infty,1)$-operad $p \colon \mathcal{O}^\otimes \to FinSet^{*/}$ as in def. \ref{InfinityOperadByCategoryOfOperators}, a [[morphism]] $f$ in $\mathcal{O}^\otimes$ is called an **inert morphism** if
1. $p(f)$ is an [[inert morphism]] in $FinSet^{*/}$ by def. \ref{FinSetPointed};
1. $f$ is a $p$-[[coCartesian morphism]].
=--
+-- {: .num_defn #MorphismOfInfinityOperads}
###### Definition
A **[[morphism of (∞,1)-operads]]** is a map between their [[(∞,1)-categories of operators]] over $FinSet^{*/}$ that preserves the inert morphisms of def. \ref{InertMorphismsInInfinityOperad}.
=--
Morphisms of operads $\mathcal{O}_1 \to \to \mathcal{O}_2$ can be understood equivalently as exhibiting an $\mathcal{O}_1$-[[algebra over an operad|algebra]] in $\mathcal{O}_2$. Therefore:
+-- {: .num_defn }
###### Definition
For $\mathcal{O}_1, \mathcal{O}_2$ to $(\infty,1)$-operads, write
$$
Alg_{\mathcal{O}_1}(\mathcal{O}_2)
\hookrightarrow
qCat_{/FinSet^{*/}}(\mathcal{O}_1, \mathcal{O}_2)
$$
for the full [[sub-(∞,1)-category]] of the [[(∞,1)-functor (∞,1)-category]] on those that are [[morphisms of (∞,1)-operads]] by def. \ref{MorphismOfInfinityOperads}.
=--
([Lurie, def. 2.1.2.7](#Lurie)).
We also have the notion of
* _[[fibration of (∞,1)-operads]]_;
* _[[coCartesian fibration of (∞,1)-operads]]_.
See there for more details.
#### Model for $(\infty,1)$-categories of operators {#ModelForinfOpera}
There is a [[model category]] that [[presentable (infinity,1)-category|presents]] the [[(∞,1)-category]] $(\infty,1)Cat_{Oper}$ of $(\infty,1)$-categories of operations.
+-- {: .num_prop}
###### Proposition
There exists a
* [[proper model category|left proper]]
* [[combinatorial model category|combinatorial]]
[[model category]] $\mathcal{P} Op_{(\infty,1)}$
* whose underlying [[category]] has
* [[object]]s are [[marked simplicial set]] $S$ equipped with a morphism $S \to N(FinSet_*)$ such that marked edges map to inert morphisms in $FinSet_*$ (those for which the preimage of the marked point contains just the marked point)
* [[morphism]]s are morphisms of [[marked simplicial set]]s $S \to T$ such that the triangle
$$
\array{
S &&\to&& T
\\
& \searrow && \swarrow
\\
&& N(FinSet_*)
}
$$
commutes;
* which is canonically an [[SSet]]-[[enriched category]];
* and whose [[model category|model structure]] is given by
* cofibrations are those morphisms whose underlying morphisms of [[simplicial set]]s ate cofibrations, hence [[monomorphism]]s
* weak equivalences are those morphisms $S \to T$ such that for all $A \to N(FinSet_*)$ that are $(\infty,1)$-categories of operations by the above definition, the morphism of [[SSet]]-[[hom object]]s
$$
\mathcal{P}Op_\infty(T,A) \to
\mathcal{P}Op_\infty(S,A)
$$
is a homotopy equivalence of simplicial sets.
* an object is fibrant if and only if it is an $(\infty,1)$-category of operations, by the above definition.
=--
This is prop 1.8 4 in
* [[Jacob Lurie]], _Commutative algebra_ ([pdf](http://www.math.harvard.edu/~lurie/papers/DAG-III.pdf))
## Examples
We list some examples of $(\infty,1)$-operads incarnated as their [[(∞,1)-categories of operators]] by def. \ref{InfinityOperadByCategoryOfOperators}.
The first basic examples to follow are in fact all given by [[1-categories]] [[category of operators|of operators]].
+-- {: .num_example}
###### Example
The [[identity]] functor on $FinSet^{*/}$ exhibits an $(\infty.1)$-operad. This is the **[[commutative operad]]**
$$
Comm^\otimes = FinSet^{*/} \stackrel{id}{\to} FinSet^{*/}
\,.
$$
The [[(∞,1)-algebras over an (∞,1)-operad]] over this $(\infty,1)$-operad are [[E-∞ algebras]].
=--
+-- {: .num_example}
###### Example
The **[[associative operad]]** has $Assoc^\otimes$ the category whose objects are the natural numbers, whose $n$-ary operations
are labeled by the [[total orders]] on $n$ elements, equivalently the elements of the [[symmetric group]] $\Sigma_n$, and whose composition is given by forming consecutive total orders in the obvious way.
The [[(∞,1)-algebras over an (∞,1)-operad]] over this $(\infty,1)$-operad are [[A-∞ algebras]]
=--
In ([Lurie](#Lurie)) this is remark 4.1.1.4.
+-- {: .num_example}
###### Example
The **[[operad for modules over an algebra]]** $LM$ is the [[colored operad|colored]] [[symmetric operad]] whose
* [[objects]] are two elements, to be denoted $\mathfrak{a}$ and $\mathfrak{n}$;
* [[multimorphisms]] $(X_i)_{i = 1}^n \to Y$ form
* if $Y = \mathfrak{a}$ and $X_i = \mathfrak{a}$ for all $i$ then: the set of [[linear orders]] on $n$ elements, equivalently the elements of the [[symmetric group]] $\Sigma_n$;
* if $Y = \mathfrak{n}$ and exactly one of the $X_i = \mathfrak{n}$ then: the set of linear order $\{i_1 \lt \cdots \lt i_n\}$ such that $X_{i_n} = \mathfrak{n}$
* otherwise: the empty set;
* [[composition]] is given by composition of linear orders as for the [[associative operad]].
The [[(∞,1)-algebras over an (∞,1)-operad]] over this $(\infty,1)$-operad are pairs consisting of [[A-∞ algebras]] with [[(∞,1)-modules]] over them.
=--
In ([Lurie](#Lurie)) this appears as def. 4.2.1.1.
+-- {: .num_defn}
###### Definition
The **[[operad for bimodules over algebras]]** $BMod$ is the [[colored operad|colored]] [[symmetric operad]] whose
* [[objects]] are three elements, to be denoted $\mathfrak{a}_-, \mathfrak{a}_+$ and $\mathfrak{n}$;
* [[multimorphisms]] $(X_i)_{i = 1}^n \to Y$ form
* if $Y = \mathfrak{a}_-$ and all $X_i = \mathfrak{a}_-$ then: the set of [[linear orders]] of $n$ elements;
* if $Y = \mathfrak{a}_*$ and all $X_i = \mathfrak{a}_*$ then again: the set of [[linear orders]] of $n$ elements;
* if $Y = \mathfrak{n}$: the set of linear orders $\{i_1 \lt \cdots \lt i_n\}$ such that there is exactly one index $i_k$ with $X_{i_k} = \mathfrak{n}$ and $X_{i_j} = \mathfrak{a}_-$ for all $j \lt k$ and $X_{i_j} = \mathfrak{a}_+$ for all $k \gt k$.
* [[composition]] is given by the composition of linear orders as for the [[associative operad]].
The [[(∞,1)-algebras over an (∞,1)-operad]] over this $(\infty,1)$-operad are pairs consisting of two [[A-∞ algebras]] with an [[(∞,1)-bimodule]] over them.
=--
## Properties
### Relation between the two definitions
{#relation}
+-- {: .standout}
At the time of this writing there is no discussion in "the literature" of the relation between the definition of $(\infty,1)$-operads in terms of dendroidal sets (Cisinski, Moerdijk, Weiss) and $(\infty,1)$-categories of operators (Lurie). The following are some tentative observations. - [[Urs Schreiber|Urs]]
update: meanwhile this has been worked out by some people. Results should appear in preprint form soon.
=--
There is an obvious way to regard a [[tree]] as an $(\infty,1)$-category of operators:
+-- {: .num_defn}
###### Definition
**(dendroidal $(\infty,1)$-category of operators)**
Let
$$
\omega : \Omega \hookrightarrow Op \stackrel{C_{(-)}}{\to}
Cat/FinSet_* \stackrel{N}{\to}
\mathcal{P}Op_{(\infty,1)}
$$
be the dendroidal object given by the following composition:
* $\Omega \hookrightarrow Op$ is the functor from the [[tree category]] $\Omega$ to the category of symmetric colored [[operad]]s (over [[Set]]) that sends a tree to the operad freely generated from it;
* $Op \stackrel{C_{(-)}}{\to} Cat/FinSet_*$ sends an [[operad]] to its [[category of operators]];
* $Cat/FinSet_*
\stackrel{N}{\to} \mathcal{P}Op_{(\infty,1)}$ takes
the [[nerve]] of this category, regarded as
a [[marked simplicial set]] over $N(FinSet_*)$,
whose marked edges are the inert morphisms in
the category of operations.
=--
Following the general pattern of [[nerve and realization]], we get:
+-- {: .num_defn}
###### Definition
**(dendroidal nerve of Lurie-$\infty$-operad)**
The functor
$$
N_d :=
Hom_{\mathcal{P}Op_{(\infty,1)}}(\omega(-), -):
\mathcal{P}Op_{(\infty,1)} \to dSet
$$
that sends a [[marked simplicial set]] $A \to N(FinSet_*)$ to the [[dendroidal set]] which sends a [[tree]] $T$ to the set of morphisms of $\omega(T)$ into $A$
$$
N_d(A) : T \mapsto Hom_{\mathcal{P}Op_{(\infty,1)}}(\omega(T), A)
$$
is the **dendroidal nerve** of $A$.
=--
+-- {: .standout}
One expects that $N_d$ induces a [[Quillen adjunction]] and indeed a [[Quillen equivalence]] between the above model category structure on $\mathcal{P}Op_{(\infty,1)}$ and the [[model structure on dendroidal sets]]. The following is as far as I think I can prove aspects of this. -[[Urs Schreiber|Urs]].
=--
+-- {: .num_prop}
###### Proposition
The dendroidal nerve functor has the following properties:
* it is the [[right adjoint]] of a [[SSet]]-[[enriched functor|enriched]] [[adjunction]]
$$
C_{(-)} : dSet \stackrel{\leftarrow}{\to} \mathcal{P}Op_{(\infty,1)}
: N_d
$$
* it sends fibrant objects to fibrant objects
i.e. it sends $(\infty,1)$-categories of
operations to $(\infty,1)$-operads
in their incarnation as "quasi-operads";
* it sends objects $\pi : A \to N(FinSet_*)$ that come
from grouplike [[symmetric monoidal (infinity,1)-category|symmetric monoidal]] [[∞-groupoid]]s to fully Kan dendroidal sets (that have the extension property with respect to all horns)
* it sends objects $\pi : A \to N(FinSet_*)$ that come
from
[[symmetric monoidal (infinity,1)-category|symmetric monoidal (∞,1)-categories]] to dendroidal sets that have the extension property with respect to at least one outer horn $\Lambda_{v} T$ for $v \in T$ an $n$-corolla, for all $n \in \mathbb{N}$.
* its [[left adjoint]] sends cofibrations to cofibrations and acyclic cofibrations with cofibrant domain to acyclic cofibrations.
=--
+-- {: .proof}
###### Proof
**respect for fibrant objects**. If $A \to N(FinSet_*)$ is fibrant, then in particual $A$ is a [[weak Kan complex]] hence has the extension property with respect to all inner [[horn]] inclusions of [[simplex|simplices]]. We need to show that this implies that $N_d(A)$ has the extension property with respect to all inner horn inclusions of [[tree]]s.
By an (at the moment unpublished) result by [[Ieke Moerdijk|Moerdijk]], right [[lifting property]] with respect to inner horn inclusions of trees is equivalend to right lifting property with respect to inclusions of [[spine]]s of trees: the union over all the corollas in a tree.
For this the extension property means that if we find a collection $\{C_{k_i} \to N_d(A)\} = Sp(T)$ of corollas in $N_d(A)$ that match at some inputs and output, then these can be composed to an image $T \to N_d(A)$ of the corresponding tree $T$ in $N_d(A)$.
An image of $T$ in $N_d(A)$ is an image of $\omega(T)$ in $A$. In the [[category of operators]] $\omega(A)$ every tree may be represented as the composite of a sequence of morphisms each of which consists of precisely one of the corollas $C_{k_i}$ in parallel to identity morphisms. This way gluing the tree from the corollas is a matter of composing a sequence of edges in $A$. But this is guaranteed to be possible if $A$ is a [[weak Kan complex]].
**symmetric monoidal product and outer horn lifting**
As described at [[cartesian morphism]], an edge $f : \Delta^1 \to A$ in $A$ is coCartesian if for all diagrams
$$
\array{
\Delta^{0,1}
\\
\downarrow & \searrow^f
\\
\Lambda^n_0 &\to & A
\\
\downarrow && \downarrow
\\
\Delta^n &\to& N(FinSet_*)
}
$$
of 0-horn lifting problems where the first edge of the horn is $f$ itself, there exists a lift
$$
\array{
\Delta^{0,1}
\\
\downarrow & \searrow^f
\\
\Lambda^n_0 &\to & A
\\
\downarrow &\nearrow & \downarrow
\\
\Delta^n &\to& N(FinSet_*)
}
\,.
$$
For $f$ the parallel application of an $n$-corolla with a collection of identity morphisms this implies that any outer horn $\Lambda_v T \to N_d(A)$ for which the vertex $v : C_n \to N_d(A)$ maps to $f$, the dendroidal set $N_d(A)$ has the extension property with respect to the inclusion $\Lambda_d T \hookrightarrow T$.
**the left adjoint and its respect for cofibrations**
By general nonsense the [[left adjoint]] to $N_d$ is given by the [[coend]]
$$
C_{(-)} : dSet \to \mathcal{P}Op_{(\infty,1)}
$$
$$
C_P = \int^{T \in \Omega} \omega(T) \cdot P(T)
\,,
$$
where in the integrand we have the tautological [[copower|tensoring]] of $\mathcal{P}Op_{(\infty,1)}$ over [[Set]].
Notice that $\omega : \Omega \to \mathcal{P}Op_{(\infty,1)}$ is an [[SSet]]-[[enriched functor]] for the ordinary category $\Omega$ regarded as a simplicially enriched category by the canonical embedding $Set \hookrightarrow SSet$. Therefore this adjunction $F \dashv N_d$ is defined entirely in [[SSet]]-[[enriched category theory]] and hence is a simplicial adjunction.
The [[model structure on dendroidal sets]] has a set of [[cofibrantly generated model category|generating cofibrations]] given by the boundary inclusions of trees. $\partial \Omega[T] \hookrightarrow \Omega[T]$. Tese evidenly map to monomorphisms of underlying simplicial sets under $F$, hence to cofibrations.
For $f : P \hookrightarrow Q$ an acyclic cofibration with cofibrant domain, we need to check that $C_f : C_X \to C_Y$ is a weak equivalence in $\mathcal{P}Op_{(\infty,1)}$. This is by definition the case if for every fibrant object $A$ the morphism
$$
\mathcal{P}Op_{(\infty,1)}(C_Y,A) \to
\mathcal{P}Op_{(\infty,1)}(C_X,A)
$$
is a weak equivalence in the standard [[model structure on simplicial sets]]. By the simplicial adjunction $F \dashv N_d$ this is equivalent to
$$
dSet(f,N_d(A))
:
dSet(Y,N_d(A)) \to
dSet(X,N_d(A))
$$
being a weak equivalence. By the above $N_d(A)$ is fibrant. By section 8.4 of the lecture notes on dendroidal sets cited at [[model structure on dendroidal sets]] a morphism between cofibrant dendroidal sets is a weak equivalence precisely if homming it into any fibrant dendroidal set produces an equivalence of homotopy categories.
Since $f$ is a weak equivalence between cofibrant objects by assumption, it follows that indeed $dSet(f,N_d(A))$ is a weak equivalence for all fibrant $A$.
> (AHM, or does it? there is a prob here, but I need to run now...)
Hence $C_f$ is a weak equivalence.
=--
## Related concepts
* [[table - models for (∞,1)-operads]]
* [[algebraic theory]] / [[Lawvere theory]] / [[(∞,1)-algebraic theory]]
* [[monad]] / [[(∞,1)-monad]]
* [[operad]] / **$(\infty,1)$-operad**, [[model structure on operads]]
* [[morphism of (∞,1)-operads]], [[fibration of (∞,1)-operads]]
* [[generalized (∞,1)-operad]], [[family of (∞,1)-operads]]
* [[algebra over an (∞,1)-operad]], [[model structure on algebras over an operad]]
* [[module over an algebra over an (∞,1)-operad]], [[model structure on modules over an algebra over an operad]]
* [[coherent (∞,1)-operad]]
* [[operadic (∞,1)-Grothendieck construction]]
* [[cohomology of operads]]
## References
The formulation in terms of [[dendroidal sets]] is due to
* [[Ieke Moerdijk]], [[Ittay Weiss]], _Dendroidal sets_ ([web](http://cat.inist.fr/?aModele=afficheN&cpsidt=20087314))
* [[Denis-Charles Cisinski]], [[Ieke Moerdijk]], _Dendroidal sets as models for homotopy operads_ ([arXiv](http://arxiv.org/abs/0902.1954)) .
Here are two blog entries on talks on this stuff:
* [Dendroidal Sets and Infinity-Operads](http://golem.ph.utexas.edu/category/2009/02/dendroidal_sets.html)
* [Moerdijk on Infinity Operads](http://golem.ph.utexas.edu/category/2009/02/moerdijk_on_infinityoperads.html)
The formulation in terms of an $(\infty,1)$-version of the [[category of operators]] is introduced in
* [[Jacob Lurie]], _[[Commutative Algebra]]_ ([pdf](http://www.math.harvard.edu/~lurie/papers/DAG-III.pdf))
{#LurieCommutative}
and further discussed in
* _[[Ek-Algebras]]_ .
Now in section 2 of the textbook
* _[[Higher Algebra]]_
{#Lurie}
The equivalence between the [[dendroidal set]]-formulation and the one in terms of $(\infty,1)$-categories of operators is shown in
* [[Gijs Heuts]], [[Vladimir Hinich]], [[Ieke Moerdijk]], _The equivalence between Lurie's model and the dendroidal model for infinity-operads_ ([arXiv:1305.3658](http://arxiv.org/abs/1305.3658))
Further equivalence to Barwick's complete Segal operads is discussed in
* Hongyi Chu, [[Rune Haugseng]], [[Gijs Heuts]], _Two models for the homotopy theory of $\infty$-operads_ ([arXiv:1606.03826](https://arxiv.org/abs/1606.03826))
For an account in terms of _analytic_ [[monads]], that is, monads that are cartesian (multiplication and unit transformations are cartesian) and the underlying endofunctor preserves sifted colimits and wide pullbacks (or equivalently all weakly contractible limits), see
* [[David Gepner]], [[Rune Haugseng]], [[Joachim Kock]], _∞-Operads as Analytic Monads_, ([arXiv:1712.06469](https://arxiv.org/abs/1712.06469))
On the [[Eckmann-Hilton argument]] for (∞,1)-operads:
* [[Tomer Schlank]], [[Lior Yanovski]], *The ∞-Categorical Eckmann-Hilton Argument*, Algebr. Geom. Topol. 19 (2019) 3119-3170 ([arXiv:1808.06006](https://arxiv.org/abs/1808.06006), [doi:10.2140/agt.2019.19.3119](https://doi.org/10.2140/agt.2019.19.3119))
[[!redirects (∞,1)-operad]]
[[!redirects (∞,1)-operads]]
[[!redirects (infinity,1)-operads]] |
(infinity,1)-presheaf | https://ncatlab.org/nlab/source/%28infinity%2C1%29-presheaf | +-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### $(\infty,1)$-Category theory
+--{: .hide}
[[!include quasi-category theory contents]]
=--
#### $(\infty,1)$-Topos Theory
+--{: .hide}
[[!include (infinity,1)-topos - contents]]
=--
=--
=--
#Contents#
* table of contents
{:toc}
## Definition ##
Write $(\infty,0)Cat$ for [[generalized the|the]] category [[∞Grpd]] of $\infty$-[[infinity-groupoid|groupoids]] regarded as an [[(∞,1)-category]].
Let $S$ be a [[simplicial set]] (which in particular may be a [[quasi-category]]).
An **$(\infty,1)$-presheaf** on $S$ is an [[(∞,1)-functor]]
$$
F : S^{op} \to (\infty,0)Cat
\,.
$$
The [[(infinity,1)-category of (infinity,1)-presheaves|(∞,1)-category of $(\infty,1)$-presheaves]] is the corresponding [[(∞,1)-category of (∞,1)-functors]]
$$
PSh(S) := Fun(S^{op}, (\infty,0)Cat)
\,.
$$
## Remarks ##
* $(\infty,1)$-presheaves are the basic building block for the definition of [[(infinity,1)-category of (infinity,1)-sheaves|(∞,1)-categories of (∞,1)-sheaves]].
## Model structures
$(\infty,1)$-presheaves can be presented by many different [[model categories]], corresponding to several of the model structures for [[(∞,1)-categories]]. These include:
* [[global model structures on simplicial presheaves]] (injective and projective)
* [[model structure for right fibrations]] of [[quasicategories]]
* [[model structure on internal simplicial presheaves]]
* [[model structure for right fibrations of Segal spaces]]
* [[model structure on simplicial presheaves over simplices]]
Various Quillen equivalences between these model structures are constructed in the references. For special cases of the domain $S$ there exist other model structures that are also Quillen equivalent to these, such as:
* The [[Reedy model structure]] on simplicial presheaves, when $S$ is a [[Reedy category]], or more generally a [[generalized Reedy category]]
* The analogous [[model structure on internal inverse diagrams]], when $S$ is an [[inverse EI (∞,1)-category]].
## Related concepts
* [[(0,1)-presheaf]]
* [[presheaf]]
* [[(2,1)-presheaf]]
* **$(\infty,1)$-presheaf**, [[(∞,1)-sheaf]], [[∞-stack]] ,
* [[simplicial presheaf]]
* [[model structure on functors]]
* [[(∞,n)-presheaf]]
## References ##
### $(\infty,1)$-categorical definition
This is in Section 5.1 of
* [[Jacob Lurie]], _[[Higher Topos Theory]]_ .
### Model structures
The various model structures, and their Quillen equivalences, can be found in the following references.
The [[global model structures on simplicial presheaves]] are a standard special case of the [[global model structures on functors]]. The fact that the injective model structure exists is a bit less classical; see [[injective model structure]] for references.
The [[model structure for right fibrations]] of [[quasicategories]] is constructed in [[Higher Topos Theory]]. It is shown there to be Quillen equivalent to the global model structure on simplicial presheaves by a [[straightening functor]]. Alternative proofs of such an equivalence can be found in
* [[Gijs Heuts]], [[Ieke Moerdijk]], *Left fibrations and homotopy colimits II*, [arXiv](http://arxiv.org/abs/1602.01274)
* [[Danny Stevenson]], *Covariant Model Structures and Simplicial Localization*, [arXiv](https://arxiv.org/abs/1512.04815)
The latter also constructs the [[model structure on simplicial presheaves over simplices]] and links it with Quillen equivalences to the other two.
The [[model structure on internal simplicial presheaves]] is constructed in
* [[Geoffroy Horel]], *A model structure on internal categories in simplicial sets*, [TAC](http://tac.mta.ca/tac/volumes/30/20/30-20abs.html)
The [[model structure for right fibrations of Segal spaces]] is constructed in
* Pedro Boavida de Brito, *Segal objects and the Grothendieck construction*, [arXiv](http://arxiv.org/abs/1605.00706)
and shown to be Quillen equivalent to both the [[model structure on internal simplicial presheaves]] and the [[model structure for right fibrations]] of quasicategories.
Finally, the [[model structure on internal inverse diagrams]] is constructed, and shown to be Quillen equivalent to the model structure for internal simplicial presheaves (hence all the others) in
* [[Mike Shulman]], *Univalence for inverse EI diagrams*, [arXiv](http://arxiv.org/abs/1508.02410).
[[!redirects (∞,1)-presheaf]]
[[!redirects (∞,1)-presheaves]]
[[!redirects (infinity,1)-presheaves]]
[[!redirects ∞-presheaf]]
[[!redirects ∞-presheaves]]
[[!redirects ∞-prestack]]
[[!redirects ∞-prestacks]]
[[!redirects (∞,1)-prestack]]
[[!redirects (∞,1)-prestacks]]
[[!redirects infinity-presheaf]]
[[!redirects infinity-presheaves]]
[[!redirects infinity-prestack]]
[[!redirects infinity-prestacks]]
[[!redirects (infinity,1)-prestack]]
[[!redirects (infinity,1)-prestacks]]
|
(infinity,1)-pretopos | https://ncatlab.org/nlab/source/%28infinity%2C1%29-pretopos | [[!redirects (∞,1)-pretopos]]
+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### $(\infty,1)$-Topos Theory
+-- {: .hide}
[[!include (infinity,1)-topos - contents]]
=--
=--
=--
#Contents#
* table of contents
{:toc}
## Idea
The concept of _$(\infty,1)$-pretopos_ ([Lurie, appendix A](#Lurie)) is a version of the concept of _[[pretopos]]_ as one passes from [[toposes]] to [[(∞,1)-toposes]]. The definition is a variant of the characterization of [[Grothendieck (∞,1)-toposes]], via the [Giraud-Rezk-Lurie axioms](infinity-topos#GiraudAxioms), asking only for [[finite (∞,1)-limits]] and [[finite (∞,1)-colimits]] with some exactness properties relating them.
## Definition
+-- {: .num_defn #InfinityPreTopos}
###### Definition
Let $\mathcal{C}$ be an [[(∞,1)-category]]. This is called an _$(\infty,1)$-pretopos_ if
1. $\mathcal{C}$ has a [[terminal object in an (∞,1)-category|terminal object]] and [[homotopy fiber products]];
1. $\mathcal{C}$ has [[finite (∞,1)-colimits]];
1. finite [[coproducts]] in $\mathcal{C}$ are [[universal colimits|universal]] and [[disjoint coproducts|disjoint]];
1. [[groupoid object in an (infinity,1)-category|groupoid objects]] in $\mathcal{C}$ are effective:
1. realization of groupoid objects is [[universal colimit|universal]].
If these conditions hold except possibly for the existence of a [[terminal object in an (∞,1)-category|terminal object]], then $\mathcal{C}$ is a _local $(\infty,1)$-pretopos._
=--
[Lurie, def. A:6.1.1](#Lurie)
## Examples
+-- {: .num_example}
###### Example
Every [[Grothendieck (∞,1)-topos]] is an $(\infty,1)$-pretopos (def. \ref{InfinityPreTopos}).
=--
([Lurie, example A.6.1.5](#Lurie))
+-- {: .num_example }
###### Example
Let $\mathbf{H}$ be a [[Grothendieck (∞,1)-topos]] then the [[full sub-(∞,1)-category]]
$$
\mathbf{H}_{coh} \hookrightarrow \mathbf{H}
$$
on the [[coherent objects]] is a local $(\infty,1)$-pretopos (def. \ref{InfinityPreTopos}).
If moreover $\mathbf{H}$ is an [[coherent (∞,1)-topos]], then $\mathbf{H}_{coh}$ is an $(\infty,1)$-pretopos.
=--
([Lurie, prop. A.6.1.6, cor. 6.1.7](#Lurie))
## Related concepts
* [[elementary (∞,1)-topos]]
## References
* {#Lurie} [[Jacob Lurie]], appendix A of _[[Spectral Algebraic Geometry]]_ ([pdf]( http://math.harvard.edu/~lurie/papers/SAG-rootfile.pdf))
|
(infinity,1)-profunctor | https://ncatlab.org/nlab/source/%28infinity%2C1%29-profunctor |
+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### $(\infty,1)$-Category theory
+--{: .hide}
[[!include quasi-category theory contents]]
=--
=--
=--
#Contents#
* table of content
{:toc}
## Idea
The concept of _$(\infty, 1)$-profunctor_ is the [[categorification]] of that of [[profunctors]] from [[category theory]] to [[(∞,1)-category theory]].
## Definition ##
If $C$ and $D$ are [[(∞,1)-categories]], then a **profunctor** from $C$ to $D$ is a [[(∞,1)-functor]] of the form
$$
F \colon D^{op}\times C \to \infty Grpd
\,.
$$
Such a profunctor is usually written as $F \,\colon\, C ⇸ D$. Composition of (∞,1)-profunctors in [[(∞,1)Prof]] is by the "tensor product of (∞,1)-functors" [[homotopy coend]] construction: if $H\colon A ⇸ B$ and $K\colon B ⇸ C$, their composite is given as a functor $C^{op}\times A \to \infty Grpd$ by
$$(c,a)\mapsto \int^{b\in B} H(b,a)\times K(c,b).$$
Every [[(∞,1)-functor]] $f\colon C\to D$ induces two (∞,1)-profunctors $D(1,f)\colon C ⇸ D$ and $D(f,1)\colon D ⇸ C$, defined by $D(1,f)(d,c) = D(d,f(c))$ and $D(f,1)(c,d) = D(f(c),d)$. (Here $D(-,-)$ denotes the [[hom functor]] of $D$ and $1$ denotes the identity (∞,1)-functor on the respective category.)
Since a profunctor is also known as a (bi)module or a distributor or a correspondence, we should expect other names to be used for $(\infty, 1)$-profunctor. In [[Higher Topos Theory]] (Definition 2.3.1.3), Lurie speaks of a **correspondence**.
## Properties
### Relation to presentable $\infty$-categories
{#RelationToPresentableInfinityCategories}
In analogy to the situation for [[profunctors]] between [[1-categories]] (see [there](profunctor#FuncsOnPresheaves)), $\infty$-profunctors
$$
\mathcal{C} ⇸ \mathcal{D}
$$
are equivalently plain but [[(infinity,1)-colimit|$\infty$-colimit]]-[[preserved limit|preserving]] [[(infinity,1)-functors|$\infty$-functors]]
$$
PSh_\infty(\mathcal{C})
\xrightarrow{\;}
PSh_\infty(\mathcal{D})
$$
between the corresponding [[(infinity,1)-categories of (infinity,1)-presheaves|$\infty$-categories of $\infty$-presheaves]].
In this way [[small (infinity,1)-categories|small $\infty$-categories]] with $\infty$-profunctors between them is a [[full sub-(infinity,1)-category|full sub-$\infty$-category]] of of the $\infty$-category [[Pr(∞,1)Cat]] that of [[presentable (infinity,1)-categories|presentable $\infty$-categories]] with [[cocontinuous functor|cocontinuous]] [[(infinity,1)-functor|$\infty$-functors]] between them.
## Related pages
* [cograph of an (∞,1)-profunctor](cograph+of+a+profunctor#in_category_theory_2)
* [[(∞,1)Prof]]
## References
* [[Emily Riehl]], [[Dominic Verity]], _Kan extensions and the calculus of modules for ∞-categories_, ([arXiv:1507.01460](https://arxiv.org/abs/1507.01460))
* [[Rune Haugseng]], _Bimodules and natural transformations for enriched ∞-categories_, ([arXiv:1506.07341](https://arxiv.org/abs/1506.07341))
[[!redirects (infinity,1)-profunctors]]
[[!redirects (∞,1)-profunctor]]
[[!redirects (∞,1)-profunctors]]
[[!redirects (infinity, 1)-profunctor]]
[[!redirects (infinity, 1)-profunctor]]
[[!redirects (\infty, 1)-profunctor]]
[[!redirects infinity1-profunctor]]
|
(infinity,1)-pullback | https://ncatlab.org/nlab/source/%28infinity%2C1%29-pullback |
+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### $(\infty,1)$-Category theory
+--{: .hide}
[[!include quasi-category theory contents]]
=--
=--
=--
#Contents#
* table of contents
{:toc}
## Idea
An **$(\infty,1)$-pullback** is a [[limit in an (∞,1)-category]] $\mathcal{C}$ over a [[diagram]] of the shape
$$
\{a \to c \leftarrow b\} \to \mathcal{C}
\,.
$$
In other words it is a [[cone]]
$$
\array{
A \times_C B &\to& B
\\
\downarrow &\cong\swArrow& \downarrow
\\
A &\to& C
}
$$
which is [[universal property|universal]] among all such cones in the $(\infty,1)$-categorical sense.
This is the analog in [[(∞,1)-category theory]] of the notion of [[pullback]] in [[category theory]].
## Incarnations
### In quasi-categories
Let $\mathcal{C}$ be a [[quasi-category]]. Recall the notion of [[limit in a quasi-category]].
The non-degenerate cells of the [[simplicial set]] $\Delta[1] \times \Delta[1]$ obtained as the [[cartesian product]] of the simplicial 1-[[simplex]] with itself look like
$$
\array{
(0,0) &\to& (1,0)
\\
\downarrow &\searrow& \downarrow
\\
(0,1) &\to& (1,1)
}
$$
A **square** in a [[quasi-category]] $C$ is an image of this in $C$, i.e. a morphism
$$
s : \Delta[1] \times \Delta[1] \to C
\,.
$$
The simplicial square $\Delta[1]^{\times 2}$ is [[isomorphism|isomorphic]], as a [[simplicial set]], to the [[join of simplicial sets]] of a 2-[[horn]] with the point:
$$
\Delta[1] \times \Delta[1]
\simeq
\{v\} \star \Lambda[2]_2
=
\left(
\array{
v &\to& 1
\\
\downarrow &\searrow& \downarrow
\\
0 &\to& 2
}
\right)
$$
and
$$
\Delta[1] \times \Delta[1]
\simeq
\Lambda[2]_0 \star \{v\}
=
\left(
\array{
0&\to& 1
\\
\downarrow &\searrow& \downarrow
\\
2 &\to& v
}
\right)
\,.
$$
If a square $\Delta[1] \times \Delta[1] \simeq \Lambda[2]_0 \star \{v\} \to C$ exhibits $\{v\} \to C$ as a [[limit in a quasi-category|quasi-categorical limit]] over $F : \Lambda[2]_0 \to C$, we say the limit
$$
v := \lim_\leftarrow F := F(1) \prod_{F(0)} F(2)
$$
is [[generalized the|the]] **quasi-categorical pullback** of the diagram $F$.
#### Pasting law {#QuasiCatPastingLaw}
We have the following quasi-categorical analog of the familiar [pasting law of pullbacks](http://ncatlab.org/nlab/show/pullback#Pasting) in ordinary [[category theory]]:
A [[pasting]] diagram of two squares is a morphism
$$
\sigma : \Delta[2] \times \Delta[1] \to \mathcal{c}
\,.
$$
Schematically this looks like
$$
\array{
a &\to& b &\to& c
\\
\downarrow && \downarrow && \downarrow
\\
d &\to& e &\to& f
}
$$
in $\mathcal{C}$.
+-- {: .un_prop}
###### Proposition
**(pasting law for quasi-categorical pullbacks)**
If the right square is a pullback diagram in $\mathcal{C}$, then the left square is precisely if the outer square is.
=--
This is [[Higher Topos Theory|HTT, lemma 4.4.2.1]]
+-- {: .proof}
###### Proof
Consider the diagram inclusions
$$
\left(
\array{
&& && c
\\
&& && \downarrow
\\
d &\to& e &\to& f
}
\right)
\;\;\to\;\;
\left(
\array{
&& b &\to& c
\\
&& \downarrow && \downarrow
\\
d &\to& e &\to& f
}
\right)
\;\;\leftarrow\;\;
\left(
\array{
&& b
\\
&& \downarrow
\\
d &\to& e
}
\right)
$$
and the induced diagram of [[over quasi-categories]]
$$
\mathcal{C}_{/\sigma(c,d,f)}
\stackrel{\phi}{\leftarrow}
\mathcal{C}_{/\sigma(b,c,d,e,f)}
\stackrel{\psi}{\to}
\mathcal{C}_{/\sigma(b,d,e)}
\,.
$$
Notice that by definition of [[limit in a quasi-category]] the quasi-categorical pullback $\sigma(c) \times_{\sigma(f)} \sigma(d)$ is [[generalized the|the]] [[terminal object in a quasi-category|terminal object]] in $\mathcal{C}_{/\sigma(c,d,f)}$, while $\sigma(d) \times_{\sigma(e)} \sigma(b)$ is the terminal object in $\mathcal{C}_{/\sigma(b,d,e)}$.
The strategy now is to show that both these morphisms $\phi$ and $\psi$ are acyclic [[Kan fibration]]s. That will imply that these terminal objects coincide as objects of $\mathcal{C}$.
First notice that the inclusion
$$
\left(
\array{
&& b
\\
&& \downarrow
\\
d &\to& e
}
\right)
\;\; \to \;\;
\left(
\array{
&& b &\to& c
\\
&& \downarrow && \downarrow
\\
d &\to& e &\to& f
}
\right)
$$
is a [[left anodyne morphism]], being the composite of [[pushout]]s of left [[horn]] inclusions
$$
\begin{aligned}
\left(
\array{
&& b
\\
&& \downarrow
\\
d &\to& e
}
\right)
& \to
\left(
\array{
&& b &\to& c
\\
&& \downarrow &&
\\
d &\to& e
}
\right)
\\
& \to
\left(
\array{
&& b &\to& c
\\
&& \downarrow && \downarrow
\\
d &\to& e && f
}
\right)
\\
& \to
\left(
\array{
&& b &\to& c
\\
&& \downarrow &\searrow& \downarrow
\\
d &\to& e && f
}
\right)
\\
& \to
\left(
\array{
&& b &\to& c
\\
&& \downarrow &\searrow& \downarrow
\\
d &\to& e &\to& f
}
\right)
\end{aligned}
\,.
$$
We could also prove this by showing that this functor is [[homotopy final functor|homotopy initial]] using the characterization in terms of slice categories, and then invoking the theorem of [[HTT]] 4.1.1.3(4) which says (in dual form) that an inclusion of simplicial sets is homotopy initial if and only if it is left anodyne.
One of the <a href="http://ncatlab.org/nlab/show/right%2Fleft+Kan+fibration#PropRightAnodyne">properties of left anodyne morphisms</a> is that restriction of [[over quasi-categories]] along left anodyne morphisms produces an acyclic [[Kan fibration]]. This shows the desired statement for $\psi$.
To see that $\phi$ is also an acyclic fibration observe that
$\phi$ can be factored as
$$
\mathcal{C}_{/\sigma(c,d,f)} \leftarrow
\mathcal{C}_{/\sigma(c,d,e,f)} \leftarrow
\mathcal{C}_{/\sigma(b,c,d,e,f)}
$$
Observe that $\mathcal{C}_{/\sigma(c,d,e,f)}\leftarrow\mathcal{C}_{/\sigma(b,c,d,e,f)}$
fits into a pullback diagram
$$
\array{
\mathcal{C}_{/\sigma(c,d,e,f)}
& \leftarrow &
\mathcal{C}_{/\sigma(b,c,d,e,f)} \\
\downarrow & & \downarrow \\
\mathcal{C}_{/\sigma(c,e,f)}
& \leftarrow &
\mathcal{C}_{/\sigma(b,c,e,f)}
}
$$
and hence is an acyclic Kan fibration since
$\mathcal{C}_{/\sigma(c,e,f)} \leftarrow \mathcal{C}_{/\sigma(b,c,e,f)}$
is one, on account of the fact that the square
$$
\array{
\sigma(b) & \to & \sigma(c) \\
\downarrow & & \downarrow \\
\sigma(e) & \to & \sigma(f)
}
$$
is a pullback in $\mathcal{C}$. Finally,
$\mathcal{C}_{/\sigma(c,d,f)} \leftarrow \mathcal{C}_{/\sigma(c,d,e,f)}$
is a trivial fibration since
$$
\array{
\left(
\array{
& & c \\
& & \downarrow \\
d & \to & f
}
\right) & \to &
\left(
\array{
& & & & c \\
& & & & \downarrow \\
d & \to & e & \to & f
}
\right)
}
$$
is left anodyne; clearly this is a pushout of
$(d\to f)\to (d\to e\to f)$ and so it suffices to show
that $\Delta^{\{0,2\}}\to \Delta^{\{0,1,2\}}$ is left
anodyne. But this map factors as
$\Delta^{\{0,2\}}\to \Lambda^2_0 \to \Delta^{\{0,1,2\}}$
and clearly $\Delta^{\{0,2\}}\to \Lambda^2_0$ is left
anodyne since it is a pushout of $\Delta^{\{0\}}\to
\Delta^{\{0,1\}}$.
=--
### In model categories
* [[homotopy pullback]]
### In derivators
* [[pullback in a derivator]]
## Examples
### Fiber sequence
If $\mathcal{C}$ has a [[terminal object]] and $* \to C \in \mathcal{C}$ is a [[pointed object]], then the **fiber** or **$(\infty,1)$-kernel** of a morphisms $f : B \to C$ is the $(\infty,1)$-pullback
$$
\array{
ker(f) &\to& *
\\
\downarrow && \downarrow
\\
B &\stackrel{f}{\to}& C
}
\,.
$$
For more on this see [[fiber sequence]].
## Related concepts
[[!include notions of pullback -- contents]]
## References
### In homotopy type theory
A formalization of homotopy pullbacks in [[homotopy type theory]] is [[Coq]]-coded in
* [[Guillaume Brunerie]], _[Hott/Coq/Limits/Pullbacks.v](https://github.com/guillaumebrunerie/HoTT/blob/master/Coq/Limits/Pullbacks.v)_
[[!redirects (∞,1)-pullback]]
[[!redirects (infinity,1)-pullbacks]]
[[!redirects (∞,1)-pullbacks]]
[[!redirects (∞,1)-fiber product]]
[[!redirects (∞,1)-fiber products]]
[[!redirects (infinity,1)-fiber product]]
[[!redirects (infinity,1)-fiber products]]
|
(infinity,1)-pushout | https://ncatlab.org/nlab/source/%28infinity%2C1%29-pushout |
The dual notion to _[[(infinity,1)-pullback]]_.
[[!redirects (infinity,1)-pushouts]]
|
(infinity,1)-quantity | https://ncatlab.org/nlab/source/%28infinity%2C1%29-quantity | See [[schreiber:∞-quantity]] for now.
[[!redirects (∞,1)-quantity]] |
(infinity,1)-quasitopos | https://ncatlab.org/nlab/source/%28infinity%2C1%29-quasitopos |
+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### $(\infty,1)$-Topos Theory
+--{: .hide}
[[!include (infinity,1)-topos - contents]]
=--
=--
=--
#Contents#
* table of contents
{:toc}
## Idea
The notion of _$(\infty,1)$-quasitopos is the [[(∞,1)-topos]]-analog of the notion of [[quasitopos]].
## Definition
+-- {: .un_def }
###### Definition
An **(∞,1)-bisite** is an [[(∞,1)-category]] $C$ together with two [[(∞,1)-Grothendieck topologies]], $J$ and $K$ such that $J \subseteq K$.
=--
+-- {: .un_def }
###### Definition
Let $C$ be an [[(∞,1)-bisite]]. Say an [[(∞,1)-presheaf]] $F \in (\infty,1)PSh(C)$ is $\left(J,K\right)$-**biseparated** if it is an [[(∞,1)-sheaf]] for $J$ and for every $K$-covering [[sieve]] $U \to X$ in $C$ we have that the induced morphism
$$
(\infty,1)PSh_C(X,F) \hookrightarrow (\infty,1)PSh_C(U,F)
$$
in [[∞Grpd]] is a [[full and faithful (∞,1)-functor]].
We say it is $n-\left(J,K\right)$-**biseparated** if
the induced morphism
$$
(\infty,1)PSh_C(X,F) \hookrightarrow (\infty,1)PSh_C(U,F)
$$
is an [[n-truncated object in an (∞,1)-category|(n-1)-truncated object]] in the [[over quasi-category|(∞,1)-overcategory]] $\left(\infty-Gpd\right)/(\infty,1)PSh_C(U,F)$.
=--
+-- {: .un_def }
###### Definition
A **(Grothendieck) $(\infty,1)$-quasitopos** is an [[(∞,1)-category]] that is [[equivalence of quasi-categories|equivalent]] to the full [[sub-(∞,1)-category]] of some $(\infty,1)PSh_C$ on the $n-\left(J,K\right)$-**biseparated** $(\infty,1)$-presheaves, on some (∞,1)-bisite $\left(C,J,K\right)$.
=--
## Examples
For $\mathbf{H}$ a [[local (∞,1)-topos]]
$$
\mathbf{H} \stackrel{\stackrel{\overset{Disc}{\leftarrow}}{\underset{\Gamma}{\to}}}{\underset{Codisc}{\leftarrow}}
\infty Grpd
$$
and $C$ be a site of definition for $\mathbf{H}$, the $(\infty,1)$-quasitopos on $C$ that factors the [[geometric embedding]] $Codisc \infty Grpd \hookrightarrow \mathbf{H}$
$$
\infty Grpd
\stackrel{\overset{\Gamma}{\leftarrow}}{\underset{Codisc}{\hookrightarrow}}
Conc(\mathbf{H})
\stackrel{\overset{concretization}{\leftarrow}}{\underset{}{\hookrightarrow}}
\mathbf{H}
$$
is that of **concrete objects** in $\mathbf{H}$, the analog of [[concrete sheaves]].
## Related concepts
* [[quasitopos]]
* **$(\infty,1)$-quasitopos**
## References
The definition as it stands, originated out of a discussion between [[Urs Schreiber]] and [[David Carchedi]]. The suggestion to rephrase the definition in terms of bisites came from [[Mike Shulman]].
[[!redirects (∞,1)-quasitopos]]
[[!redirects (∞,1)-quasitoposes]]
[[!redirects (∞,1)-quasitopoi]]
[[!redirects Grothendieck (∞,1)-quasitopos]]
[[!redirects Grothendieck (∞,1)-quasitoposes]]
[[!redirects Grothendieck (∞,1)-quasitopoi]]
[[!redirects (infinity,1)-quasitoposes]]
[[!redirects (infinity,1)-quasitopoi]]
[[!redirects Grothendieck (infinity,1)-quasitopos]]
[[!redirects Grothendieck (infinity,1)-quasitoposes]]
[[!redirects Grothendieck (infinity,1)-quasitopoi]]
[[!redirects (∞,1)-bisite]]
|
(infinity,1)-semitopos | https://ncatlab.org/nlab/source/%28infinity%2C1%29-semitopos |
+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### $(\infty,1)$-Category theory
+--{: .hide}
[[!include quasi-category theory contents]]
=--
#### $(\infty,1)$-Topos Theory
+--{: .hide}
[[!include (infinity,1)-topos - contents]]
=--
=--
=--
#Contents#
* table of contents
{:toc}
## Definition
+-- {: .num_defn}
###### Definition
An [[(∞,1)-category]] $C$ is an **$(\infty,1)$-semitopos if
1. it is [[presentable (∞,1)-category]];
1. it has [[universal colimits]];
1. for every [[morphism]] the corresponding [[Cech nerve]] [[groupoid object in an (∞,1)-category|groupoid object]] is effective.
=--
This appears as [[Higher Topos Theory|HTT, def. 6.2.3.1]].
## Properties
+-- {: .num_prop}
###### Proposition
If $C$ is an $(\infty,1)$-semitopos and $X \in C$ is any object, then also the [[over-(∞,1)-category]] is an $(\infty,1)$-semitopos.
=--
This appears as ([[Higher Topos Theory|HTT, remark 6.2.3.3]]).
## Examples
* Of course every [[(∞,1)-topos]] is an $(\infty,1)$-semitopos.
## Related concepts
* [[semitopos]]
* [[(∞,1)-topos]]
[[!redirects (∞,1)-semitopos]]
[[!redirects (∞,1)-semitoposes]]
[[!redirects (infinity,1)-semitoposes]] |
(infinity,1)-sheaf | https://ncatlab.org/nlab/source/%28infinity%2C1%29-sheaf |
+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### $(\infty,1)$-Topos Theory
+--{: .hide}
[[!include (infinity,1)-topos - contents]]
=--
#### Locality and descent
+--{: .hide}
[[!include descent and locality - contents]]
=--
=--
=--
#Contents#
* table of contents
{:toc}
## Idea
The notion of **$(\infty,1)$-sheaf** (or _[[∞-stack]]_ or _[[geometric homotopy type]]_) is the analog in [[(∞,1)-category]] theory of the notion of [[sheaf]] ([[geometric type]]) in ordinary [[category theory]].
See [[(∞,1)-category of (∞,1)-sheaves]] for more.
## Definition
Given an [[(∞,1)-site]] $C$, let $S$ be the class of [[monomorphism in an (∞,1)-category|monomorphisms]] in the [[(∞,1)-category of (∞,1)-presheaves]] $PSh_{(\infty,1)}(C)$ that correspond to [[covering]] [[(∞,1)-sieve]]s
$$
\eta : U \hookrightarrow j(c)
$$
on objects $c \in C$, where $j$ is the [[(∞,1)-Yoneda embedding]].
Then an [[(∞,1)-presheaf]] $A \in PSh_{(\infty,1)}(C)$ is an **$(\infty,1)$-sheaf** if it is an $S$-[[local object]]. That is, if for all such $\eta$ the morphism
$$
A(c) \simeq PSh_C(j(c),A) \stackrel{PSh_C(\eta,A)}{\to}
PSh(U,A)
$$
is an [[equivalence in a quasi-category|equivalence]]. For a presheaf $A : C^{\op} \to E$ with values in an arbitrary ∞-category, we say it is a sheaf iff $E(e, A(-))$ is a sheaf for every object $e$ of $E$.
This is the analog of the ordinary sheaf condition for covering sieves. The [[∞-groupoid]] $PSh_C(U,A)$ is also called the [[descent]]-[[∞-groupoid]] of $A$ relative to the covering encoded by $U$.
As in the 1-categorial case, the sheaf condition for a covering sieve can be translated into a condition on a covering family that generates it:
+-- {: .num_defn}
###### Proposition
Let $\{ u_i \to c \}$ be a family of morphisms of $C$ that generate the sieve corresponding to $\eta : U \hookrightarrow j(c)$, and let $r_\bullet : \mathbf{\Delta}^{\op} \to PSh_C$ be the [[groupoid object in an (infinity,1)-category#cech_nerves|Čech nerve]]
of $\amalg_i j(u_i) \to j(c)$.
Then a presheaf $A$ is local with respect to $\eta$ iff the induced map $A(c) \to \lim A(r_\bullet)$ is an equivalence.
=--
Thus, a presheaf $A$ is a sheaf iff every covering sieve contains
a generating family satisfying this condition. Spelling out the description of the Čech nerve, the condition is that we have
$$
A(c) \simeq \lim\left(
\prod_i A(u_i)
\stackrel{\to}{\to}
\prod_{i,j} PSh_C(j(u_i) \times_{j(c)} j(u_j), A)
\stackrel{\to}{\stackrel{\to}{\to}}
\cdots
\right)
$$
If $C$ has pullbacks, this simplifies to
$$
A(c) \simeq \lim\left(
\prod_i A(u_i)
\stackrel{\to}{\to}
\prod_{i,j} A(u_i \times_c u_j)
\stackrel{\to}{\stackrel{\to}{\to}}
\cdots
\right)
$$
and furthermore this formulation applies to presheaves with values
in an arbitrary ∞-category.
+-- {: .proof}
###### Proof
Taking colimits of Čech nerve computes $(-1)$-truncations in $(PSh_C)/j(X)$,
so $\colim(r_\bullet)$ is the subobject of $j(c)$ corresponding
to the sieve $\eta$. We have
$$
PSh_C(\colim(r_\bullet), A) \simeq \lim PSh_C(r_\bullet, A)
$$
and so the theorem follows.
=--
## Terminology
An **($\infty$,1)-sheaf** is also called an [[∞-stack]] with values in [[∞-groupoid]]s.
The practice of writing "$\infty$-sheaf" instead of [[∞-stack]] is a rather reasonable one, since a [[stack]] is nothing but a _2-sheaf_.
Notice however that there is ambiguity in what precisely one may mean by an $\infty$-stack: it can be an $(\infty,1)$-sheaf or more specifically a [[hypercomplete]] $(\infty,1)$-sheaf. This is a distinction that only appears in [[(∞,1)-topos]] theory, not in [[(n,1)-topos]] theory for finite $n$.
## Related concepts
* [[sheaf]]
* [[2-sheaf]] / [[stack]]
* **$(\infty,1)$-sheaf** / [[∞-stack]]
* [[sheaf of spectra]]
* [[sheaf of L-∞ algebras]]
* [[(∞,2)-sheaf]]
* [[(∞,n)-sheaf]]
[[!include homotopy n-types - table]]
## References
Section 6.2.2 in
* [[Jacob Lurie]], _[[Higher Topos Theory]]_
{#Lurie}
[[!redirects (infinity,1)-sheaves]]
[[!redirects infinity-sheaf]]
[[!redirects infinity-sheaves]]
[[!redirects ∞-sheaf]]
[[!redirects ∞-sheaves]]
[[!redirects (∞,1)-sheaf]]
[[!redirects (∞,1)-sheaves]] |
(infinity,1)-sheafification | https://ncatlab.org/nlab/source/%28infinity%2C1%29-sheafification |
+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### $(\infty,1)$-Topos theory
+--{: .hide}
[[!include (infinity,1)-topos - contents]]
=--
=--
=--
#Contents#
* table of contents
{:toc}
## Idea
As for ordinary [[sheafification]], _$(\infty,1)$-sheafification_ is the [[exact (∞,1)-functor|left exact]] [[left adjoint|left]] [[adjoint (∞,1)-functor]] to the inclusion of [[(∞,1)-sheaves]] into [[(∞,1)-presheaves]].
For more details see [[(∞,1)-category of (∞,1)-sheaves]].
## Properties
An $(n,1)$-presheaf can be $(n,1)$-sheafified by
* applying the [[plus-construction on presheaves]] $n+1$ times (see [Lurie, section 6.5.3](#Lurie))
* applying a single "[[hypercover]]-plus-construction" (see [DHI, theorem 7.6](#DHI))
## Related concepts
* [[sheafification]], [[plus-construction on presheaves]]
* **(∞,1)-sheafification** / [[∞-stackification]]
## References
The iterated plus-construction is mentioned in section 6.5.3 of
* [[Jacob Lurie]], _[[Higher Topos Theory]]_
{#Lurie}
The "hypercover-plus construction" is discussed around theorem 7.6 of
* [[Dan Dugger]], [[Sharon Hollander]], [[Dan Isaksen]], _Hypercovers and simplicial presheaves_ ([pdf](http://www.math.uiuc.edu/K-theory/0563/spre.pdf))
{#DHI}
[[!redirects (infinity,1)-sheafifications]]
[[!redirects (∞,1)-sheafification]]
[[!redirects (∞,1)-sheafifications]] |
(infinity,1)-site | https://ncatlab.org/nlab/source/%28infinity%2C1%29-site | +-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### $(\infty,1)$-Topos Theory
+--{: .hide}
[[!include (infinity,1)-topos - contents]]
=--
=--
=--
#Contents#
* table of contents
{:toc}
## Idea
The structure of an $(\infty,1)$-site on an [[(∞,1)-category]] $C$ is precisely the data encoding an [[(∞,1)-category of (∞,1)-sheaves]]
$$
Sh(C) \hookrightarrow PSh(C)
$$
inside the [[(∞,1)-category of (∞,1)-presheaves]] on $C$.
The notion is the analog in [[(∞,1)-category]] theory of the notion of a [[site]] in 1-[[category theory]].
## Definition
The definition of $(\infty,1)$-sites parallels that of 1-categorical [[site]]s closely. In fact the structure of an $(\infty,1)$-site on an $(\infty,1)$-category is equivalent to that of a 1-categorical site on its [[homotopy category of an (infinity,1)-category|homotopy category]] (see below).
+-- {: .un_defn}
###### Definition
**($(\infty,1)$-Grothendieck topology)**
A **[[sieve]] in** an [[(∞,1)-category]] $C$ is a full [[sub-(∞,1)-category]] $D \subset C$ which is closed under precomposition with morphisms in $C$.
A **sieve on** an [[object]] $c \in C$ is a sieve in the [[over quasi-category|overcategory]] $C_{/c}$.
Equivalently, a sieve on $c$ is an [[equivalence class]] of [[monomorphism in an (infinity,1)-category|monomorphisms]] $U \to j(c)$ in the [[(∞,1)-category of (∞,1)-presheaves]] $PSh(C)$, with $j : C \to PSh(C)$ the [[(∞,1)-Yoneda embedding]]. (See below for the proof of this equivalence).
For $S$ a sieve on $c$ and $f : d \to c$ a [[morphism]] into $c$, we take the **pullback sieve** $f^* S$ on $d$ to be that spanned by all those morphisms into $d$ that become equivalent to a morphism in $S$ after postcomposition with $f$.
A **[[Grothendieck topology]]** on the $(\infty,1)$-category $C$ is the specification of a collection of sieves on each object of $C$ -- called the **covering sieves** , subject to the following conditions:
1. _the trivial sieve covers_ -- For each object $c \in C$ the overcategory $C_{/c}$ regarded as a maximal subcategory of itself is a covering sieve on $c$. Equivalently: the monomorphism $Id : j(c) \to j(c)$ covers.
1. _the pullback of a sieve covers_ -- If $S$ is a covering sieve on $c$ and $f : d \to c$ a morphism, then the pullback sieve $f^* S$ is a covering sieve on $d$. Equivalently, the [[pullback]]
$$
\array{
f^* U &\to& U
\\
\downarrow && \downarrow
\\
d &\stackrel{f}{\to}& c
}
$$
in $PSh(C)$ is covering.
1. _a sieve covers if its pullbacks cover_ -- For $S$ a covering sieve on $c$ and $T$ any sieve on $c$, if the pullback sieve $f^* T$ for every $f \in S$ is covering, then $T$ itself is covering.
An $(\infty,1)$-category equipped with a Grothendieck topology is an **$(\infty,1)$-site**.
=--
## Properties
### Of sieves
+-- {: .un_lemma}
###### Lemma
A sieve $S'$ on $c$ that contains a covering sieve $S \subset S'$ is itself covering.
=--
+-- {: .proof}
###### Proof
For every $f : d \to c$ an object of $S \subset C_{/c}$, the pullback sieve
$f^* S'$ equals the pullback sieve $f^* S$. So it covers $d$ by the second axiom on sieves. So by the third axiom $S'$ itself is covering.
=--
+-- {: .un_proposition}
###### Proposition
There is a natural bijection between sieves on $c$ in $C$ and equivalence class of [[monomorphism in an (infinity,1)-category|monomorphisms]] $U \to j(C)$ in $PSh(C)$.
=--
This is [[Higher Topos Theory|HTT, prop. 6.2.2.5]].
+-- {: .proof}
###### Proof
First observe that equivalence classes of $(-1)$-[[truncated]] object of $PSh(C_{/c})$ are in bijection with sieves on $c$:
An $(\infty,1)$-presheaf $F$ is $(-1)$-truncated if its value on any object is either the empty [[∞-groupoid]] $\emptyset$ or a [[contractible]] $\infty$-groupoid. The full subcategory of $C_{/c}$ on those objects on which $F$ takes a contractible value is evidently a sieve (because there is no morphism from a contractible to the empty $\infty$-groupoid). Conversely, given a sieve $S$ on $c$ we obtain a (-1)-truncated presheaf fixed by the demand that it takes the value $* = \Delta[0] \in \infty Grpd$ on those objects that are in $S$, and $\emptyset$ otherwise.
Now, as described at <a href="http://ncatlab.org/nlab/show/(infinity%2C1)-category+of+(infinity%2C1)-presheaves#WithOvercategories">Interaction of presheaves and overcategories</a> we have an equivalence
$$
PSh(C_{/c})
\simeq
PSh(C)_{/j(c)}
\,.
$$
Under this equivalence our bijection above maps to the statement that there is a bijection between sieves on $c$ and equivalence class of $(-1)$-[[truncated]] objects in $PSh(C)_{/j (c)}$. But such a (-1)-truncated object is precisely a [[monomorphism in an (infinity,1)-category|monomorphism]] $U \to j(c)$.
=--
### Of coverages
\begin{lemma}
The set of Grothendieck topologies on an $(\infty,1)$-category $C$ is in natural bijection with the set of Grothendieck topologies on its [[homotopy category of an (infinity,1)-category|homotopy category]].
\end{lemma}
This is [[Higher Topos Theory|HTT, remark 6.2.2.3]].
+-- {: .proof}
###### Proof
Because picking full sub-1-categories as well as full sub-$(\infty,1)$-categories amounts to picking sub-sets/sub-classes of the set of equivalence classes of objects.
=--
+-- {: .un_corollary}
###### Corollary
If the $(\infty,1)$-category $C$ happens to be an ordinary [[category]] (for instance in its incarnation as a [[quasi-category]] it is the [[nerve]] of an ordinary [[category]]), then the structure of an $(\infty,1)$-site on it is the same as the 1-categorical structure of a [[site]] on it.
=--
### Of sites
+-- {: .un_prop}
###### Proposition
Structures of $(\infty,1)$-sites on an [[(∞,1)-category]] $C$ correspond bijectively to [[topological localization]]s of the [[(∞,1)-category of (∞,1)-presheaves]] to a [[(∞,1)-category of (∞,1)-sheaves]]. See there for more details.
=--
## Incarnations and models
If [[(∞,1)-categories]] are incarnated as [[simplicially enriched categories]], then an $(\infty,1)$-site appears as an
* [[sSet-site]]
If $(\infty,1)$-categories are [[presentable (∞,1)-category|presented]] by [[model categories]], then the notion of $(\infty,1)$-site appears as that of
* [[model site]].
## Examples
* The **trivial Grothendieck-topology** on an $(\infty,1)$-category is that where the only covering sieve on each object $c$ is $C_{/c}$ itself. Equivalently, where the only covering monomorphisms $U \to j(c)$ in $PSh(C)$ are the equivalences.
The [[(∞,1)-category of (∞,1)-sheaves]] on this site is just the [[(∞,1)-category of (∞,1)-presheaves]] itself. The localization is an equivalence.
* [[étale (∞,1)-site]]
## Related concepts
* [[site]]
* [[2-site]], [[(2,1)-site]]
* **$(\infty,1)$-site**
* [[model site]], [[simplicial site]]
*[[infinity-cohesive site]]
* [[internal site]] / [[internal (infinity,1)-site]]
## References
* [[Jacob Lurie]], Section 6.2.2 of: _[[Higher Topos Theory]]_ (2009)
* [[Raffael Stenzel]], *Notions of $(\infty,1)$-sites and related formal structures* [[arXiv:2306.06619](https://arxiv.org/abs/2306.06619)]
* [[Raffael Stenzel]], *Higher sites and their higher categorical logic*, talk at *[[Homotopy Type Theory Electronic Seminar Talks|HoTT Electronic Seminar]]* (18 November 2021) [video:[YT](https://www.youtube.com/watch?v=hUJlK8XgZxQ), slides:[pdf](https://www.uwo.ca/math/faculty/kapulkin/seminars/hottestfiles/Stenzel-2021-11-18-HoTTEST.pdf)]
[[!redirects (∞,1)-site]]
[[!redirects (infinity,1)-sites]]
[[!redirects (∞,1)-sites]]
[[!redirects (∞,1)-Grothendieck topology]]
[[!redirects (∞,1)-Grothendieck topologies]] |
(infinity,1)-topos | https://ncatlab.org/nlab/source/%28infinity%2C1%29-topos |
+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### $(\infty,1)$-Category theory
+-- {: .hide}
[[!include quasi-category theory contents]]
=--
#### $(\infty,1)$-Topos Theory
+-- {: .hide}
[[!include (infinity,1)-topos - contents]]
=--
=--
=--
# Contents
* table of contents
{: toc}
## Idea
Recall the following familiar 1-categorical statement:
* Working in the 1-[[category]] [[Set]] of [[0-category|0-categories]] amounts to doing [[set theory]]. The point of [[sheaf topos|sheaf]] [[toposes]] is to pass to _parameterized_ [[0-category|0-categories]], namely [[presheaf]] categories. Although these [[topos|topoi]] behave much like the 1-topos [[Set]], their objects are generalized [[spaces]] that may carry more structure. For instance, a (pre)[[sheaf]] on [[Diff]] is a [[generalized smooth space]].
The idea of $(\infty,1)$-toposes is to generalize the above situation from $1$ to $(\infty,1)$ (recall the notion of [[(n,r)-category]] and see the general discussion at [[∞-topos]]):
* Working in the [[(∞,1)-category]] [[∞Grpd]] of [[infinity-groupoid|(∞,0)-categories]] amounts to doing [[homotopy theory]]. The point of [[(∞,1)-sheaves]] is to pass to _parameterized_ [[(∞,0)-categories]], namely [[(∞,1)-presheaf]] categories. Although these $(\infty,1)$-topoi behave much like the $(\infty,1)$-topos [[∞Grpd]], their objects are generalized [[spaces]] with higher [[homotopies]] that may carry more structure. More generally we have topoi of [[sheaves]], and $(\infty,1)$-topoi of [[(∞,1)-sheaves]]. For instance, an [[∞-Lie groupoid]] is an [[(∞,1)-sheaf]] on [[CartSp]].
## Definition {#Definition}
### As a geometric embedding into a $(\infty,1)$-presheaf category
{#AsAGeometricEmbedding}
Recall that [[sheaf toposes are equivalently the left exact reflective subcategories of presheaf toposes]] and that the inclusion functor is necessarily an [[accessible functor]]. This characterization has the following immediate generalization to a definition in [[(∞,1)-category theory]], where the only subtlety is that accessibility needs to be explicitly required:
+-- {: .num_defn #ToposByLocalization}
###### Definition
A [[Alexander Grothendieck|Grothendieck]]--[[Charles Rezk|Rezk]]--[[Jacob Lurie|Lurie]] **$(\infty,1)$-topos** $\mathbf{H}$ is an [accessible](reflective%20sub-%28infinity,1%29-category#AccessibleReflectiveSubcategory) [left exact](reflective+sub-%28infinity%2C1%29-category#ExactLocalizations) [[reflective sub-(∞,1)-category]] of an [[(∞,1)-category of (∞,1)-presheaves]]
$$
\mathbf{H}
\stackrel{\overset{lex}{\leftarrow}}{\hookrightarrow}
PSh_{(\infty,1)}(C)
\,.
$$
If the above localization is a [[topological localization]] then $\mathbf{H}$ is an **[[(∞,1)-category of (∞,1)-sheaves]]**.
=--
### By Giraud-Rezk-Lurie axioms
{#GiraudAxioms}
Equivalently:
\begin{proposition}
**(Giraud-Rezk-Lurie axioms)**
\linebreak
An $(\infty,1)$-topos $\mathbf{H}$ is
an [[(∞,1)-category]] that satisfies the $(\infty,1)$-category-theoretic analogs of [[Giraud's axioms]]:
* $\mathbf{H}$ is [[presentable (infinity,1)-category|presentable]];
* [[limit in quasi-categories|(∞,1)-colimits]] in $\mathbf{H}$ [[universal colimits|are universal]];
* [[coproducts]] in $\mathbf{H}$ are [[disjoint coproduct|disjoint]];
* every [[groupoid object in an (infinity,1)-category|groupoid object]] in $\mathbf{H}$ is [[groupoid objects in an (∞,1)-topos are effective|effective]] (i.e. has a [[delooping]]).
\end{proposition}
This is part of the statement of [[Higher Topos Theory|HTT, theorem 6.1.0.6]].
This is derived from the following equivalent one:
+-- {: .num_prop #CharacterizationByObjectClassifier}
###### Proposition
An [[(∞,1)-topos]] is
* a [[presentable (∞,1)-category]] with [[universal colimits]]
* that has [[object classifiers]].
=--
+-- {: .num_remark #ReflectonOnCharacterizationByObjectClassifier}
###### Remark
An [[object classifier]] is a (small) _self-reflection_ of the $\infty$-topos inside itself ([[type of types]], internal [[universe]]). See also ([WdL, book 2, section 1](Science+of+Logic#WesenAlsReflexionInIhmSelbst)).
=--
A further equivalent one (essentially by an invocation of the [[adjoint functor theorem]]) is:
+-- {: .num_prop}
###### Proposition
An [[(∞,1)-topos]] is
* a [[presentable (∞,1)-category]]
* in which all colimits are [[van Kampen colimits]].
=--
### Morphisms
A [[morphism]] between $(\infty,1)$-toposes is an [[(∞,1)-geometric morphism]].
The [[(∞,1)-category]] of all $(\infty,1)$-topos is [[(∞,1)Toposes]].
## Types of $(\infty,1)$-toposes
### Topological localizations / $(\infty,1)$-sheaf toposes
for the moment see
* [[topological localization]]
### Hypercomplete $(\infty,1)$-toposes
for the moment see
* [[hypercomplete (∞,1)-topos]]
### Cubical type theory
The Cartesian cubical model of [[cubical type theory]] and [[homotopy type theory]] is [conjectured](https://groups.google.com/d/msg/homotopytypetheory/RQkLWZ_83kQ/s6iazlFdBgAJ) to be an (∞,1)-topos not equivalent to (∞,1)-groupoids.
## Models
Another main theorem about $(\infty,1)$-toposes is that
[[models for ∞-stack (∞,1)-toposes]] are given by the [[model structure on simplicial presheaves]]. See there for details
## Properties
### Global sections geometric morphism
{#GlobalSectionsGeometricMorphism}
Every [[∞-stack]] $(\infty,1)$-topos $\mathbf{H}$ has a canonical [[(∞,1)-geometric morphism]] to the terminal $\infty$-stack $(\infty,1)$-topos [[∞Grpd]]: the [[direct image]] is the [[global section]]s [[(∞,1)-functor]] $\Gamma$, the [[inverse image]] is the [[constant ∞-stack]] functor
$$
(LConst \dashv \Gamma)
\;\colon\;
\mathbf{H}
\underoverset
{\underset{\Gamma}{\longrightarrow}}
{\overset{LConst}{\longleftarrow}}
{\;\;\; \bot \;\;\;}
\infty Grpd
\,.
$$
In fact, this is unique, up to [[equivalence in an (infinity,1)-category|equivalence]]: Since every $\infty$-groupoid is an [[(infinity,1)-colimit|$(\infty,1)$-colimit]] (namely over itself, by [this Prop.](infinity1-limit#EveryInfinityGroupoidIsHomotopyColimitOfConstantFunctorOverItself)) of the [[point]] (hence of the [[terminal object]]), and since the [[inverse image]] $\infty$-functor $LConst$ needs to preserve these $\infty$-colimits (being a [[left adjoint]]) as well as the point (being a [[lex functor]]).
### Closed monoidal structure {#ClosedMonoidalStructure}
+-- {: .num_prop}
###### Proposition
Every $(\infty,1)$-topos is a [[cartesian closed (∞,1)-category]].
=--
+-- {: .proof}
###### Proof
By the fact that every $(\infty,1)$-topos $\mathbf{H}$ has [[universal colimits]] it follows that for every object $X$ the [[(∞,1)-functor]]
$$
X \times (-) : \mathbf{H} \to \mathbf{H}
$$
preserves all [[(∞,1)-colimit]]s. Since every $(\infty,1)$-topos is a [[locally presentable (∞,1)-category]] it follows with the [[adjoint (∞,1)-functor theorem]] that there is a [[right adjoint|right]] [[adjoint (∞,1)-functor]]
$$
(X \times (-) \dashv [X,-]) : \mathbf{H} \stackrel{\overset{X \times (-)}{\leftarrow}}{\underset{[X,-]}{\to}} \mathbf{H}
\,.
$$
=--
+-- {: .num_prop}
###### Proposition
For $C$ an [[(∞,1)-site]] for $\mathbf{H}$ we have that the [[internal hom]] ([[mapping stack]]) $[X,-]$ is given on $A \in \mathbf{H}$ by the [[(∞,1)-sheaf]]
$$
[X,A] \,\colon\, U \mapsto \mathbf{H}(X \times L y(U), A)
\,,
$$
where $y : C \to \mathbf{H}$ is the [[(∞,1)-Yoneda embedding]] and $L : PSh_C \to \mathbf{H}$ denotes [[∞-stackification]].
=--
+-- {: .proof}
###### Proof
The argument is entirely analogous to that of the [[closed monoidal structure on sheaves]].
We use the [[full and faithful (∞,1)-functor|full and faithful]] [[geometric embedding]] $(L \dashv i) : \mathbf{H} \hookrightarrow PSh_C$ and the [[(∞,1)-Yoneda lemma]] to find for all $U \in C$ the value
$$
[X,A](U) \simeq PSh_C(y U, [X,A])
$$
and then the fact that [[∞-stackification]] $L$ is [[left adjoint]] to inclusion to get
$$
\cdots \simeq \mathbf{H}(L y(U), [X,A])
\,.
$$
Then the defining adjunction $(X \times (-) \dashv [X,-])$ gives
$$
\cdots \simeq \mathbf{H}(X \times L y(U) , A)
\,.
$$
=--
[[!include powering of ∞-toposes over ∞-groupoids -- section]]
### Slice-$(\infty,1)$-toposes
+-- {: .num_prop}
###### Proposition
For $\mathbf{H}$ an $(\infty,1)$-topos and $X \in \mathbf{H}$ an object, the [[slice-(∞,1)-category]] $\mathbf{H}_{/X}$ is itself an $(\infty,1)$-topos -- an **[[over-(∞,1)-topos]]**. The projection $\pi_! : \mathbf{H}_{/X} \to \mathbf{H}$ part of an [[essential geometric morphism]]
$$
\pi : \mathbf{H}_{/X}
\stackrel{\overset{\pi_!}{\to}}{\stackrel{\overset{\pi^*}{\leftarrow}}{\underset{\pi_*}{\to}}}
\mathbf{H}
\,.
$$
=--
This is [[Higher Topos Theory|HTT, prop. 6.3.5.1]].
The $(\infty,1)$-topos $\mathbf{H}_{/X}$ could be called the [[gros topos]] of $X$. A [[geometric morphism]] $\mathbf{K} \to \mathbf{H}$ that factors as $\mathbf{K} \xrightarrow{\simeq} \mathbf{H}_{/X} \stackrel{\pi}{\to} \mathbf{H}$ is called an [[etale geometric morphism]].
### Syntax in univalent homotopy type theory
$(\infty,1)$-Toposes provide [[categorical semantics]] for [[homotopy type theory]] with a [[univalence|univalent]] Tarskian [[type of types]] (which inteprets as the [[object classifier]]).
For more on this see at
* _[[homotopytypetheory:model of type theory in an (infinity,1)-topos]]_
* _[relation between type theory and category theory -- Univalent homotopy type theory and infinity-toposes](relation+between+type+theory+and+category+theory#HomotopyWithUnivalence)_
## $(\infty,1)$-Topos theory {#ToposTheory}
Most of the standard constructions in [[topos theory]] have or should have immediate generalizations to the context of $(\infty,1)$-toposes, since all notions of [[category theory]] exist for [[(∞,1)-categories]].
For instance there are evident notions of
* [[geometric morphism]]s between $(\infty,1)$-toposes, such as the [[global section]] geometric morphism to the [[terminal object|terminal]] [[(∞,1)-category of (∞,1)-sheaves|(∞,1)-sheaf]] $(\infty,1)$-topos [[∞Grpd]].
Moreover, it turns out that $(\infty,1)$-toposes come with plenty of internal structures, more than canonically present in an ordinary topos.
Every $(\infty,1)$-topos comes with its intrinsic notion of
* [[cohomology|cohomology in an (∞,1)-topos]]
and with an intrinsic notion of
* [[homotopy groups in an (∞,1)-topos|homotopy in an (∞,1)-topos]].
In classical topos theory, cohomology and homotopy of a topos $E$ are defined in terms of [[simplicial object]]s in $C$. If $E$ is a [[sheaf topos]] with [[site]] $C$ and [[point of a topos|enough point]]s, then this classical construction is secretly really a model for the intrinsic cohomology and homotopy in the above sense of the [[hypercomplete (∞,1)-topos]] of [[∞-stack]]s on $C$.
The beginning of a list of all the structures that exist intrinsically in a big $(\infty,1)$-topos is at
* [[cohesive (∞,1)-topos]].
But **$(\infty,1)$-topos theory** in the style of an $\infty$-analog of the [[Elephant]] is only barely beginning to be conceived.
There are some indications as to what the
* [[internal logic of an (∞,1)-topos]]
should be.
## Related concepts
* [[elementary (∞,1)-topos]], [[(∞,1)-pretopos]]
* [[model topos]]
* [[structured (∞,1)-topos]]
* [[compact topos]], [[coherent (∞,1)-topos]]
* [[category object in an (∞,1)-topos]]
* [[tangent (∞,1)-topos]]
* [[doubly monoidal (∞,1)-topos]]
[[!include flavors of higher toposes -- list]]
[[!include locally presentable categories - table]]
## References
### General
{#ReferencesGeneral}
In retrospect, at least the [[homotopy category of an (∞,1)-category|homotopy categories]] of [[(∞,1)-toposes]] have been known since
* [[Kenneth Brown]], *[[BrownAHT|Abstract homotopy theory and generalized sheaf cohomology]]*, Transactions of the American Mathematical Society, Vol. 186 (1973), 419-458 ([jstor:1996573](http://www.jstor.org/stable/1996573)).
presented there via [[categories of fibrant objects]] among [[simplicial presheaves]]. The enhancement of this to [[model categories]] [[model structure on simplicial presheaves|of simplicial presheaves]] originates wit:h
* [[André Joyal]], Letter to [[Alexander Grothendieck]], 11. 4. 1984, ([pdf scan](http://webusers.imj-prg.fr/~georges.maltsiniotis/ps/lettreJoyal.pdf)).
* {#JardineLecture} [[John F. Jardine]], _Simplicial presheaves_, Journal of Pure and Applied Algebra 47 (1987), 35-87 ([pdf](https://core.ac.uk/download/pdf/82485559.pdf))
A more intrinsic characterization of these "[[model toposes]]" ([Rezk 2010](#Rezk10)) as $\infty$-toposes (the term seems to first appear here in [Simpson 1999](#Simpson99)) is due to:
* {#Simpson99} [[Carlos Simpson]], *A Giraud-type characterization of the simplicial categories associated to closed model categories as $\infty$-pretopoi* ([arXiv:math/9903167](http://arxiv.org/abs/math/9903167))
The generalization of these [[model toposes]] from 1-sites to [[simplicial site|simplicial]] [[model sites]] is due to
* {#ToenVezzosi05} [[Bertrand Toën]], [[Gabriele Vezzosi]], _Homotopical Algebraic Geometry I: Topos theory_, Advances in Mathematics **193** 2 (2005) 257-372 [[arXiv:math.AG/0207028](http://arxiv.org/abs/math.AG/0207028), [doi:10.1016/j.aim.2004.05.004](https://doi.org/10.1016/j.aim.2004.05.004)]
The term *$\infty$-topos* is due to
* {#Lurie03} [[Jacob Lurie]], *On $\infty$-Topoi* ([arXiv:math/0306109](https://arxiv.org/abs/math/0306109))
The term *[[model topos]]* was later coined in:
* {#Rezk10} [[Charles Rezk]], _Toposes and homotopy toposes_, 2010 ([pdf](http://www.math.uiuc.edu/~rezk/homotopy-topos-sketch.pdf), [[Rezk_HomotopyToposes.pdf:file]])
A comprehensive conceptualization and discussion of [[(∞,1)-toposes]] is then due to
* [[Jacob Lurie]], section 6.1 of: _[[Higher Topos Theory]]_, Annals of Mathematics Studies 170, Princeton University Press 2009 ([pup:8957](https://press.princeton.edu/titles/8957.html), [pdf](https://www.math.ias.edu/~lurie/papers/HTT.pdf))
building on [Rezk 2010](#Rezk10). There is is also proven that the Brown-Joyal-Jardine-Toën-Vezzosi models indeed precisely model $\infty$-stack $(\infty,1)$-toposes. Details on this relation are at [[models for ∞-stack (∞,1)-toposes]].
Overview:
* [[Denis-Charles Cisinski]], _Catégories supérieures et théorie des topos_, Séminaire Bourbaki, 21.3.2015, [pdf](http://www.math.univ-toulouse.fr/~dcisinsk/1097.pdf).
* {#Rezk19} [[Charles Rezk]], _Lectures on Higher Topos Theory_, Leeds (2019) [[pdf](https://rezk.web.illinois.edu/leeds-lectures-2019.pdf), [[RezkHigherToposTheory2019.pdf:file]]]
A useful collection of facts of [[simplicial homotopy theory]] and [[(infinity,1)-topos theory]] is in
* [[Zhen Lin Low]], _[[Notes on homotopical algebra]]_
A quick introduction to the topic is in
* [[André Joyal]], _[[A crash course in topos theory -- The big picture]]_, lecture series at [Topos à l'IHES](https://indico.math.cnrs.fr/event/747/), November 2015, Paris
On $\infty$-toposes [[category object in an (infinity,1)-category|internal to]] other $\infty$-toposes;
* [[Louis Martini]], [[Sebastian Wolf]], *Internal higher topos theory* [[arXiv:2303.06437](https://arxiv.org/abs/2303.06437)]
### Giraud-Rezk-Lurie axioms
A discussion of the $(\infty,1)$-[[universal colimits]] in terms of [[model category]] presentations is due to
* [[Charles Rezk]], _Fibrations and homotopy colimits of simplicial sheaves_ ([pdf](http://www.math.uiuc.edu/~rezk/rezk-sharp-maps.pdf))
More on this with an eye on [[associated ∞-bundles]] is in
* [[Matthias Wendt]], _Classifying spaces and fibrations of simplicial sheaves_ ([arXiv](http://arxiv.org/abs/1009.2930))
### Homotopy type theory
Proof that all [[∞-stack]] [[(∞,1)-topos]] have [[presentable (∞,1)-category|presentations]] by [[model categories]] which interpret (provide [[categorical semantics]]) for [[homotopy type theory]] with [[univalence|univalent]] [[type universes]]:
* {#Shulman19} [[Michael Shulman]], _All $(\infty,1)$-toposes have strict univalent universes_ ([arXiv:1904.07004](https://arxiv.org/abs/1904.07004)).
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|
(infinity,1)-topos - contents | https://ncatlab.org/nlab/source/%28infinity%2C1%29-topos+-+contents |
**[[(∞,1)-topos theory]]**
## Background {#sidebar_background}
* [[sheaf and topos theory]]
* [[(∞,1)-category]]
* [[(∞,1)-functor]]
* [[(∞,1)-presheaf]]
* [[(∞,1)-category of (∞,1)-presheaves]]
## Definitions {#sidebar_definitions}
* [[elementary (∞,1)-topos]]
* [[(∞,1)-site]]
* [[reflective sub-(∞,1)-category]]
* [[localization of an (∞,1)-category]]
* [[topological localization]]
* [[hypercompletion]]
* [[(∞,1)-category of (∞,1)-sheaves]]
* [[(∞,1)-sheaf]]/[[∞-stack]]/[[derived stack]]
* [[(∞,1)-topos]]
* [[(n,1)-topos]], [[n-topos]]
* [[truncated|n-truncated object]]
* [[connected|n-connected object]]
* [[topos|(1,1)-topos]]
* [[presheaf]]
* [[sheaf]]
* [[(2,1)-topos]], [[2-topos]]
* [[(2,1)-presheaf]]
* [[(∞,1)-quasitopos]]
* [[separated (∞,1)-presheaf]]
* [[quasitopos]]
* [[separated presheaf]]
* [[(2,1)-quasitopos]]
* [[separated (2,1)-presheaf]]
* [[(∞,2)-topos]]
* [[(∞,n)-topos]]
## Characterization {#sidebar_characterization}
* [[universal colimits]]
* [[object classifier]]
* [[groupoid object in an (∞,1)-category|groupoid object in an (∞,1)-topos]]
* [[effective epimorphism]]
## Morphisms {#sidebar_morphisms}
* [[(∞,1)-geometric morphism]]
* [[(∞,1)Topos]]
* [[Lawvere distribution]]
## Extra stuff, structure and property {#sidebar_extra}
* [[hypercomplete (∞,1)-topos]]
* [[hypercomplete object]]
* [[Whitehead theorem]]
* [[over-(∞,1)-topos]]
* [[n-localic (∞,1)-topos]]
* [[locally n-connected (n,1)-topos]]
* [[structured (∞,1)-topos]]
* [[geometry (for structured (∞,1)-toposes)]]
* [[locally ∞-connected (∞,1)-topos]], [[∞-connected (∞,1)-topos]]
* [[local (∞,1)-topos]]
* [[concrete (∞,1)-sheaf]]
* [[cohesive (∞,1)-topos]]
## Models {#sidebar_models}
* [[models for ∞-stack (∞,1)-toposes]]
* [[model category]]
* [[model structure on functors]]
* [[model site]]/[[sSet-site]]
* [[model structure on simplicial presheaves]]
* [[descent for simplicial presheaves]]
* [[Verity on descent for strict omega-groupoid valued presheaves|descent for presheaves with values in strict ∞-groupoids]]
## Constructions {#sidebar_constructions}
**structures in a [[cohesive (∞,1)-topos]]**
* [[shape of an (∞,1)-topos|shape]] / [[coshape of an (∞,1)-topos|coshape]]
* [[cohomology]]
* [[homotopy groups in an (∞,1)-topos|homotopy]]
* [[fundamental ∞-groupoid in a locally ∞-connected (∞,1)-topos]]/[[fundamental ∞-groupoid of a locally ∞-connected (∞,1)-topos|of a locally ∞-connected (∞,1)-topos]]
* [[categorical homotopy groups in an (∞,1)-topos|categorical]]/[[geometric homotopy groups in an (∞,1)-topos|geometric]] homotopy groups
* [[Postnikov tower in an (∞,1)-category|Postnikov tower]]
* [[Whitehead tower in an (∞,1)-topos|Whitehead tower]]
* [[rational homotopy theory in an (∞,1)-topos|rational homotopy]]
* [[dimension]]
* [[homotopy dimension]]
* [[cohomological dimension]]
* [[covering dimension]]
* [[Heyting dimension]]
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+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### $(\infty,1)$-Category theory
+--{: .hide}
[[!include quasi-category theory contents]]
=--
#### $(\infty,1)$-Topos Theory
+--{: .hide}
[[!include (infinity,1)-topos - contents]]
=--
=--
=--
#Contents#
* table of contents
{:toc}
## Idea
The theory of [[(∞,1)-toposes]], generalizing [[topos theory]] from [[category theory]] to [[(∞,1)-category theory]]: "[[geometric homotopy theory]]".
## Related concepts
* [[topos theory]]
* [[2-topos theory]]
* **$(\infty,1)$-topos theory**
* [[higher topos theory]]
* [[Awodey's conjecture]]
## References
An quick introduction is in part 3, 4 of
* [[André Joyal]], _[[A crash course in topos theory -- The big picture]]_, lecture series at [Topos à l'IHES](https://indico.math.cnrs.fr/event/747/), November 2015, Paris
For origins of the notion of $(\infty,1)$-topos itself see the references at [[(∞,1)-topos]].
Early frameworks for [[Grothendieck topos|Grothendieck]] (as opposed to "elementary") $(\infty,1)$-topoi are due [[Charles Rezk]] via [[model categories]]
* {#Rezk10} [[Charles Rezk]], _Toposes and homotopy toposes_, 2010 ([pdf](http://www.math.uiuc.edu/~rezk/homotopy-topos-sketch.pdf))
and due to [[Bertrand Toen|Toën]]--[[Gabriele Vezzosi|Vezzosi]] in two versions (preprints 2002), via [[simplicially enriched category|simplically enriched categories]] and via [[Segal categories]]:
* [[Bertrand Toën]], [[Gabriele Vezzosi]], _Homotopical algebraic geometry. I. Topos theory_, Adv. Math. 193 (2005) no. 2, 257--372 [doi](https://doi.org/10.1016/j.aim.2004.05.004) [arXiv:math.AT/0207028](http://arxiv.org/abs/math/0207028)
* [[Bertrand Toën]] , [[Gabriele Vezzosi]], _Segal topoi and stacks over Segal categories_, [arXiv:math.AG/0212330](http://arXiv.org/abs/math/0212330)
A general abstract conception of $(\infty,1)$-topos theory in terms of [[(∞,1)-category theory]] was given in
* [[Jacob Lurie]], _[[Higher Topos Theory]]_ .
The analog of the [[Elephant]] for $(\infty,1)$-topos theory is still to be written. |
(infinity,1)-Yoneda extension | https://ncatlab.org/nlab/source/%28infinity%2C1%29-Yoneda+extension |
#Contents#
* automatic table of contents goes here
{:toc}
## Idea ##
In ordinary [[category theory]] the [[Yoneda extension]] of a [[functor]] $F : C \to D$ is its left [[Kan extension]] through the [[Yoneda embedding]] of its domain to a functor $\hat F : PSh(C) \to D$.
In [[higher category theory]] there should be a corresponding version of this construction.
In particular with categories replaced by [[(∞,1)-category|(∞,1)-catgeories]] there should be a version with the category of [[presheaf|presheaves]] replaced by a [[(∞,1)-category of (∞,1)-presheaves]], corresponding to the [[Yoneda lemma for (∞,1)-categories]].
This in turn should have a [[presentable (infinity,1)-category|presentation]] in terms of the global [[model structure on simplicial presheaves]].
## Statement ##
+-- {: .standout}
[[Urs Schreiber]]: this here is something I thought about. Check. Even to the extent that this is right, it is clearly not yet a full answer, but at best a step in the right direction.
=--
Let $C$ be a [[category]] and write $[C^{op}, SSet]_{proj} = SPSh(C)_{proj}$ for the projective [[nLab:model structure on simplicial presheaves|model structure on simplicial presheaves]] on $C$. Let $\mathbf{D}$ be any [[combinatorial simplicial model category]].
+-- {: .un_proposition }
###### Proposition
If $F$ takes values in cofibrant objects of $\mathbf{D}$ then the [[SSet]]-[[enriched category theory|enriched]] [[Yoneda extension]] $\hat F$ of $F$ is the [[nLab:left adjoint|left adjoint]] part of an [[SSet]]-[[Quillen adjunction]]
$$
\hat F : SPSh(C)_{proj}
\stackrel{\leftarrow}{\to}
\mathbf{D}
:
R
\,.
$$
=--
Accordingly, if $F$ does not take values in cofibrant objects but where a cofibrant replacement functor $Q : \mathbf{D} \to \mathbf{D}$ is given, the [[Yoneda extension]] $\widehat{Q F}$ of $Q F$ is an $(\infty,1)$-extension up to weak equivalence of $F$.
### Proof ###
We prove this in two steps.
+-- {: .num_lemma}
###### Lemma
The [[Yoneda extension]] $F : SPSh(C)_{proj} \to \mathbf{D}$ preserves cofibrations and acyclic cofibrations.
=--
+-- {: .proof}
###### Proof
Recall that the [[Yoneda extension]] of $F : SPSh(C)_{proj} \to \mathbf{D}$ is given by the [[nLab:coend|coend]] formula
$$
\hat F : X \mapsto \int^{U \in C} F(U) \cdot X(U)
\,,
$$
where in the integrand we have the [[copower|tensoring]] of the object $F(U) \in \mathbf{D}$ by the [[simplicial set]] $X(U)$.
The lemma now rests on the fact that this [[coend]] over the [[copower|tensor]]
$$
\int (-)\cdot (-)
:
[C,\mathbf{D}]_{inj}
\cdot
[C^{op}, SSet]_{proj}
\to
\mathbf{D}
$$
is a [[Quillen bifunctor]] using the injective and projective [[global model structure on functors]] as indicated. This is [[Higher Topos Theory|HTT prop. A.2.9.26 & rmk. A.2.9.27]] and recalled at [[Quillen bifunctor]].
Since by assumption $F(U)$ is cofibrant for all $U$ we have that $\hat F$ itself is cofibrant regarded as an object of $[C,\mathbf{D}]_{inj}$. From the definition of [[Quillen bifunctor]]s it follows that
$$
\hat F = \int^U F(U) \cdot (-)(U) : SPSh(C)_{proj} \to \mathbf{D}
$$
preserves cofibrations and acyclic cofibrations.
=--
+-- {: .un_lemma }
###### Lemma
The functor $\hat F$ has an [[enriched functor|enriched]] [[right adjoint]]
$$
R : \mathbf{D} \to \mathrm{SPSh}(C)
$$
given by
$$
R(A)
=
\mathbf{D}(F(-), A)
\,.
$$
=--
+-- {: .proof}
###### Proof
This is a standard argument.
We demonstrate the Hom-isomorphism that characterizes the [[adjunction]]:
Start with the above [[coend]] description of $\hat F$
$$
\mathbf{D}({\hat F}(X), A)
\simeq
\mathbf{D}(
\int^{U \in S}
F(U) \cdot X(U)
,
A
)
\,.
$$
Then use the continuity of the enriched Hom-functor to pass it through the coend and obtain the following [[end]]:
$$
\cdots \simeq
\int_{U \in S}
\mathbf{D}({\hat F}(U) \cdot X(U), A)
\,.
$$
The defining property of the tensoring operation
implies that this is equivalent to
$$
\simeq
\int_{U \in S}
SSet( X(U), \mathbf{D}(F(U),A))
\,.
$$
But this is the [[enriched functor category|end-formula]] for the $SSet$-object of
[[natural transformation]]s between [[simplicial presheaf|simplicial presheaves]]:
$$
\cdots
\simeq
[C^{op},SSet](X, \mathbf{D}(\Pi(-), A))
\,.
$$
By definition this is the desired right hand of the hom isomorphism
$$
\cdots
= [C^{op}, SSet](X, R(A))
\,.
$$
=--
These two lemmas together constitute the proof of the proposition.
[[!redirects (∞,1)-Yoneda extension]] |
(infinity,1)Cat | https://ncatlab.org/nlab/source/%28infinity%2C1%29Cat |
+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### $(\infty,1)$-Category theory
+--{: .hide}
[[!include quasi-category theory contents]]
=--
#### [[categories of categories - contents|categories of categories]]
+-- {: .hide}
[[!include categories of categories - contents]]
=--
=--
=--
**$(\infty,1)Cat$** is the [[(∞,2)-category]] of all small [[(∞,1)-categories]].
Its full [[subcategory]] on [[∞-groupoid]]s is [[∞Grpd]].
#Contents#
* table of contents
{:toc}
## The $(\infty,2)$-category
### As an $SSet$-category
One incarnation of [[(∞,2)-categories]] is given by [[quasi-category]]-enriched categories (see there for details). As such $(\infty,1)Cat$ is the full [[SSet]]-[[enriched category|enriched]] [[subcategory]] of [[SSet]] on those [[simplicial set]]s that are [[quasi-categories]]. By the fact described at [[(∞,1)-category of (∞,1)-functors]] this is indeed a [[quasi-category]]-enriched category.
### As an enriched model category
The [[model category]] presenting this [[(∞,2)-category]] is the Joyal [[model structure for quasi-categories]] $sSet_{Joyal}$. Its full [[sSet]]-[[subcategory]] is the [[quasi-category]] enriched category of quasi-categories from above.
## The $(\infty,1)$-category
Sometimes it is useful to consider inside the full $(\infty,2)$-catgeory of $(\infty,1)$-categories just the maximal $(\infty,1)$-category and discarding all non-invertible [[k-morphism|2-morphisms]]. This is the [[(∞,1)-category of (∞,1)-categories]].
### As an $SSet$-category
As an [[SSet]]-[[enriched category]] the [[(∞,1)-category of (∞,1)-categories]] is obtained from the quasi-category-enriched version by picking in each [[hom-object]] simplicial set of $(\infty,1)Cat$ the maximal [[Kan complex]].
### As an enriched model category
One [[model category]] structure presenting this is the [[model structure on marked simplicial over-sets|model structure on marked simplicial sets]]. As a plain [[model category]] this is [[Quillen equivalence|Quillen equivalent]] to $sSet_{Joyal}$, but as an [[enriched model category]] it is $sSet_{Quillen}$ enriched, so that its full [[SSet]]-subcategory on fibrant-cofibrant objects presents the $(\infty,1)$-category of $(\infty,1)$-categories.
### Properties
#### Limits and colimits in $(\infty,1)$Cat
{#LimitsAndColimits}
[[limit in a quasi-category|Limits and colimits]] over a [[(∞,1)-functor]] with values in $(\infty,1)Cat$ may be reformulation in terms of the [[universal fibration of (infinity,1)-categories]] $Z \to (\infty,1)Cat^{op}$
Then let $X$ be any [[(∞,1)-category]] and
$$
F : X \to (\infty,1)Cat
$$
an [[(∞,1)-functor]]. Recall that the [[Cartesian fibration|coCartesian fibration]] $E_F \to X$ classified by $F$ is the pullback of the [[universal fibration of (∞,1)-categories]] $Z$ along F:
$$
\array{
E_F &\to& Z
\\
\downarrow && \downarrow
\\
X &\stackrel{F}{\to}& (\infty,1)Cat
}
$$
+-- {: .un_prop }
###### Proposition
Let the assumptions be as above. Then:
* The colimit of $F$ is equivalent to $E_F$:
$$
E_F \simeq colim F
$$
* The limit of $F$ is equivalent to the [[(infinity,1)-category of cartesian section]] of $E_F \to X$
$$
\Gamma_X(E_F) \simeq lim F
\,.
$$
=--
+-- {: .proof}
###### Proof
This is [[Higher Topos Theory|HTT, section 3.3]].
=--
#### Automorphisms
{#Automorphisms}
+-- {: .un_theorem }
###### Theorem
The full subcategory of the [[(∞,1)-category of (∞,1)-categories]] $Func((\infty,1)Cat, (\infty,1)Cat)$ on those [[(∞,1)-functor]]s that are equivalences is equivalent to $\{Id, op\}$: it contains only the identity functor and the one that sends an $(\infty,1)$-category to its [[opposite (infinity,1)-category]].
=--
+-- {: .proof}
###### Proof
This is due to
* [[Bertrand Toen]], _Vers une axiomatisation de la théorie des catégories supérieures_ , K-theory 34 (2005), no. 3,
233-263.
It appears as [[Higher Topos Theory|HTT, theorem 5.2.9.1]] ([arxiv v4+](http://arxiv.org/abs/math.CT/0608040) only)
First of all the statement is true for the ordinary category of [[poset]]s. This is [prop. 5.2.9.14](http://arxiv.org/PS_cache/math/pdf/0608/0608040v4.pdf#page=311).
From this the statement is deduced for $(\infty,1)$
-categories by observing that posets are characterized by the fact that two parallel functors into them that are objectwise equivalent are already equivalent, [prop. 5.2.9.11](http://arxiv.org/PS_cache/math/pdf/0608/0608040v4.pdf#page=310), which means that posets $C$ are characterized by the fact that
$$
\pi_0 (\infty,1)Cat(D,C)
\to
Hom_{Set}(
\pi_0 (\infty,1)Cat(*,D)
,
\pi_0 (\infty,1)Cat(*,C)
)
$$
is an injection for all $D \in (\infty,1)Cat$.
This is preserved under automorphisms of $(\infty,1)Cat$, hence any such automorphism preserves posets, hence restricts to an automorphism of the category of posets, hence must be either the identity or $(-)^{op}$ there, by the above statement for posets.
Now finally the main point of the proof is to see that the linear posets $\Delta \subset (\infty,1)Cat$ are [[dense functor|dense]] in $(\infty,1)Cat$, i.e. that the identity transformation of the inclusion functor $\Delta \hookrightarrow (\infty,1)Cat$ exhibits $Id_{(\infty,1)Cat}$ as the left [[Kan extension]]
$$
\array{
\Delta &\hookrightarrow& (\infty,1)Cat
\\
\downarrow & \nearrow_{Lan = \mathrlap{Id}}
\\
(\infty,1)Cat
}
\,.
$$
=--
## Presentations
[[!include table - models for (infinity,1)-operads]]
## Related concepts
* [[formal (infinity,1)-category theory|formal $(\infty,1)$-category theory]]
* [[Pos]]
* [[Set]]
* [[Grpd]], [[∞Grpd]]
* [[Cat]], [[Operad]]
* [[2Cat]]
* **$(\infty,1)$Cat**, [[(∞,1)Operad]]
* [[homotopy 2-category of (∞,1)-categories]]
* [[(∞,n)Cat]]
* [[(infinity, 1)Prof]]
category: category
[[!redirects (∞,1)Cat]]
[[!redirects (∞,1)Categories]]
[[!redirects (infinity,1)Categories]]
[[!redirects (infinity,2)-category of (infinity,1)-categories]]
|
(infinity,1)Prof | https://ncatlab.org/nlab/source/%28infinity%2C1%29Prof | [[!redirects (infinity, 1)Prof]]
#Contents#
* table of contents
{:toc}
## Definition
$\mathbf{(\infty, 1)Prof}$ is the [[(∞,2)-category]] of [[(∞,1)-categories]], [[(\infty, 1)-profunctor|(∞,1)-profunctors]], and [[natural transformations]].
Recall that a (∞,1)-profunctor from $A$ to $B$ is a (∞,1)-functor $B^{op}\times A\to \infty Grpd$. Composition of (∞,1)-profunctors in $(\infty, 1)Prof$ is by the "tensor product of (∞,1)-functors" [[homotopy coend]] construction: if $H\colon A ⇸ B$ and $K\colon B ⇸ C$, their composite is given as a functor $C^{op}\times A \to \infty Grpd$ by
$$(c,a)\mapsto \int^{b\in B} H(b,a)\times K(c,b).$$
The identity on an (∞,1)-category $A$ is its hom-functor $Hom_A(-,-)$.
## Properties
Note that every (∞,1)-functor $f\colon A\to B$ gives two [[representable functor|representable]] (∞,1)-profunctors $B(f-,-)$ and $B(-,f-)$. This defines two [[(∞,2)-functors]] $(\infty,1)Cat \to (\infty,1)Prof$ that are the identity on objects. The relationship between [[(∞,1)Cat]] and $(\infty,1)Prof$ encoded in this way makes them into an $(\infty, 1)$-version of an [[equipment]], see ([Haugseng 15](#Haugseng15)).
$(\infty, 1)Prof$ can act as a classifying object for kinds of (∞,1)-functors; see [higher exponentiable functor](Conduché+functor#highercategorical_versions).
$(\infty,1)Prof$ is equivalent to the full subcategory of [[Pr(∞,1)Cat]] whose objects are presheaf categories.
## Related concepts
* [[Prof]]
* [[Pr(∞,1)Cat]]
##References
* {#Haugseng15} [[Rune Haugseng]], _Bimodules and natural transformations for enriched ∞-categories_, ([arXiv:1506.07341](https://arxiv.org/abs/1506.07341))
[[!redirects (∞,1)Prof]] |
(infinity,1)Topos | https://ncatlab.org/nlab/source/%28infinity%2C1%29Topos |
+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### $(\infty,1)$-Topos Theory
+--{: .hide}
[[!include (infinity,1)-topos - contents]]
=--
=--
=--
#Contents#
* table of contents
{:toc}
## Idea
By **$(\infty,1)Topos$** is denoted the collection of all [[(∞,1)-topos]]es. This is the [[(∞,1)-category]]-theory analog of [[Topos]].
## Definition
**$(\infty,1)\,Topos$** is the (non-full) [[sub-(∞,1)-category]] of [[(∞,1)Cat]] on [[(∞,1)-toposes]] and [[(∞,1)-geometric morphisms]] between them.
Morally, this should actually be an [[(∞,2)-category]], just as [[Topos]] is a [[2-category]], but since the technology of $(\infty,2)$-categories is not well developed, this point of view is not often taken yet.
## Properties
In the following, $(\infty,1)Cat$ will refer to the (superlarge)
$(\infty,1)$ category of _large_ $(\infty, 1)$-categories.
### Existence of sites of definition
{#SitesOfDefinition}
(...)
### Existence of limits and colimits
{#ExistenceOfLimitsAndColimits}
We discuss existence of [[(∞,1)-limit]]s and [[(∞,1)-colimit]]s in $(\infty,1)Topos$.
+-- {: .num_prop}
###### Proposition
The $(\infty,1)$-category $(\infty,1)Topos$ has all small $(\infty,1)$-colimits and functor
$$
(\infty,1)Topos^{op} \to (\infty,1)Cat
$$
preserves small limits.
=--
This is [[Higher Topos Theory|HTT, prop. 6.3.2.3]]. In the notation there, $LTop$ is the $(\infty,1)$-category of toposes whose arrows are the inverse image morphisms, and thus opposite to $(\infty,1)Topos$.
+-- {: .num_prop}
###### Proposition
The $(\infty,1)$-category $(\infty,1)Topos$ has [[filtered (infinity,1)-category|filtered]] [[(∞,1)-limit]]s and the inclusion
$$
(\infty,1)Topos \to (\infty,1)Cat
$$
preserves these.
=--
This is [[Higher Topos Theory|HTT, prop. 6.3.3.1]].
+-- {: .num_prop}
###### Proposition
The $(\infty,1)$-category $(\infty,1)Topos$ has all small [[(∞,1)-limit]]s.
=--
This is [[Higher Topos Theory|HTT, prop. 6.3.4.7]].
+-- {: .num_remark}
###### Remark
The $(\infty,1)$-limits in $(\infty,1)Topos$ coincide actually with the proper $(\infty,2)$-limits.
=--
This is [[Higher Topos Theory|HTT, remark 6.3.4.10]].
+-- {: .num_prop}
###### Proposition
The [[terminal object in an (infinity,1)-category|terminal object]] in
[[(∞,1)Topos]] is [[∞Groupoids]].
=--
This is [[Higher Topos Theory|HTT, Prop. 6.3.4.1]], for details see at *[[terminal geometric morphism]]*.
### Computation of limits and colimits
{#ComputationOfLimitsAndColimits}
We discuss more or less explicit descriptions of
[[(∞,1)-limits]] and [[(∞,1)-colimits]] in $(\infty,1)Topos$.
+-- {: .num_prop}
###### Proposition
Let
$$
\array{
&& \mathcal{X}
\\
&& \downarrow^{\mathrlap{(g^* \dashv g_*)}}
\\
\mathcal{Y}
&\stackrel{(f^* \dashv f_*)}{\to}&
\mathcal{Z}
}
$$
be a [[diagram]] of [[(∞,1)-toposes]]. Then its [[(∞,1)-pullback]] is computed, roughly, by the [[(∞,1)-pushout]] of their [[(∞,1)-sites]] of definition (see [above](#SitesOfDefinition)).
More in detail: there exist [[(∞,1)-sites]] $\tilde \mathcal{D}$, $\mathcal{D}$, and $\mathcal{C}$ with [[finite limit|finite]] [[(∞,1)-limit]] and [[morphisms of sites]]
$$
\array{
&& \mathcal{D}
\\
&& \uparrow^{\mathrlap{g}}
\\
\tilde \mathcal{D}
&\stackrel{f}{\leftarrow}&
\mathcal{C}
}
$$
such that
$$
\left(
\array{
&& \mathcal{X}
\\
&& \downarrow^{\mathrlap{(g^* \dashv g_*)}}
\\
\mathcal{Y}
&\stackrel{(f^* \dashv f_*)}{\to}&
\mathcal{Z}
}
\right)
\,\,\,
\simeq
\,\,\,
\left(
\array{
&& Sh_{(\infty,1)}(\mathcal{D})
\\
&& \downarrow^{\mathrlap{(Lan_g \dashv (-)\circ g)}}
\\
Sh_{(\infty,1)}(\tilde \mathcal{D})
&\stackrel{(Lan_f \dashv (-)\circ f)}{\to}&
Sh_{(\infty,1)}(\mathcal{C})
}
\right)
\,.
$$
Let then
$$
\array{
\tilde \mathcal{D} \coprod_{\mathcal{C}} \mathcal{D} &\stackrel{f'}{\leftarrow}& \mathcal{D}
\\
{}^{\mathllap{g'}}\uparrow &\swArrow_{\simeq}& \uparrow^{\mathrlap{g}}
\\
\tilde \mathcal{D}
&\stackrel{f}{\leftarrow}&
\mathcal{C}
}
\,\,\,\,\,
\in
(\infty,1)Cat^{lex}
$$
be the [[(∞,1)-pushout]] of the underlying [[(∞,1)-categories]] in the [[full sub-(∞,1)-category]] [[(∞,1)Cat]]${}^{lex} \subset (\infty,1)Cat$ of $(\infty,1)$-categories with finite $(\infty,1)$-limits.
Let moreover
$$
Sh_{(\infty,1)}(\tilde \mathcal{D} \coprod_{\mathcal{C}} \mathcal{D})
\hookrightarrow
PSh_{(\infty,1)}(\tilde \mathcal{D} \coprod_{\mathcal{C}} \mathcal{D})
$$
be the [[reflective sub-(∞,1)-category]] obtained by [[localization]] at the class of morphisms generated by the inverse image $Lan_{f'}(-)$ of the [[covering]]s of $\mathcal{D}$ and the inverse image $Lan_{g'}(-)$ of the coverings of $\tilde \mathcal{D}$.
Then
$$
\array{
Sh_{(\infty,1)}(\tilde \mathcal{D} \coprod_{\mathcal{C}} \mathcal{D})
&\to& \mathcal{X}
\\
\downarrow &\swArrow_{\simeq}& \downarrow^{\mathrlap{(g^* \dashv g_*)}}
\\
\mathcal{Y}
&\stackrel{(f^* \dashv f_*)}{\to}&
\mathcal{Z}
}
$$
is an [[(∞,1)-pullback]] square.
=--
This is [[Higher Topos Theory|HTT, prop. 6.3.4.6]].
### Colimits of over-toposes
{#ColimitsOfOverToposes}
+-- {: .num_prop}
###### Proposition
For any $(\infty,1)$ topos $\mathbf{H}$, there is a colimit preserving functor $\mathbf{H} \to (\infty,1)Topos$ sending an object $X$ to its over-topos $\mathbf{H}_{/X}$, and sending an arrow $f : X \to Y$ to the essential geometric morphism $f_* : \mathbf{H}_{/X} \to \mathbf{H}_{/Y}$.
=--
+-- {: .proof}
###### Proof
[[Higher Topos Theory|HTT, Prop. 6.1.3.9]] implies the "terminal vertex" Cartesian fibration $\mathbf{H}^{[1]} \to \mathbf{H}$ is classified by a limit preserving functor $\mathbf{H}^{op} \to Pr^L$, the $(\infty,1)$-category of locally presentable $(\infty,1)$-categories and colimit-preserving functors between them.
This functor factors through the subcategory $(\infty,1)Topos^{op} \subseteq Pr^L$ that sends a geometric morphism to its inverse image part. By [HTT, Prop. 6.3.2.3] and [HTT, Prop. 5.5.1.13], it follows that this is also a limit preserving functor.
The opposite category is then formed by taking right adjoints.
=--
## Related concepts
* [[Topos]]
* **$(\infty,1)$Topos**
## References
section 6.3 in
* [[Jacob Lurie]], _[[Higher Topos Theory]]_
{#Lurie}
category: category
[[!redirects (infinity,1)Topos]]
[[!redirects (infinity,1)Topoi]]
[[!redirects (infinity,1)Toposes]]
[[!redirects (infinity,1)-Topos]]
[[!redirects (infinity,1)-Topoi]]
[[!redirects (infinity,1)-Toposes]]
[[!redirects (∞,1)Topos]]
[[!redirects (∞,1)Topoi]]
[[!redirects (∞,1)Toposes]]
[[!redirects (∞,1)-Topos]]
[[!redirects (∞,1)-Topoi]]
[[!redirects (∞,1)-Toposes]]
[[!redirects (infinity,1)Toposes]]
|
(infinity,2)-Categories and the Goodwillie Calculus | https://ncatlab.org/nlab/source/%28infinity%2C2%29-Categories+and+the+Goodwillie+Calculus | [[!redirects (∞,2)-Categories and the Goodwillie Calculus]]
this entry is about the article
* [[Jacob Lurie]], _$(\infty,2)$-categories and the Goodwillie calculus_ ([pdf](http://www.math.harvard.edu/~lurie/papers/GoodwillieI.pdf), [arXiv:0905.0462](http://arxiv.org/abs/0905.0462))
on
1. [[(∞,1)-category|(∞,1)-categories]] of [[(∞,n)-categories]] via [[category objects in an (∞,1)-category]] in general and
1. of [[(∞,2)-categories]] in particular.
1. and about [[Goodwillie calculus]].
category: reference
[[!redirects (∞,2)-Categories and the Goodwillie Calculus]]
|
(infinity,2)-category | https://ncatlab.org/nlab/source/%28infinity%2C2%29-category |
+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Higher category theory
+--{: .hide}
[[!include higher category theory - contents]]
=--
=--
=--
#Contents#
* table of contents
{:toc}
## Idea
An __$(\infty,2)$-category__ is the special case of $(\infty,n)$-[[(infinity,n)-category|category]] for $n=2$.
It is best known now through a [[geometric definition of higher category]].
Models include:
* the definition by [[Carlos Simpson]] and Tamsamani;
* the definition in terms of [[n-fold Segal spaces]];
* a definition in terms of scaled simplicial sets, following Verity's [[simplicial model for weak omega-categories]] by Jacob Lurie (see reference below)
See also the list of all definitions of higher categories at [[(∞,n)-category]].
## Properties
### Models for the $(\infty,1)$-category of $(\infty,2)$-categories {#TotMod}
In _[[(∞,2)-Categories and the Goodwillie Calculus]]_ [[Jacob Lurie]] discusses a variety of [[model category]] structures, all [[Quillen equivalence|Quillen equivalent]], that all model the [[(∞,2)-category]] of $(\infty,1)$-categories, in generalization of the standard model category models for [[(∞,1)-category|(∞,1)-categories]] themselves (see there for details).
Recall that
* the standard [[model structure on simplicial sets]] models [[∞-groupoid]]s.
A [[simplicial model category|simplicially]] [[enriched model category]] with respect to the standard model structure on simplicial sets hence models [[∞Grpd]]-enriched categories, hence [[(∞,1)-category|(∞,1)-categories]].
Along this pattern $(\infty,2)$-categories should be modeled by [[enriched category|categories enriched]] in the [[model structure on simplicial sets|Joyal model structure]] that models the [[(∞,1)-category of (∞,1)-categories]].
Write $SSet^J$ for [[SSet]] equipped with the Joyal model structure. Then, indeed, there is a diagram of [[Quillen equivalence]]s of [[model category]] structures
$$
SSet^J Cat \to SSet SegSp \to [\Delta^{op}, SSet^J]
$$
between Joyal-$SSet$-enriched categories, Joyal-$SSet$-enriched [[complete Segal space]]s and simplicial Joyal-simplicial sets.
This is [remark 0.0.4, page 5](http://arxiv.org/PS_cache/arxiv/pdf/0905/0905.0462v2.pdf#page=5) of the article.
There are many more models. See there for more.
## Examples
Classes of examples include
* [[(∞,2)-toposes]]
* For $\mathcal{C}$ a suitable [[monoidal (∞,1)-category]] there is the $(\infty,2)$-category $Mod(\mathcal{C})$ of $\infty$-algebras and $\infty$-bimodules in $\mathcal{C}$. See at _[bimodule - Properties - The (∞,2)-category of ∞-algebras and ∞-bimodules](bimodule#Infinity2CategoryOfInfinityAlgebrasAndBimodules)_.
## Related concepts
* [[0-category]], [[(0,1)-category]]
* [[category]]
* [[2-category]]
* [[3-category]]
* [[n-category]]
* [[(∞,0)-category]]
* [[(∞,1)-category]]
* **(∞,2)-category**
* [[(∞,n)-category]]
* [[(n,r)-category]]
##References
* [[Jacob Lurie]], _[[(∞,2)-Categories and the Goodwillie Calculus]]_
[[!redirects (infinity,2)-categories]]
[[!redirects (∞,2)-category]]
[[!redirects (∞,2)-categories]] |
(infinity,2)-sheaf | https://ncatlab.org/nlab/source/%28infinity%2C2%29-sheaf |
+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Higher category theory
+--{: .hide}
[[!include higher category theory - contents]]
=--
#### Higher topos Theory
+--{: .hide}
[[!include (infinity,1)-topos - contents]]
=--
=--
=--
#Contents#
* table of contents
{:toc}
## Idea
An _$(\infty,2)$-sheaf_ or _$(\infty,2)$-stack_ is the higher analog of an [[(∞,1)-sheaf]] / [[∞-stack]].
For $\mathcal{C}$ an [[(∞,1)-category]] equipped with the structure of an [[(∞,1)-site]], an $(\infty,2)$-sheaf on $\mathcal{C}$ is an [[(∞,1)-functor]]
$$
X : \mathcal{C}^{op} \to Cat_{(\infty,1)}
$$
to [[(∞,1)Cat]], that satisfies [[descent]]: hence which is a [[local object]] with respect to the [[covering]] [[sieve]] inclusions in $Func(\mathcal{C}^{op}, Cat_{(\infty,1)})$.
The [[(∞,2)-category]] of $(\infty,2)$-sheaves
$$
Sh_{(\infty,2)}(\mathcal{C})
$$
is an [[(∞,2)-topos]], the [[homotopy theory]]-generalization of a [[2-topos]] of [[2-sheaves]].
## Examples
### Codomain fibration / canonical $(\infty,2)$-sheaf
{#CodomainFibration}
Let $\mathcal{X}$ be an [[(∞,1)-topos]], regarded as a ([[large site|large]]) [[(∞,1)-site]] equipped with the [[canonical topology]]. Then an [[(∞,1)-functor]]
$$
A : \mathcal{X}^{op} \to CAT_{(\infty,1)}
$$
is an $(\infty,2)$-sheaf precisely if it preserves [[(∞,1)-limits]] (takes [[(∞,1)-colimits]] in $\mathcal{X}$ to [[(∞,1)-limits]] in [[(∞,1)Cat]]).
+-- {: .num_prop }
###### Propositon
For $\mathcal{X}$ an $(\infty,1)$-topos, the functor
$$
Cod : \mathcal{X}^{op} \to CAT_{(\infty,1)}
$$
$$
Cod : A \mapsto \mathcal{X}_{/A}
$$
is a ([[universe enlargement|large]]) $(\infty,2)$-sheaf on $\mathcal{X}$ , regarded as a [[(∞,1)-site]] equipped with the [[canonical topology]]. Here $\mathcal{X}_{/A}$ is the [[slice (∞,1)-topos]] over $A$.
=--
This is a special case of ([Lurie, lemma 6.1.3.7](#Lurie)).
+-- {: .num_remark }
###### Remark
The functor $Cod$ [[(∞,1)-Grothendieck construction|classifies]] the [[codomain fibration]]. Its fiberwise [[stabilization]] to the [[tangent (∞,1)-category]] is the $(\infty,2)$-sheaf of [[quasicoherent sheaves]] on $\mathcal{X}$.
=--
## Related concepts
* [[sheaf]]
* [[2-sheaf]], [[(2,1)-sheaf]]
* [[(∞,1)-sheaf]] / [[∞-stack]]
* **$(\infty,2)$-sheaf**
* [[(∞,n)-sheaf]]
## References
* {#Lurie} [[Jacob Lurie]], section 6.1.3 of _[[Higher Topos Theory]]_
Discussion of a local [[model structure on simplicial presheaves]] $[S^op, sSet_{Joyal}]_{loc}$ with respect to the [[Joyal model structure]] $sSet_{Joyal}$ for [[quasicategories]] is in
* {#Meadows15} [[Nicholas Meadows]], _The Local Joyal Model Structure_ ([arXiv:1507.08723](http://arxiv.org/abs/1507.08723))
and with respect to the [[model structure for complete Segal spaces]] in
* [[Nicholas Meadows]], _Local Complete Segal Spaces_ ([arXiv:1607.05794](http://arxiv.org/abs/1607.05794))
[[!redirects (∞,2)-sheaf]]
[[!redirects (∞,2)-sheaves]]
[[!redirects (infinity,2)-sheaves]]
|
(infinity,2)-site | https://ncatlab.org/nlab/source/%28infinity%2C2%29-site | [[!redirects (∞,2)-site]]
+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### $(\infty,2)$-Category theory
+--{: .hide}
[[!include 2-category theory - contents]]
=--
#### $(\infty,2)$-Topos theory
+--{: .hide}
[[!include (infinity,2)-topos theory - contents]]
=--
=--
=--
An [[(n,r)-site]] for $n = \infty$, $r = 2$.
[[!redirects (infinity,2)-sites]]
|
(infinity,2)-topos | https://ncatlab.org/nlab/source/%28infinity%2C2%29-topos |
+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Higher category theory
+--{: .hide}
[[!include higher category theory - contents]]
=--
#### Higher topos Theory
+--{: .hide}
[[!include (infinity,1)-topos - contents]]
=--
=--
=--
#Contents#
* table of contents
{:toc}
## Idea
An [[(∞,2)-category]] of [[(∞,2)-sheaves]]. The joint higher generalization of the notion of [[(∞,1)-topos]] and [[2-topos]].
The archetypical example is the [[(infinity,2)-category of (infinity,1)-categories]] $Cat_{(\infty,1)}$. See also at *[[formal category theory]]*.
## Related concepts
* [[formal (infinity,1)-category theory|formal $(\infty,1)$-category theory]]
* [[directed homotopy type theory]]
[[!include flavors of higher toposes -- list]]
## References
Discussion of, potentially, the [[internal language]] of $(\infty,2)$-toposes as a form of [[directed homotopy type theory]]:
* {#CisinskiEtAl23} [[Denis-Charles Cisinski]], [[Hoang Kim Nguyen]], Tashi Walde: *Univalent Directed Type Theory*, lecture series in the *[CMU Homotopy Type Theory Seminar](https://www.cmu.edu/dietrich/philosophy/hott/seminars/index.html)* (13, 20, 27 Mar 2023) [[web](https://www.cmu.edu/dietrich/philosophy/hott/seminars/index.html#230313)]
> **Abstract:** We will introduce a version of [[dependent type theory]] that is suitable to develop a [[synthetic mathematics|synthetic]] theory of [[1-categories]]. The [[axioms]] are both a [[fragment]] and an extension of ordinary [[dependent type theory]]. The [[axioms]] are chosen so that [[(∞,1)-category theory]] (in the form of [[quasi-categories]] or [[complete Segal spaces]]) gives a [[categorical semantics|semantic interpretation]], in a way which extends [[Vladimir Voevodsky|Voevodsky]]'s interpretation of [[univalence axiom|univalent]] [[dependent type theory]] in the [[homotopy theory]] of [[Kan complexes]]. More generally, using a slight generalization of [Shulman's methods](relation+between+type+theory+and+category+theory#Shulman19), we should be able to see that the theory of [[internal (∞,1)-category|(∞,1)‑categories internally]] in any [[(∞,1)-topos|∞‑topos]] (as [developed by Martini and Wolf](internal+infinity1-category#ReferencesInternalToInfinityTopos)) is a [[categorical semantics|semantic interpretation]] as well (hence so is [[Parametrized Higher Category Theory and Higher Algebra|parametrized higher category theory]] introduced by Barwick, Dotto, Glasman, Nardin and Shah). There are of course strong links with [[∞-cosmoi]] of Riehl and Verity as well as with [[cubical type theory|cubical HoTT]] (as strongly suggested by the work of Licata and Weaver), or [[simplicial type theory|simplicial HoTT]] (as in the work of Buchholtz and Weinberger). We will explain the axioms in detail and have a glimpse at basic theorems and constructions in this context ([[(∞,1)-Yoneda lemma|Yoneda Lemma]], [[(∞,1)-Kan extension|Kan extensions]], [[localization of an (∞,1)-category|Localizations]]). We will also discuss the perspective of reflexivity: since the theory speaks of itself (through directed [[univalence axiom|univalence]]), we can use it to justify new deduction rules that express the idea of working up to [[equivalence]] natively (e.g. we can produce a logic by rectifying the idea of having a locally cartesian type). In particular, this logic can be used to produce and study [[categorical semantics|semantic interpretations]] of [[HoTT]].
[[!redirects (infinity,2)-toposes]]
[[!redirects (∞,2)-topos]]
[[!redirects (∞,2)-toposes]]
|
(infinity,2)-topos theory - contents | https://ncatlab.org/nlab/source/%28infinity%2C2%29-topos+theory+-+contents |
**[[(∞,2)-topos]]**
* [[(∞,2)-presheaf]], [[(∞,2)-site]],
**[[(∞,2)-sheaf]]**
* [[codomain fibration]], [[tangent (∞,1)-category]], [[quasicoherent ∞-stack]]
## Truncations
**[[2-topos]]**
* [[2-pretopos]]
* [[2-presheaf]], [[2-site]], [[2-sheaf]], [[stack]]
* [[n-localic 2-topos]]
* [[2-Giraud theorem]]
**[[(2,1)-topos]]**
* [[(2,1)-presheaf]], [[(2,1)-site]], [[(2,1)-sheaf]]
**[[(∞,1)-topos]]**
* [[(∞,1)-presheaf]], [[(∞,1)-site]], [[(∞,1)-sheaf]]
**[[1-topos]]**
* [[presheaf]], [[site]], [[sheaf]]
|
(infinity,n)-category | https://ncatlab.org/nlab/source/%28infinity%2Cn%29-category |
+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Higher category theory
+--{: .hide}
[[!include higher category theory - contents]]
=--
=--
=--
#Contents#
* table of contents
{:toc}
## Idea ##
In [[higher category theory]] an _$(\infty,n)$-category_ may be thought of as
* an [[n-category]] up to [[coherence|coherent]] [[homotopy theory|homotopy]];
* an [[(n,r)-category|(r,n)-category]] for $r = \infty$;
* a [[weak omega-category]] for which all [[k-morphisms]] with $k \gt n$ are [[equivalences]].
Accordingly, the notion of $(\infty,n)$-categories is a joint generalization of _[[categories]]_, _[[2-categories]]_, _[[3-categories]]_, _[[4-categories]]_, etc. and of _[[∞-groupoids]]_ / _[[homotopy types]]_ and _[[(∞,1)-categories]]_. From the point of [[homotopy theory]] they are a generalization to _[[directed homotopy theory]]_, from the point of view of [[homotopy type theory]] they are a generalization to _[[directed homotopy type theory]]_.
There are two main [[recursion|recursive]] definitions of $(\infty,n)$-categories:
1. by iterated [[enriched (∞,1)-category|(∞,1)-enrichment]]
$$
Cat_{(\infty,n)} \simeq (\cdots((Cat_{(\infty,0)} Cat) Cat) \cdots) Cat
$$
1. by iterated [[category object in an (∞,1)-category|(∞,1)-internalization]]
$$
Cat_{(\infty,n)} \simeq Cat(\cdots(Cat(Cat_{(\infty,0)}))\cdots)
\,.
$$
There is also a fairly simple axiomatization of the [[(∞,1)-category]] [[(infinity,n)Cat|$Cat_{(\infty,n)}$]] itself, as something _[[generators and relations|generated]]_ by [[strict n-categories]].
Then there is also a plethora of [[model category]] structures that [[presentable (∞,1)-category|present]] the [[(∞,1)-category]] $Cat_{(\infty,n)}$ of all $(\infty,n)$-categories, which means that there are many (and many different) very explicit ways to describe them.
A central result of $(\infty,n)$-category theory is the proof of the [[cobordism hypothesis]], which revolves around the _[[(∞,n)-category of cobordisms]]_. This turns out to be the [[free construction|free]] _[[symmetric monoidal (∞,n)-category]]_ _[[(∞,n)-category with duals|with duals]]_ and provides deep relations between [[algebraic topology]], [[higher algebra]] and [[extended topological quantum field theory]].
Other fundamental examples of $(\infty,n)$-categories, also in this context, are [[(∞,n)-categories of spans]] and of [[(∞,n)-vector spaces]].
While the subject is still young, visible at the horizon is its role in [[higher topos theory]]. Where [[(∞,1)-toposes]] regarded as [[(∞,1)-categories of (∞,1)-sheaves]]/[[∞-stacks]] are by now fairly well understood, it is clear that the [[(∞,2)-categories]] of [[(∞,2)-sheaves]] -- such as the [[codomain fibration]]/[[indexed category|self-indexing]] of an [[(∞,1)-topos]] -- will form an [[(∞,2)-topos]] in generalization of the non-homotopic notion of [[2-topos]]. And so on.
## Introduction
Here are some introductory words for readers unfamiliar with the general idea. Other readers should skip [ahead](#Definition).
* _[Introduction for 1-category theorists](#1CatIntro)_
* _[Introduction for homotopy theorists](#HomotopyIntro)_
### For 1-category theorists
{#1CatIntro}
This section assumes that the reader is well familiar with [[category theory]] and maybe with [[strict omega-categories]] but in need of some introductory words on $(\infty,n)$-categories.
Ordinary [[category theory]] provides various powerful tools for generating higher order structures, among them notably
1. [[enriched category theory|enrichment]]
1. [[internalization]].
Here we are interested in higher order _[[categories]]_, so we consider [[Cat]] itself as a 1-categorical context for either of these procedures. Since [[Cat]] naturally a [[cartesian monoidal category]]
$$
(\mathcal{V}, \otimes) \coloneqq (Cat, \times)
$$
we may form the [[category of V-enriched categories]] $\mathcal{V}Cat \coloneqq Cat Cat$. A $Cat$-category consists of
* a collection of [[objects]];
* for each pair of objects $A$, $B$ a _category_ of morphisms, hence to be thought of as collection of ordinary morphisms $A \to B$ together with _morphisms between these morphisms_: [[2-morphisms]];
* such that composition is a _functor_ on these hom-categories.
This is the structure of a _[[strict 2-category]]_. We have that
$$
Cat Cat \simeq Str 2 Cat
\,.
$$
is the category of strict 2-categories.
By general results of [[enriched category theory]] (or by immediate inspection), this is still a [[cartesian monoidal category]] and so we may iterate this and consider now the enriching category
$$
(\mathcal{V}, \otimes) \coloneqq (Cat Cat, \times)
$$
and construct again $\mathcal{V}Cat$, which now is
$$
(Cat Cat) Cat \simeq Str 3 Cat
$$
the category of strict _[[3-categories]]_. It continues this way, and so for every $n \in \mathbb{N}$ the $n$-fold iterated enrichment of $Cat$ is the category
$$
Str n Cat \simeq (\cdots ((Cat Cat)Cat) \cdots) Cat
$$
of _[[strict n-categories]]_. The [[inductive limit]] of this construction finally is the category of [[strict omega-categories]].
While this easily generates [[higher category theory|higher categorical structures]], it does so, as the terminology indicates, only in a very restrictive way: while every [[2-category]] still happens to be [[equivalence of 2-categories|equivalent]] to a [[strict 2-category]], already the general [[3-category]] is no longer equivalent to a strict 3-category, and the discrepancy only increases with $n$.
But inspection in the case of [[2-categories]] already shows what the problem is: in a [[bicategory|weak 2-category]] structural relations such as [[associativity]] and [[unitality]] no longer hold as equations but only _up to_ an _invertible_ [[2-morphism]], whereas objects in $Str 2 Cat \simeq Cat Cat$, by definition of [[enriched category]], satisfy these relations _strictly_ -- therefore the name.
But this problem directly corresponds to an evident shortcoming of the very starting point of the above recursive construction: that construction regarded [[Cat]] as a 1-category in order to fit it into the standard formulation of [[enriched category theory]]; however [[Cat]] is naturally rather a [[2-category]] itself. The enrichment procedure should be allowed to make use of this extra structure. On the other hand, as we have just seen, the failure of $Cat Cat$ to model all of [[2Cat]] is only in the lack of _invertible_ 2-morphisms. Therefore what should really matter for the improved enrichment is just the [[(2,1)-category]] underlying [[Cat]], which is the 2-category consisting of all [[categories]], all [[functors]] between them, but only _[[natural isomorphism]]_ instead of all [[natural transformations]] between those.
This way one does arrive at a suitable refined notion of enrichment over the [[(2,1)-category]] $Cat$, and interpreted this way one does finds that $Cat Cat$ then indeed produces all of [[2Cat]].
However, this only fixed the first step of the above recursive definition. In the next step we want $(2 Cat)Cat$ to produce all [[3-categories]], but their associativity and unitalness now involves invertible [[coherence]] _[[3-morphisms]]_ which do not appear in enriched $(2,1)$-category theory. And so on, as the recursion proceeds.
This shows that the natural starting point for a construction of [[n-categories]] by recursive enrichment must be a conception of 1-[[category theory]] which knows already about _invertible_ [[k-morphisms]] for all $k$. The notion of category where all 1-categorical operations are relaxed up to _invertible_ higher morphisms is that of _[[(∞,1)-category]]_. And this now turns out to be a good starting point for producing $n$-categories by recursive enrichment.
If we then just replace in the above the naive [[Cat]] with [[(∞,1)Cat]], then the simple formula
$$
Cat_{(\infty,n)} \coloneq (\cdots ((Cat_{(\infty,1)} Cat_{(\infty,1)})Cat_{(\infty,1)}) \cdots) Cat_{(\infty,1)}
$$
does produce a good general notion of $n$-categories, these are the _$(\infty,n)$-categories_ discussed here.
There is also an alternative road to the same conclusion: another standard procedure for producing higher order structures from the 1-category [[Cat]] is to consider [[internal categories]] in $Cat$. For $E$ a category with [[finite limits]], write $Cat(E)$ for the category of $E$-[[internal categories]], and hence $Cat(Cat)$ for the category of $Cat$-internal categories.
This gives _[[double categories]]_
$$
DoubleCat \simeq Cat(Cat)
$$
and hence again not quite the [[2-categories]] that we are after. But it is of interest to note that now there are _two_ problems, not just the one above: while a $Cat$-internal category again has strict [[associativity]] and [[unitality]], instead of the desired version up to an invertible 2-morphism, in another direction it is more general than a [[strict 2-category]]: the latter only corresponds to those special double categories for which the "vertical" and the "horizontal" 1-morphisms come from the same 1-category and have sufficiently many degenerate 2-morphisms between them.
The first problem turns out to be solved as before: instead of working with the 1-category [[Cat]] we should already regard that as a [[(2,1)-category]] and then formulate _internal (2,1)-categories_ in straightforward generalization of the ordinary notion. For the second problem it turns out that one needs to slightly enhance that straightforward generalization and add a condition known (somewhat undescriptively) as _[[complete Segal space|completeness]]_. But if this is understood then (as discussed in detail at [[internal category in an (∞,1)-category]]) the simple idea of iterated internalization does work out and we obtain $(\infty,n)$-categories by
$$
Cat_{(\infty,n)} \simeq Cat(\cdots(Cat(Cat_{(\infty,0)}))\cdots)
\,.
$$
### For homotopy theorists
{#HomotopyIntro}
This section assumes that the reader is well familiar with [[homotopy theory]] and maybe with [[(∞,1)-category theory]] but in need of some introductory words on $(\infty,n)$-categories.
A fundamental insight of [[homotopy theory]] is, of course, that the [[geometric shapes for higher structures|cellular shape]] of [[simplices]] naturally serves to model paths and higher [[homotopies]] in "spaces", which here really means: in [[homotopy types]]/[[∞-groupoids]]. In fact, the simplices see a bit more: since $\Delta[n]$ is naturally identified with the [[total order|linear category]] $\{0 \to 1 \to 2 \to \cdots \to n\}$ on $(n+1)$-objects, there is a _direction_ on the paths which form the [[simplicial skeleton|1-skeleton]] of a map $\Delta^n \to X$.
If $X$ is a [[topological space]]/[[simplicial set]]/[[homotopy type]], then this directedness in a way "disappears up to equivalence", in that for every such directed path there is also the reverse path, which is an inverse up to equivalence.
But it is straightforward to consider a slight generalization of this situation where we take $X$ to be such that _not_ all paths in it have inverses. Still thinking of $X$ as a homotopy type this may be thought of as modelling a _[[directed homotopy theory|directed homotopy type]]_. For $X$ instead modeled as a simplicial set, this has been formalized by the concept of a _[[quasi-category]]_ or _[[(∞,1)-category]]_. These are combinatorial models for _[[directed homotopy theory|directed homotopy types]]_ in direct generalization of how [[Kan complexes]] are combinatorial models for ordinary [[homotopy types]].
As the notation already suggests, the idea of $(\infty,n)$-category theory is that this generalization from [[∞-groupoids]] ("($\infty,0$)-categories") to [[(∞,1)-categories]] is but the first step in a tower of higher generalizations, where in step $n$ one considers "directed homotopy" up to and including dimension $n$.
It is natural that such _$(\infty,n)$-categories_ should be probed by corresponding higher dimensional analogs of the objects in the [[simplex category]], the linear categories $\Delta[n] = \{0 \to 1 \to 2 \to \cdots \to n\}$ that support traditional homotopy theory. There are many such generalizations which one could consider. One which has proven to be useful are the objects in the $n$th [[Theta-category]] $\Theta_n$. Where the linear categories as above arise from gluing -- [[pasting]] -- of cellular intervals, the objects of $\Theta_n$ arise from [[pasting]] of $n$-dimensional cellular _[[globes]]_ (an interval being a 1-dimensional globe).
Accordingly, just as an [[∞-groupoid]]/[[homotopy type]] may be presented by a [[simplicial set]], hence a [[presheaf]] on the [[simplex category]] -- or more generally by a [[bisimplicial set|simplicial space]]-- satisfying some ([[Kan complex|Kan filler]]-)condition that encodes the existence of composites and inverses, so an [[(∞,n)-category]] may be presented by a presheaf of spaces on the $n$th [[Theta-category]], similarly subject to some conditions that ensure the existence of composites and inverses -- but only of inverses above dimension $n$.
## Definitions
{#Definition}
There are various different ways of defining $(\infty,n)$-categories, which are all natural in their own right, and all equivalent to each other.
There is an axiomatic characterization of the [[(∞,1)-category]] of $(\infty,n)$-categories by [[generators and relations|generation]] from [[strict n-categories]]:
* [Definition via generation from strict n-categories](#AxiomaticCharacterization)
Among the more direct definitions of $(\infty,n)$-categories one can roughly distinguish two flavors, those that build $(\infty,n)$-categories by _[[enriched category|enrichment]]_ over $(\infty,n-1)$-categories
* [Definitions via enrichment](#ViaEnrichment)
and those that build them by [[internalization]] in the collection of $(\infty,n-1)$-categories
* [Definitions via internalization](#ViaInternalization).
### Via generation by strict $n$-categories
{#AxiomaticCharacterization}
We discuss a characterization of the [[(∞,1)-category]] of $(\infty,n)$-categories as an $(\infty,1)$-category [[generators and relations|generated]] by [[strict n-categories]], due to ([Barwick, Schommer-Pries](#BarwickSchommerPries)).
The blueprint for the following construction is the traditional fact that a [[category]] is characterized by the fact that its [[nerve]] is a [[simplicial set]] which satisfies the [[Segal conditions]], which reflect the existence of composition in a category. Since the simplicial nerve is induced from the [[total order|linear categories]] $\Delta[n] = \{0 \to 1 \to 2 \to \cdots \to n\}$ this can be taken as saying that these linear categories _generate_ [[Cat]], subject to the condition that there exists composites.
The following discussion takes this point of view and generalizes it to a similar presentation of $(\infty,n)$-categories by very simple [[strict n-categories]].
#### Strict $n$-categories
The main definition is def. \ref{AxiomaticDefinition} below, which roughly says that the collection of $(\infty,n)$-categories is _generated_ from [[strict n-categories]] in a certain sense. Therefore we first need to fix some terminology and notions about strict $n$-categories and about the relevant notion of generation.
+-- {: .num_defn #GauntStrictNCategories}
###### Definition
Write $Str n Cat$ for the 1-[[category]] of [[strict n-categories]].
Write
$$
Str n Cat_{gaunt} \hookrightarrow Str n Cat
$$
for the [[full subcategory]] on the _[[gaunt category|gaunt]] $n$-categories_, those $n$-categories whose only invertible [[k-morphisms]] are the identities.
=--
This subcategory was considered in ([Rezk](#Rezk)). The term "gaunt" is due to ([Barwick, Schommer-Pries](#BarwickSchommerPries)). See prop. \ref{GauntIs0Truncated} below for a characterization intrinsic to $(\infty,n)$-categories.
+-- {: .num_example #Globes}
###### Example
For $k \leq n $ the $k$-[[globe]] is gaunt, $G_k \in Str n Cat_{gaunt} \hookrightarrow \in Str n Cat$.
Write
$$
\mathbb{G}_{\leq n} \hookrightarrow Str n Cat_{gaunt}
$$
for the [[full subcategory]] of the [[globe category]] on the $k$-globes for $k \leq n$.
Being a [[subobject]] of a gaunt $n$-category, also the [[boundary]] of a globe $\partial G_k \hookrightarrow G_k$ is gaunt, i.e. the $(k-1)$-[[skeleton]] of $G_k$.
=--
+-- {: .num_defn #Suspension}
###### Definition
Write
$$
\sigma_k : Str (k) Cat \to Str (k+1) Cat
$$
for the "categorical suspension" functor which sends a strict $k$-category to the object $\sigma(X) \in Str (k+1) Cat \simeq (Str k Cat)Cat$ which has precisely two objects $a$ and $b$, has $\sigma(C)(a,a) = \{id_a\}$, $\sigma(C)(b,b) = \{id_b\}$, $\sigma(C)(b,a) = \emptyset$ and
$$
\sigma(C)(a,b) = C
\,.
$$
=--
We usually suppress the subscript $k$ and write $\sigma^i = \sigma_{k+i} \circ \cdots \circ \sigma_{k+1} \circ \sigma_k$, etc.
+-- {: .num_example }
###### Example
The $k$-[[globe]] $G_k$ is the $k$-fold suspension of the 0-globe (the point)
$$
G_k = \sigma^k(G_0)
\,.
$$
The [[boundary]] $\partial G_k$ of the $k$-globe is the $k$-fold suspension of the
empty category
$$
\partial G_k = \sigma^k(\emptyset)
\,.
$$
Accordingly, the boundary inclusion $\partial G_k \hookrightarrow G_k$ is the $k$-fold suspension of the initial morphism $\emptyset \to G_0$
$$
(\partial G_k \hookrightarrow G_k) = \sigma^k(\emptyset \to G_0)
\,.
$$
=--
+-- {: .num_prop }
###### Proposition
The category $Str n Cat_{gaunt}$ is a [[locally presentable category]] and in fact a [[locally finitely presentable category]].
=--
([B-PS, lemma 3.5](#BarwickSchommerPries))
We are going to be interested in a full [[subcategory]] $Str n Cat_{gen} \hookrightarrow Str n Cat_{gaunt}$, given below in def. \ref{nCatGen}, which knows about the higher [[profunctors]]/[[correspondence|correspondences]] between $n$-categories.
\begin{remark}
For $A,B$ two categories, a [[profunctor]] $A^{op} \times B \to Set$ is equivalently a category [[slice category|over]] the 1-globe functor, hence a functor
$$
\array{
K
\\
\downarrow
\\
G_1 & = \Delta[1]
}
$$
equipped with an identification $A \simeq K_0$ and $B \simeq K_1$.
\end{remark}
This motivates the following definition.
+-- {: .num_defn }
###### Definition
A _$k$-profunctor_ / $k$-correspondence of strict $n$-categories is a morphism
$K \to G_k$ in $Str n Cat$. The category of $k$-correspondences is the [[slice category]] $Str n Cat/ G_k$.
=--
+-- {: .num_defn }
###### Definition
The categories $Str n Cat_{gaunt}/G_k$ of $k$-correspondences between gaunt $n$-categories are [[cartesian closed category]].
=--
([B-SP, cor. 5.4](#BarwickSchommerPries))
+-- {: .num_remark}
###### Remark
By standard facts, in a [[locally presentable category]] $\mathcal{C}$ with [[finite limits]], a [[slice category|slice]] $\mathcal{C}/X$ is cartesian closed precisely if [[pullback]] along all morphisms $f : Y \to X$ with codomain $X$ preserves [[colimits]] (see at _[[locally cartesian closed category]]_ the section _[Cartesian closure in terms of base change and dependent product](locally%20cartesian%20closed%20category#EquivalentCharacterizations)_).
=--
+-- {: .num_example}
###### Example
Without the restriction that the codomain of $f$ in the above is a [[globe]], the pullback $f^*$ in $Str n Cat$ will in general fail to preserves colimits. For a simple example of this, consider the [[pushout]] diagram in [[Cat]] $\hookrightarrow Cat_{(\infty,1)}$ given by
$$
\array{
\Delta[0] &\stackrel{\delta_1}{\to}& \Delta[1]
\\
{}^{\mathllap{\delta_0}}\downarrow && \downarrow^{\mathrlap{\delta_0}}
\\
\Delta[1] &\stackrel{\delta_2}{\to}& \Delta[2]
}
\,.
$$
Notice that this is indeed also a [[homotopy pushout]]/[[(∞,1)-pushout]] since, by remark \ref{GauntIs0Truncted}, all objects involved are 0-truncated.
Regard this canonically as a pushout diagram in the [[slice category]] $Cat_{/\Delta[2]}$ and consider then the pullback $\delta_1^* : Cat_{/\Delta[1]} \to Cat_{/\Delta[1]}$ along the remaining face $\delta_1 : \Delta[1] \to \Delta[2]$. This yields the diagram
$$
\array{
\emptyset &\stackrel{}{\to}& \emptyset
\\
{}^{}\downarrow && \downarrow^{}
\\
\emptyset &\stackrel{}{\to}& \Delta[1]
}
\,,
$$
which evidently no longer is a pushout.
=--
(See also the discussion [here](http://golem.ph.utexas.edu/category/2011/11/the_1category_of_ncategories.html#c040335)).
The definition of $Cat_{(\infty,n)}$ below, def. \ref{AxiomaticDefinition}, will take this property to be one of the characteristic properties. Therefore consider
+-- {: .num_defn #nCatGen}
###### Definition
Write
$$
Str n Cat_{gen} \hookrightarrow Str n Cat_{gaunt}
$$
for the smallest [[full subcategory]] that
1. contains the [[globe category]] $\mathbb{G}_{\leq n}$, example \ref{Globes};
1. is closed under [[retracts]] in $Str n Cat_{gaunt}$;
1. has all [[fiber products]] over [[globes]] (equivalently: such that all [[slice categories]] over globes have [[products]]).
=--
([B-SP, def. 5.6](#BarwickSchommerPries))
+-- {: .num_example }
###### Example
The following categories are naturally [[full subcategories]] of $Str n Cat_{gen}$
* the $n$-fold [[simplex category]] $\Delta^{\times n}$;
* the $n$th [[Theta-category]].
=--
This is discussed in more detail below in _[Presentation by Theta-spaces and by n-fold Segal spaces](#PresentationByThetaSpaces)_.
+-- {: .num_defn #FundamentalPushouts}
###### Definition
The following [[pushouts]] in $Str n Cat$ we call the **fundamental pushouts**
1. Gluing two $k$-[[globes]] along their [[boundary]] gives the boundary of the $(k+1)$-globle
$$G_k \coprod_{\partial C_{k-1}} G_k \simeq \partial G_{k+1}$$
1. Gluing two $k$-globes along an $i$-face gives a [[pasting]] composition of the two globles
$$G_k \coprod_{G_i} G_k $$
1. The [[fiber product]] of globes along non-degenerate morphisms $G_{i+j} \to G_i$ and $G_{i+k} \to G_i$ is built from gluing of globes by
$$
G_{i+j} \times_{G_i} G_{i+k}
\simeq
(G_{i+j} \coprod_{G_i} G_{i+k}) \coprod_{\sigma^{i+1}(G_{j-1} \times G_{k-1})} (G_{i+k} \coprod_{G_i} G_{i+j})
$$
1. The [[interval groupoid]] $(a \stackrel{\simeq}{\to} b)$ is obtained by forcing in $\Delta[3]$ the morphisms $(0\to 2)$ and $(1 \to 3)$ to be identities and it is equivalent, as an $n$-category, to the 0-globe
$ \Delta[3] \coprod_{\{0,2\} \coprod \{1,3\}} (\Delta[0] \coprod \Delta[0])
\stackrel{\sim}{\to} G_0$
and the analog is true for all suspensions of this relation
$$
\sigma^k(\Delta[3]) \coprod_{\sigma^k\{0,2\} \coprod \sigma^k\{1,3\}} (G_k\coprod G_k)
\stackrel{\sim}{\to} G_k
\,.
$$
We say a functor $i$ on $Str n Cat$ _preserves_ the fundamental pushouts if it preserves the first three classes of pushouts, and if for the last one the morphism
$i(\sigma^k(\Delta[3])) \coprod_{i(\sigma^k\{0,2\}) \coprod i(\sigma^k\{1,3\})} (i(G_k \coprod G_k)) \to i(G_k)$ is an equivalence.
=--
#### Generation by strict $n$-categories
Def. \ref{AxiomaticDefinition} considers an $(\infty,1)$-category _generated_ from $Str n Cat_{gen}$ in the following sense
+-- {: .num_defn #StrongGeneration}
###### Definition
For $\mathcal{D}$ an [[(∞,1)-category]] with all small [[(∞,1)-colimits]], say that an [[(∞,1)-functor]]
$$
f : \mathcal{C} \to \mathcal{D}
$$
_strongly generates_ $\mathcal{D}$ if its $(\infty,1)$-[[Yoneda extension]] on the [[(∞,1)-category of (∞,1)-presheaves]]
$$
f : \mathcal{C} \stackrel{y}{\hookrightarrow} PSh_\infty(\mathcal{C}) \stackrel{Lan_y}{\to} \mathcal{D}
$$
is the reflector of a [[reflective sub-(∞,1)-category]]
$$
\mathcal{D} \stackrel{\overset{Lan_y}{\leftarrow}}{\hookrightarrow}
PSh_\infty(\mathcal{C})
\,.
$$
=--
+-- {: .num_remark}
###### Remark
By definition, a strongly generated $(\infty,1)$-category is in particular a
[[presentable (∞,1)-category]].
=--
+-- {: .num_defn #AxiomaticDefinition}
###### Definition
An **$(\infty,1)$-category of $(\infty,n)$-categories** $Cat_{(\infty,n)}$ is an [[(∞,1)-category]] equipped with a [[full and faithful functor]]
$$
i : Str n Cat_{gen} \hookrightarrow \tau_{\leq 0}Cat_{(\infty,n)}
$$
from the generating strict $n$-categories, def. \ref{nCatGen} into its category of [[n-truncated object in an (infinity,1)-category|0-truncated]] objects, such that
1. $Str n Cat_{gen} \to \tau_{\leq 0} Cat_{(\infty,n)} \hookrightarrow Cat_{(\infty,n)} $ [strongly generates](#StrongGeneration) $\mathcal{C}$;
1. $i$ preserves the [fundamental pushout](#FundamentalPushouts) relations;
1. the [[base change]] [[adjoint triple]] in $Cat_{(\infty,n)}$ exists along morphisms with codomain a [[globe]];
and such that $\mathcal{C}$ is [[universal property|universal]] with respect to these properties in that for any other $j : Str n Cat_{gen} \hookrightarrow \mathcal{C}$ satisfying these three conditions it factors through $i$
$$
j : Str n Cat \stackrel{i}{\to} Cat_{(\infty,n)} \stackrel{L}{\to} \mathcal{C}
$$
by an [[(∞,1)-functor]] $L$ which is the reflector of a [[reflective sub-(∞,1)-category|reflective inclusion]] $\mathcal{C} \hookrightarrow Cat_{(\infty,n)}$.
=--
([B-SP, def. 6.8](#BarwickSchommerPries))
+-- {: .num_remark }
###### Remark
By the first axiom, the localization demanded in the universal property is essentially unique. In particular, therefore, $Cat_{(\infty,n)}$ is defined uniquely, up to [[equivalence of (∞,1)-categories]].
For more on this see prop. \ref{AutomorphismInfinityGroup} below.
=--
+-- {: .num_remark #GauntIs0Truncted}
###### Remark
The gaunt $n$-categories, def. \ref{GauntStrictNCategories} are indeed among the [[n-truncated object in an (∞,1)-category|0-truncated]] objects: since we are looking at just the [[(∞,1)-category]] of $(\infty,n)$-categories, instead of more generally the $(\infty,n+1)$-category the non-invertible [[transfors]] between $n$-categories are disregarded and so if an object $X \in Cat_{(\infty,n)}$ has no non-trivial invertible cells, then for every other objeyt $Y$, the hom-$\infty$-groupoid $Cat_{(\infty,n)}(Y,X)$ is 0-truncated, hence is a set.
=--
+-- {: .num_remark }
###### Remark
The first axiom in particular says that $Cat_{(\infty,n)}$ is a [[presentable (∞,1)-category]], and hence so are all its [[over-(∞,1)-category|slices]]. In view of this the [[adjoint (∞,1)-functor theorem]] says that the third condition is equivalent to [[(∞,1)-pullbacks]]
$$
f^* : Cat_{(\infty,n)}/_{i(G_k)} \to Cat_{(\infty,n)}/X
$$
along morphisms of the form $X \to i(G_k)$ preserving [[(∞,1)-colimits]].
=--
#### Universal presentation
By def. \ref{AxiomaticDefinition} $Cat_{(\infty,n)}$ is [[equivalence of (∞,1)-categories|equivalent]] to a [[localization of an (∞,1)-category|localization]] of the [[(∞,1)-category of (∞,1)-presheaves]] on $Str n Cat_{gen}$. In fact, various subcategories of $Str n Cat_{gen}$ are already sufficient, notable the [[Theta-category]] $\Theta_n \hookrightarrow Str n Cat$ (discussed below in \ref{PresentationByThetaSpaces}). Here we discuss these [[presentable (∞,1)-category|presentations]].
+-- {: .num_defn #UniversalLocalizingClass}
###### Definition
Let $S_{0} \subset Mor(PSh_\infty(Str n Cat_{gen}))$
be the class of morphism generated under fiber product $X \times_{G_k} (-)$ with objects $X \in Str n Cat_{gen}$ over globes by
1. the morphisms that witness the [fundamental pushout relations](#FundamentalPushouts)
1. the initial morphism $\emptyset \to i(\emptyset)$ into presheaf represented by the empty category (which coincides with the initial presheaf on all objects except on the empty category, where it is the singleton).
Write $S$ for the strongly saturated class of morphisms (see [[reflective sub-(∞,1)-category]]) generated by $S_0$.
=--
+-- {: .num_prop }
###### Proposition
The [[localization of an (∞,1)-category|localization]] of the [[(∞,1)-category of (∞,1)-presheaves]] over $Str n Cat_{gen}$, def. \ref{nCatGen} at the class of morphism $S$ from def. \ref{UniversalLocalizingClass} is a [[presentable (∞,1)-category|presentation]] of $Cat_{(\infty,n)}$, def. \ref{AxiomaticDefinition}:
$$
Cat_{(\infty,n)} \simeq PSh_\infty(Str n Cat_{gen})[S^{-1}]
\,.
$$
=--
([B-SP, theorem 7.6](#BarwickSchommerPries)).
+-- {: .proof}
###### Proof
The three axioms of def. \ref{AxiomaticDefinition} are satisfied effectively by construction of $S$ (...). Conversely, every localization satisfying the second and third axiom must invert the morphisms in $S$, hence must be a sub-localization.
=--
+-- {: .num_remark }
###### Remark
This construction shows that the [fundamental pushout relations](#FundamentalPushouts) encode the _composition_ of [[k-morphisms]] in an $(\infty,n)$-category.
Let $X \in PSh_\infty(Str n Cat)$ be some object.
First, by the [[(∞,1)-Yoneda lemma]] the value of this [[(∞,1)-presheaf]] on a strict $n$-category $C$ is the $\infty$-groupoid of $(\infty,n)$-functors $C \to X$, [[natural equivalences]] between them, and so on.
And if $X$ is an $S$-[[local object]] then it has in particular the property that all the morphisms
$$
Cat_{(\infty,n)}(
i(G_k) \coprod_{i(G_j)} i(G_k)
\to
i(G_jk \coprod_{ G_j } G_k)
,
X
)
$$
are equivalences of $\infty$-groupoids. So by the [[(∞,1)-Yoneda lemma]] this is equivalent to
$$
X(G_k) \times_{X(G_i)} X(G_k)
\to
Cat_{(\infty,n)}(i(G_jk \coprod_{ G_j } G_k), X)
$$
being an equivalence. On the left this is the collection of all those pairs of $k$-globes in $X$ that touch at an $i$-boundary. On the right this is the collection of all [[k-morphisms]] in $X$ equipped with a choice of decomposing them into two $k$-morphisms touching at an $i$-boundary. So the statement that this morphism is an equivalence says that _composition_ of $k$-morphisms along $i$-boundaries exists in $X$.
=--
Various other presentations of $Cat_{(\infty,n)}$ are obtained by localizations over subcategories of
$i : Str n Cat_{restr} \hookrightarrow Str n Cat_{gen}$ at a set of morphisms $T \subset Mor(PSh_\infty(R))$. Write
$$
PSh_\infty(Str n Cat_{restr})
\stackrel{\overset{i_!}{\to}}{\stackrel{\overset{i^*}{\leftarrow}}{\underset{i_*}{\to}}}
PSh_\infty(Str n Cat_{gen})
$$
for the induced [[essential geometric morphism]].
+-- {: .num_prop #SufficientConditionsForPresentation}
###### Proposition
The following conditions are sufficient in order that
$$
Cat_{(\infty,n)} \simeq
PSh_\infty(Str n Cat_{gen})[S^{-1}]
\stackrel{i^*}{\to}
PSh_\infty(Str n Cat_{restr})[T^{-1}]
$$
is an [[equivalence of (∞,1)-categories]]:
1. $i^*(S_0) \subset T$
1. $i_!(T_0) \subset S$
1. [[generalized the|the]] [[unit of an adjunction|counit]] $id \to i^* i_!$ has components in $T$;
1. the $k$-[[globe]] $G_k$ is in the essential image of $i$, for each $0 \leq k \leq n$.
=--
([B-SP, theorem 9.2](#BarwickSchommerPries))
#### Presentation by $\Theta_n$-spaces and $n$-fold complete Segal spaces
{#PresentationByThetaSpaces}
We discuss now presentations of $Cat_{(\infty,n)}$ over subcategories of $Str n Cat_{gen}$, according to prop. \ref{SufficientConditionsForPresentation}.
+-- {: .num_prop }
###### Proposition
The $n$th [[Theta category]] is a [[full subcategory]]
$$
\Theta_n \hookrightarrow Str n Cat_{gen}
$$
and the localization of $PSh_\infty(\Theta_n)$ that defines the $(\infty,1)$-category $\Theta_n Space$ of $(\infty,n)$-[[Theta-spaces]] satisfies the conditions of prop. \ref{SufficientConditionsForPresentation}.
Hence $(\infty,n)$-[[Theta-spaces]] are a model for $(\infty,n)$-categories, in the sense of def. \ref{AxiomaticDefinition}:
$$
\Theta_n Space \simeq Cat_{(\infty,n)}
\,.
$$
=--
([B-SP, theorem 11.15](#BarwickSchommerPries))
There is a further restriction from the objects of $\Theta_n$ to _$n$-fold simplices_ regarded as _[grid object](Theta+category#EmbeddingOfGrids)_, under the canonical embedding
$$
\delta_n : \Delta^{\times n} \to \Delta^{\wr n} \simeq \Theta_n
$$
induced by the identification of the $n$th[[Theta-category]] (see there) with the $n$-fold [[categorical wreath product]] of the [[simplex category]] with itself.
+-- {: .num_prop #CompleteSegalOvernFoldSimplicialSetsIsPresentation}
###### Proposition
The inclusion
$$
\Delta^{\times n} \stackrel{\delta_n}{\to}
\Theta_n \hookrightarrow Str n Cat_{gen}
$$
and the localization of $PSh_\infty(\Delta^{\times n})$ that defines the $(\infty,1)$-category $CSS(\Delta^{\times n})$ of _[[n-fold complete Segal spaces]]_ satisfies the conditions of prop. \ref{SufficientConditionsForPresentation}.
Hence [[n-fold complete Segal spaces]] are a model for $(\infty,n)$-categories, in the sense of def. \ref{AxiomaticDefinition}:
$$
CSS(\Delta^{\times n}) \simeq Cat_{(\infty,n)}
\,.
$$
=--
([B-SP, theorem 12.6](#BarwickSchommerPries))
+-- {: .num_remark}
###### Remark
Below in [Via ∞-Internalization -- Presentation by complete Segal spaces](#PresentationByCompleteSegal) is discussed that $n$-fold complete Segal spaces also naturally model an alternative definition of $(\infty,n)$-categories by [iterated ∞-internalization](#ViaInternalization). Then prop. \ref{CompleteSegalOvernFoldSimplicialSetsIsPresentation} serves to show that this is equivalent to def. \ref{AxiomaticDefinition} above.
=--
### Via $\infty$-enrichment
{#ViaEnrichment}
#### General
There should be a general notion of _[[enriched (∞,1)-category]]_ (see there) over a [[monoidal (∞,1)-category]] $\mathcal{V}$. Write $\mathcal{V}Cat$ for the [[(∞,1)-category]] of $\mathcal{V}$-enriched $(\infty,1)$-categories.
+-- {: .num_defn #EnrichementDefinition}
###### Definition
For $n \in \mathbb{N}$ write
$$
Cat_{(\infty,n)} \coloneqq
(((\infty Grpd Cat) Cat) \cdots) Cat
\,.
$$
=--
#### Presentation by Segal $n$-categories
The notion of _[[Segal n-categories]]_ is a realization of the
idea of _weak enrichment_ in a suitable [[model category]].
For nice enough model categories this can be further
strictfied to just the notion of [[enriched model category]],
discussed _[below](#ByEnrichedModelCategories)_
(...)
#### Presentation by enriched model categories
{#ByEnrichedModelCategories}
(...)
### Via $\infty$-internalization
{#ViaInternalization}
#### General
There is a general notion of _[[internal category in an (∞,1)-category]]_ $\mathcal{C}$ provided that
1. $\mathcal{C}$ has [[finite limit|finite]] [[(∞,1)-limits]] -- in order to formulate the [[Segal condition]];
1. $\mathcal{C}$ is equipped with a "choice of internal [[∞-groupoids]]" -- in order to formulate the [completeness condition](complete%20Segal%20space#CompleteSegalSpaces).
We can use this to define $Cat_{(\infty,n)}$ by iterative internalization.
+-- {: .num_defn }
###### Proposition
Write $Grpd(Cat_{(\infty,0)})$ for the catgeory of [[groupoid objects in an (∞,1)-category]] in $Cat_{(\infty,0)} \simeq $ [[∞Grpd]].
Assume we have already defined $Cat_{(\infty,n)}$, either by one of the methods above, or by the induction in the following. Then the canonical inclusion
$$
Grpd(Cat_{(\infty,0)}) \hookrightarrow PreCat_{Grpd(Cat_{(\infty,0)})} (Cat_{(\infty,n)})
$$
into the $(\infty,1)$-category of [[simplicial objects]] $X_\bullet$ in $Cat_{(\infty,n)}$ that
1. satisfy the [[Segal conditions]]
1. such that $X_0 \in Cat_{(\infty,0)}$
has a [[right adjoint|right]] [[adjoint (∞,1)-functor]] $Core$.
=--
([Lurie, prop. 1.1.14](#Lurie)).
+-- {: .num_defn #IteratedInternalization}
###### Definition
An $(\infty,n+1)$-category is an object $X \in PreCat_{Grpd(Cat_{(\infty,0)})}$
such that $Core(X) \in \infty Grpd \hookrightarrow Grpd(Cat_{(\infty,0)})$.
For $n \in \mathcal{N}$ the $(\infty,1)$-category of
$(\infty,n)$-categories is
$$
Cat_{(\infty,n)}
\coloneqq
Cat^n(\infty Grpd)
\coloneqq
Cat(\cdots Cat(Cat_{(\infty,0)}) \cdots)
\,.
$$
=--
([Lurie, prop. 1.1.14](#Lurie)).
+-- {: .num_prop }
###### Proposition
The $(\infty,1)$-category $Cat_{(\infty,n)}$ given by
def. \ref{IteratedInternalization} is [[equivalence of (∞,1)-categories|equivalent]] to that given by
def. \ref{AxiomaticDefinition}.
=--
This is prop. \ref{CompleteSegalOvernFoldSimplicialSetsIsPresentation}
in view of the presentation discussed [below](#PresentationByCompleteSegal).
#### Presentation by $n$-fold complete Segal spaces
{#PresentationByCompleteSegal}
By the discussion [here](category+object+in+an+%28infinity,1%29-category#ModelCategoryPresentations) at _[[category object in an (∞,1)-category]]_ we have
+-- {: .num_prop }
###### Proposition
Write $cSegal_0 \coloneqq sSet_{Quillen}$ for the standard [[model structure on simplicial sets]]. Then recursively for $n \in \mathbb{N}$, $n \geq 1$, there is a model structure on
$$
cSegal_n \coloneqq [\Delta^{op}, cSegal_{n-1}]
$$
which [[presentable (infinity,1)-category|presents]] $Cat^n(\infty Grpd)$.
=--
+-- {: .num_prop }
###### Proposition
$cSegal_n$ is equivalent to the $CSS(\Delta^{\times n})$
from prop. \ref{CompleteSegalOvernFoldSimplicialSetsIsPresentation}.
=--
(...)
## Properties
### Generators
{#Generators}
+-- {: .num_prop }
###### Proposition
The [[(∞,1)-category]] $Cat_{(\infty,n)}$ is generated under [[(∞,1)-colimits]] from the $k$-[[globes]] $G_k$ for $k \leq n$: every object is the [[(∞,1)-colimit]] over a diagram of globes.
=--
([B-SP, cor. 8.4](#BarwickSchommerPries))
+-- {: .num_prop }
###### Proposition
[[equivalence in an (∞,1)-category|Equivalences]] in the [[(∞,1)-category]] $Cat_{(\infty,n)}$ are detected on [[globes]]: a morphism $f : X \to Y$ in $Cat_{(\infty,n)}$ is an equivalence precisely if for all globes $G_{k \leq n}$ the induced morphism on [[derived hom-space|(∞,1)-categorical hom-spaces]]
$$
Cat_{(\infty,n)}(G_k, f) : Cat_{(\infty,n)}(Y, f) \to Cat_{(\infty,n)}(X, f)
$$
is an equivalence of [[∞-groupoids]].
=--
([B-SP, cor. 8.5](#BarwickSchommerPries))
### Truncated objects
+-- {: .num_prop #GauntIs0Truncated}
###### Proposition
The [[truncated object in an (∞,1)-category|truncated objects]] in the [[(∞,1)-category]] $Cat_{(\infty,n)}$ are precisely the [gaunt](#GauntStrictNCategories) [[strict n-categories]]
=--
([B-SP, cor. 8.6](#BarwickSchommerPries))
+-- {: .num_remark }
###### Remark
That 0-truncated objects in the $Cat_{(\infty,n)}$ regarded as an $(\infty,1)$-category are gaunt is effectively the definition of 0-truncation in the absence of non-invertibles [[transfors]]. That these gaunt $(\infty,n)$-categories are then necessarily _[[strict n-category|strict]]_ reflects the fact that all the weakening, namely all the [[associators]] and [[unitors]] as well as all there [[coherence|coherences]] need to be invertible [[k-morphisms]], and hence must be trivial if there are no non-trivial such.
=--
### Moduli
+-- {: .num_prop #AutomorphismInfinityGroup}
###### Proposition
Let $Models_{(\infty,n)} \hookrightarrow \hat Cat_{(\infty,1)}$
be the [[core]] (maximal [[∞-groupoid]] inside) the full [[sub-(∞,1)-category]]
of [[(∞,1)Cat]] on those that satisfy the definition \ref{AxiomaticDefinition}.
This is [[equivalence of (∞,1)-categories|equivalent]] to
$$
Models_{(\infty,n)} \simeq B (\mathbb{Z}_2)^n,
$$
the [[delooping]] [[groupoid]] of the [[group]] $(\mathbb{Z}_2)^n$, the $n$-fold [[product]] of the [[group of order 2]] with itself.
The nontrivial element $\sigma \in \mathbb{Z}_2$ in the $k$th slot acts by passing to the $k$-opposite $(\infty,n)$-category.
=--
([B-SP, theorem 8.13](#BarwickSchommerPries))
+-- {: .num_remark }
###### Remark
This means that
1. the $(\infty,1)$-category $Cat_{(\infty,n)}$ from def. \ref{AxiomaticDefinition} is uniquely defined, up to [[equivalence of (∞,1)-categories]];
1. the [[automorphism ∞-group]] of $Cat_{(\infty,n)}$ in $\hat Cat_{(\infty,1)}$ is $(\mathbb{Z}_2)^n$, hence the only auto-equivalences are given by forming the $n$ analogs of forming an [[opposite (∞,1)-category]].
=--
+-- {: .proof}
###### Proof
The idea is this:
One first observes that $Str n Cat_{gaunt}$ from def. \ref{GauntStrictNCategories} has $(\mathbb{Z}_2)^{\times n}$ worth of automorphisms, given by
reversing the directions of the [[k-morphisms]].
For this,
1. observe that the identity is the only [[natural transformation]] [[endomorphism]] on $Id : Str n Cat_{gaunt} \to Str n Cat_{gaunt}$: this can be checked on [[globes]] for which one observes that if a functor $G_n \to G_n$ is the identity on $\partial G_n$, then it is so also on the unique $n$-cell. ([B-SP, lemma 4.1](#BarwickSchomerPries))
1. observe that every autoequivalence of $Str n Cat_{gaunt}$ restricts to one on the [[globe category]] $\mathbb{G}_n$ ([B-SP, lemma 4.4](#BarwickSchomerPries)).
1. observe that the only autoequivalences of $\mathbb{G}_n$ are those that reverse the direction of the $k$-morphisms for $1 \leq k \neq n$, which with the above implies the same for all of $Str n Cat_{gaunt}$ ([B-SP, lemma 4.5](#BarwickSchomerPries)).
Now us that, by the [above discussion](#Generators), $Str n Cat_{gaunt}$ generates all of $Cat_{(\infty,n)}$ under [[(∞,1)-colimits]].
=--
### Web of Quillen equivalent model category presentations
{#WebOfQuillenEquivalences}
We list [[model category]] structures that [[presentable (infinity,1)-category|present]] $Cat_{(\infty,n)}$ and [[Quillen equivalences]] between them.
In the following $A$ is an [[model category]] presenting $Cat_{(\infty,n-1)}$ that is an "absolute distributor" in the sense discussed at _[[category object in an (∞,1)-category]]_. (That includes most of the model structures in the table, so that one can recurse over these constructions.)
| [[model category]] | [[Quillen equivalence]] | [[model category]] | $n$Lab page | literature |
|---|--|---|---|---|
| [[model structure for Segal categories|projective structure for]] $A$-[[Segal categories]] | $\stackrel{identity}{\leftrightarrow}$ | [[model structure for Segal categories|injective structure for]] $A$-[[Segal categories]] | | ([Lurie, prop. 2.3.9](#Lurie))|
| [[model structure for Segal categories|projective structure for]] $A$-[[Segal categories]] | $\stackrel{inclusion}{\leftarrow}$ | $A$-[[enriched categories]] | | [Lurie, theorem 2.2.16](#Lurie) |
| [[model structure for Segal categories|injective structure for]] $A$-[[Segal categories]] | $\stackrel{UnPre}{\to}$ | [[model structure for complete Segal spaces|complete Segal space objects]] in $A$ | | [Lurie, prop 2.3.1](#Lurie) |
| [[Theta-space|Theta-(n-1)-space]]-[[Segal categories]] | | [[Theta-space|Theta-(n-1)-space]]-[[enriched categories]] | | ([Bergner-Rezk, prop. 7.2](#BergnerRezk)) |
|$\vdots$| |$\vdots$| | |
## Examples ##
### Special cases
* [[∞-groupoid]] = $(\infty,0)$-category
* [[(∞,1)-category]]
* [[(∞,2)-category]]
* etc ...
In addition,
* [[(n,r)-category|(m,n)-categories]] can be obtained as particular $(\infty,n)$-categories whose $k$-cells are trivial for $k\gt m$.
* In particular, [[n-categories]] = $(n,n)$-categories can be so obtained.
### Specific examples
* One motivating example for $(\infty,n)$-categories is the [[(∞,n)-category of cobordisms]] which plays a central role in the formalization of the [[cobordism hypothesis]].
* Another class of examples are [[(∞,n)-categories of spans]].
## Extra structure and properties
We discuss extra [[structure]] that an [[(∞,n)-category]] can carry
and extra [[properties]] that it may enjoy.
### $\mathcal{O}$-Monoidal $(\infty,n)$-categories
* [[monoidal (∞,1)-category]]
* [[symmetric monoidal (∞,n)-category]]
### $(\infty,n)$-Categories with all adjoints
* [[(∞,n)-category with all adjoints]]
## Related concepts
* [[0-category]], [[(0,1)-category]]
* [[category]]
* [[2-category]]
* [[3-category]]
* [[n-category]]
* [[(∞,0)-category]]
* [[(∞,1)-category]]
* [[table - models for (∞,1)-categories]]
* [[(∞,2)-category]]
* **(∞,n)-category**
* [[category object in an (∞,1)-category]]
* [[n-category object in an (∞,1)-category]]
* [[n-fold complete Segal space]]
* [[Theta-space]], [[n-quasicategory]], [[model structure on cellular sets]]
* [[symmetric monoidal (∞,n)-category]]
* [[(n,r)-category]]
* [[(∞,n)-sheaf]], [[(∞,n)-topos]]
* [[directed homotopy type theory]]
## References ##
Definition in terms of [[n-fold complete Segal spaces]] and [[Segal n-categories]] are due to the (unpublished) thesis
* [[Clark Barwick]], _$(\infty,n)$-$Cat$ as a closed model category_ PhD (2005)
The definition in terms of [[Theta spaces]] is due to
* {#Rezk} [[Charles Rezk]], _A cartesian presentation of weak n-categories_ ([arXiv:0901.3602](http://arxiv.org/abs/0901.3602))
An iterartive definition in terms of [[n-fold complete Segal spaces]] is given in
* {#Lurie} [[Jacob Lurie]], _$(\infty,2)$-Categories and the Goodwillie Calculus I_ ([arXiv:0905.0462](http://arxiv.org/abs/0905.0462))
A summary of definitions and some known comparison results can be found in
* [[Julie Bergner]], _Models for $(\infty,n)$-Categories and the Cobordism Hypothesis_ , in [[Urs Schreiber]], [[Hisham Sati]] (eds.) _[[Mathematical Foundations of Quantum Field and Perturbative String Theory]]_, Proceedings of Symposia in Pure Mathematics, volume 83
AMS (2011) ([arXiv:1011.0110](http://arxiv.org/abs/1011.0110))
A textbook account focusing on [[n-fold complete Segal spaces]] and related models is in
* {#Paoli19} [[Simona Paoli]], _Simplicial Methods for Higher Categories -- Segal-type Models of Weak $n$-Categories_, Springer 2019 ([doi:10.1007/978-3-030-05674-2](https://doi.org/10.1007/978-3-030-05674-2), [toc pdf](https://link.springer.com/content/pdf/bfm%3A978-3-030-05674-2%2F1.pdf))
One axiomatic characterization is in
* {#BarwickSchommerPries}[[Clark Barwick]], [[Chris Schommer-Pries]], _On the Unicity of the Homotopy Theory of Higher Categories_ ([arXiv:1112.0040](http://arxiv.org/abs/1112.0040), [slides](http://prezi.com/w0ykkhh5mxak/the-uniqueness-of-the-homotopy-theory-of-higher-categories/))
Comparison of models ($\Theta_{n+1}$-spaces and [[enriched (infinity,1)-categories]] in $\Theta_n$-spaces) is in
* {#BergnerRezk} [[Julie Bergner]], [[Charles Rezk]], _Comparison of models for $(\infty,n)$-categories_ ([arXiv:1204.2013](http://arxiv.org/abs/1204.2013))
* {#BergnerRezk14} [[Julie Bergner]], [[Charles Rezk]], _Comparison of models for $(\infty,n)$-categories II_ ([arXiv:1406.4182](http://arxiv.org/abs/1406.4182))
* [[Rune Haugseng]], *On the equivalence between $\Theta_n$-spaces and iterated Segal spaces*, [arXiv](https://arxiv.org/abs/1604.08480)
* [[Julie Bergner]], _A survey of models for $(\infty,n)$-categories_ ([arXiv:1810.10052](https://arxiv.org/abs/1810.10052))
A model for $(\infty,n)$-categories in terms of [[(∞,1)-sheaves]] on variant of a [[site]] of $n$-[[dimension|dimensional]] [[manifolds]] with [[embeddings]] between them is discussed in
* [[David Ayala]], [[Nick Rozenblyum]], _Weak $n$-categories are sheaves on iterated submersions of $\leq n$-manifolds_ (in preparation)
previewed in
* [[David Ayala]], _Higher categories are sheaves on manifolds_, talk at _[FRG Conference on Topology and Field Theories](http://www.nd.edu/~cmnd/conferences/topology/)_, U. Notre Dame (2012) ([video](http://www.youtube.com/watch?v=8nm2ByS5NnY))
**Abstract** [[topological chiral homology|Chiral]]/[[factorization homology]] gives a procedure for constructing a [[topological field theory]] from the data of an [[En-algebra]]. I'll explain a mulit-object version of this construction which produces a topological field theory from the data of an $n$-category with adjoints. This construction is a consequence of a more primitive result which asserts an equivalence between [[(infinity,n)-category|n-categories]] with adjoints and "transversality sheaves" on [[framed manifold|framed]] $n$-[[manifolds]] - of which there is an abundance of examples.
This lends itself to a model of _[[(∞,n)-category with adjoints]]_. See there for more.
The first globular and algebraic models of $(\infty,n)$-categories is in
* [[Camell Kachour]], Algebraic Definition of weak $(\infty,n)$-Categories, Published in Theory and Applications of Categories (2015), Volume 30, No. 22, pages 775-807: [journal web site](http://tac.mta.ca/tac/volumes/30/22/30-22abs.html)
A model structure using [[complicial sets]] is in
* [[Viktoriya Ozornova]], [[Martina Rovelli]], *Model structures for $(\infty,n)$-categories on (pre)stratified simplicial sets and prestratified simplicial spaces}, Algebr. Geom. Topol. **20** (2020) 1543-1600[[arxiv:1809.10621](https://arxiv.org/abs/1809.10621), [doi:10.2140/agt.2020.20.1543](https://doi.org/10.2140/agt.2020.20.1543)]
[[!redirects (infinity,r)-category]]
[[!redirects (infinity,k)-category]]
[[!redirects (infinity,r)-categories]]
[[!redirects (infinity,k)-categories]]
[[!redirects (∞,n)-category]]
[[!redirects (∞,n)-categories]]
[[!redirects (infinity,n)-categories]]
[[!redirects (∞,r)-category]]
[[!redirects (∞,r)-categories]]
[[!redirects (∞,k)-category]]
[[!redirects (∞,k)-categories]] |
(infinity,n)-category of cobordisms | https://ncatlab.org/nlab/source/%28infinity%2Cn%29-category+of+cobordisms |
> under construction
+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Higher category theory
+--{: .hide}
[[!include higher category theory - contents]]
=--
#### Manifolds and Cobordisms
+--{: .hide}
[[!include manifolds and cobordisms - contents]]
=--
#### Cobordism theory
+--{: .hide}
[[!include cobordism theory -- contents]]
=--
#### Functorial Quantum Field Theory
+--{: .hide}
[[!include functorial quantum field theory - contents]]
=--
=--
=--
#Contents#
* table of contents
{:toc}
## Idea
$n$-Dimensional [[manifolds]] (possibly and usually equipped with certain structure, notably for instance with [[orientation]], [[framing]]-structure or more general [[G-structure]]) should naturally form an [[(∞,n)-category]] of [[extended cobordisms]] whose
* [[objects]] are 0-dimensional (oriented) manifolds (disjoint unions of (oriented) points);
* 1-[[morphisms]] are (oriented) [[cobordisms]] between disjoint unions of (oriented) points;
* 2-morphisms are cobordisms between 1-dimensional cobordisms
* etc.
* (n+1)-morphisms are _diffeomorphisms_ between $n$-dimensional cobordisms;
* (n+2)-morphisms are smooth homotopies of these;
* etc.
The $(\infinity,n)$-category of cobordisms is the subject of the [[cobordism hypothesis]].
## Definition
### As an $n$-fold complete Segal space
Here is an outline of the idea of the definition of
$Bord_{(\infty,n)}$ as given in ([Lurie](#LurieQFT)) where the main point, apart from the [[(∞,n)-category]] machinery in the background, is definition 2.2.9.
The idea is to start with thinking of $n$-dimensional cobordisms as forming something like an [[n-fold category]] by simply saying that the collection of composites of cobordisms is given by big cobordisms with markings on them, indicating where we think of them as being composed.
Let's first do this for composition in one direction, as in an ordinary 1-category of $n$-dimensional cobordisms.
Consider a [[manifold]] $X \hookrightarrow V \times \mathbb{R}$ embedded in a [[vector space]] of the form $V \times \mathbb{R}$. We can think of this as a manifold canonically equipped with a coordinate function $\phi : X \hookrightarrow V \times \mathbb{R} \to \mathbb{R}$ that measures the "height" or maybe better the "length" of the embedded manifold.
We can pick a bunch of numbers $\{t_j \in \mathbb{R}\}$ and think of these as marking a bunch of slices of $X$, the preimages $\phi^{-1}(t_j)$. We can think of these slices as being the $(n-1)$-dimensional boundary manifolds at which a sequence of manifolds have been glued together to produce $X$.
> (there is an obvious picture to be drawn and uploaded here, maybe somebody finds the time and energy)
In this way an embedded manifold $X \hookrightarrow V \times \mathbb{R} $ and a set of $k$-numbers $\{t_i\}$ may represent an element in the space of sequences of composable cobordisms. To make this work as expected, the markings on $X$ may not be too irregular, so we should impose some conditions on what qualifies as a marked manifold. The precise statement is given further below.
The collection of these tuples, consisting of an embedded manifold $X \hookrightarrow V \times \mathbb{R}$ and a collection of $k$ numbers $\{t_i \in \mathbb{R}\}$ naturally form a [[simplicial set]], which is like the [[nerve]] of the 1-category of $n$-dimensional cobordisms under composition in one direction.
To generalize this from just a 1-categorical structure to an $n$-categorical structure, we simply take a manifold $X$ as before, but now draw markings on it in $n$ transversal directions, thereby putting a kind of grid on it that subdivides the manifold into cubical slices. A manifold with such subdivision on it may then be regarded as giving an element in the space of $n$-dimensional pasting diagrams in an $n$-fold category.
To formalize this more general case, we embed $X$ not just into a $V \times \mathbb{R}$, but a $V \times \mathbb{R}^n$. This then provides us with $n$ different coordinate functions $\phi_i : X \hookrightarrow V \times \mathbb{R}^n \stackrel{p_i}{\to} \mathbb{R}$ on $X$, each running along one of the directions in which we may think of $X$ as having been glued from smaller manifolds.
A collection of markings indicating such gluing is now a collection of numbers $\{t_j^1\}, \;\{t_j^2\}, \; \cdots \{t_j^n\}$, one for each of these directions.
For each direction this yields a [[simplicial set]] of such structures, to be thought of as the [[nerve]] of the category of cobordisms under composition in one of these directions. Taken together this is an $n$-fold simplicial set
$$
\Delta^{op} \times \Delta^{op} \times \cdots \times \Delta^{op}
\to Set
$$
which is like the nerve of an $n$-fold category of cobordisms.
When suitable regularity conditions are imposed on this data, there is naturally a [[topology]] on each of these sets of embedded marked cobordisms, that makes this into an $n$-fold simplicial [[topological space]]
$$
\Delta^{op} \times \Delta^{op} \times \cdots \times \Delta^{op}
\to Top
\,.
$$
To get rid of the dependence of this construction on $V$, we can let $V$ "grow arbitrarily large" by taking the [[colimit]] of the above $n$-fold cosimplicial spaces as $V$ ranges over the finite dimensional subspaces of $\mathbb{R}^\infty$.
The resulting $n$-fold simplicial topological space obtained by this colimit then is essentially the [[(∞,n)-category]] $Bord_n$ that we are after. It turns out that it actually is an $n$-fold Segal space. We just formally _complete_ it to an [[n-fold complete Segal space]]
$$
Bord_n : (\Delta^{op})^n \to Top
\,.
$$
This, then, is a model for the [[(∞,n)-category]] of extended $n$-dimensional cobordisms.
### As a blob $n$-category
There is a definition of a [[blob n-category]] of $n$-cobordisms. See there for more details.
## Examples
### $Bord_2^{fr}$
{#Framed2Bordisms}
Some comments on 2-[[framed manifold|framed]] 2-cobordisms.
Consider the pictures in ([Schommer-Pries 13, figure 5](#SchommerPries13)).
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> Somebody should produce pictures like this here...
>> Well, it worked in the preview; a little help?
Let $\gamma$ be a 1-dimensional manifold of the form of the interval $[0,1]$. A 2-[[framing]] of $\gamma$ is a trivialization of $T\gamma \oplus \mathbb{R}$. Let $\{1\} \subset \mathbb{R}$ be the canonical [[basis]] of $\mathbb{R}$. If we think of the plane $\mathbb{R}^2$ as equipped with its canonical 2-framing, then a 2-framing of $\gamma$ is induced by embedding $\gamma$ into the plane and shading one of its two sides. This identifies at each point $x \in \gamma$ the tangent space to $\gamma$ at that point with the tangent vector to the embedding of $\gamma$ as a vector in $\mathbb{R}^2$ and identifies $1\in \mathbb{R}$ with the vector in $\mathbb{R}^2$ orthogonal to this tangent vector and pointing into the shaded region.
This shows that if $\gamma$ is regarded with its two endpoints both as incoming or both as outgoing, then the induced 2-framing of these endpoints is opposite to each other. This way such an arc is a morphism from the union of the "positive point" and the "negative point" to the empty 0-manifold, hence is a unit/counit exhibiting these as [[dual objects]].
## Properties
{#Properties}
### Adjoints
$Bord_n$ is an [[(∞,n)-category with all adjoints]].
### Relation to Thom spectrum
For $n \to \infty$ we have that $Bord_{(\infty,\infty)}$ is the [[symmetric monoidal (infinity,1)-category|symmetric monoidal]] [[∞-groupoid]] ($\simeq$ [[infinite loop space]]) $\Omega^\infty M O$ that underlies the [[Thom spectrum]].
Its [[homotopy group]]s are the [[cobordism ring]]s
$$
\pi_n Bord_{(\infty,\infty)} \simeq \Omega_n
\,.
$$
Therefore a symmetric monoidal $\infty$-functor
$$
Bord_{(\infty,\infty)} \to S
$$
to some symmetric monoidal $\infty$-groupoid $S$ is a [[genus]].
## Related concepts
* [[cobordism]], [[extended cobordism]]
* [[category of cobordisms]]
* **(∞,n)-category of cobordisms**
* [[cobordism hypothesis]]
## References
### General
A specific realization of this idea in terms of [[(∞,n)-category]] modeled as [[n-fold complete Segal space]] is in (definition 2.2.9, page 36)
* {#LurieQFT} [[Jacob Lurie]], _[[On the Classification of Topological Field Theories]]_ .
* [[Jacob Lurie]], [Video Stream](http://www.youtube.com/watch?v=Bo8GNfN-Xn4)
In that article a proof of the [[cobordism hypothesis]] is indicated. A review is in
* [[Julie Bergner]], _Models for $(\infty,n)$-Categories and the Cobordism Hypothesis_ , in [[Hisham Sati]], [[Urs Schreiber]] (eds.) _[[Mathematical Foundations of Quantum Field and Perturbative String Theory]]_ Available [on the arxiv](http://arxiv.org/abs/1011.0110).
Other discussions of higher categories of cobordisms are
* [[Marco Grandis]], _[[Cospans in Algebraic Topology]]_
* [[Eugenia Cheng]] and [[Nick Gurski]], _Toward an $n$-category of cobordisms_ , Theory and Applications of Categories 18 (2007), 274-302. ([tac](http://www.tac.mta.ca/tac/volumes/18/10/18-10abs.html))
### In dimension 2
A detailed construction of the [[(n,r)-category|(2,2)-category]] of 2-dimensional cobordisms is
* [[Chris Schommer-Pries]], _[[2-category of 2-dimensional cobordisms]]_.
* {#SchommerPries13} [[Chris Schommer-Pries]], _Dualizability in Low-Dimensional Higher Category Theory_ ([arXiv:1308.3574](http://arxiv.org/abs/1308.3574))
Discussion of a [[2-category]] of complex cobordisms, aimed at formalizing chiral [[2d CFT]]:
* [[André Henriques]], *The complex cobordism 2-category*, 2021 ([video](http://andreghenriques.com/ComplexCob2CatandCentralExt.mp4))
### In dimension 3
* [[Bruce Bartlett]], [[Christopher Douglas]], [[Christopher Schommer-Pries]], [[Jamie Vicary]], _Extended 3-dimensional bordism as the theory of modular objects_ ([arXiv:1411.0945](http://arxiv.org/abs/1411.0945))
### In dimension $\infty$
For a discussion of the relation of $Bord_{(\infty,\infty)}$ to the [[Thom spectrum]] and the [[cobordism ring]] see also
* [[John Francis]], _Cobordisms_ (notes by [[Owen Gwilliam]]) ([pdf](http://math.northwestern.edu/~jnkf/classes/mflds/2cobordism.pdf))
[[!redirects (∞,n)-category of cobordisms]]
[[!redirects (∞,n)-categories of cobordisms]]
[[!redirects (infinity,n)-categories of cobordisms]]
|
(infinity,n)-category of correspondences | https://ncatlab.org/nlab/source/%28infinity%2Cn%29-category+of+correspondences | [[!redirects (infinity,n)-category of spans]]
+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Higher category theory
+--{: .hide}
[[!include higher category theory - contents]]
=--
=--
=--
#Contents#
* table of contents
{:toc}
## Idea
The generalization of the [[bicategory]] _[[Span]]_ to [[(∞,n)-categories]]:
An _$(\infty,n)$-category of correspondences_ in [[∞-groupoid]] is an [[(∞,n)-category]] whose
* [[objects]] are [[∞-groupoids]];
* [[morphism]]s $X \to Y$ are [[correspondences]]
$$
\array{
&& Z
\\
& \swarrow && \searrow
\\
X &&&& Y
}
$$
in [[∞Grpd]]
* [[2-morphisms]] are correspondences of correspondences
$$
\array{
&& Z
\\
& \swarrow &\uparrow& \searrow
\\
X &&Q&& Y
\\
& \nwarrow &\downarrow& \nearrow
\\
&& Z'
}
$$
(where the triangular sub-[[diagram]]s are filled with [[2-morphism]]s in [[∞Grpd]] which we do not display here)
* and so on up to [[k-morphism|n-morphism]]s
* $k \gt n$-morphisms are equivalences of order $(k-n)$ of higher correspondences.
Using the symmetric monoidal structure on [[∞Grpd]] this becomes a [[symmetric monoidal (∞,n)-category]].
More generally, for $C$ some [[symmetric monoidal (∞,n)-category]], there is a symmetric monoidal $(\infty,n)$-category of correspondences over $C$, whose
* [[object]]s are [[∞-groupoid]]s $X$ equipped with an [[(∞,n)-functor]] $X \to C$;
* [[morphism]]s $X \to Y$ are [[correspondences]] in [[(∞,1)Cat]] over $C$
$$
\array{
&& Z
\\
& \swarrow && \searrow
\\
X &&\swArrow&& Y
\\
& \searrow && \swarrow
\\
&& C
}
$$
* and so on.
Even more generally one can allow the [[∞-groupoid]]s $X, Y, \cdots$ to be [[(∞,n)-categories]] themselves.
## Definition
{#Definition}
### Direct definition
The [[(∞,2)-category]] of correspondences in [[∞Grpd]] is discussed in some detail in ([Dyckerhoff-Kapranov 12, section 10](#DyckerhoffKapranov12)).
A sketch of the definition for all $n$ was given in ([Lurie, page 57](#Lurie)). A fully detailed version of this definition is in ([Haugseng 14](#Haugseng14)).
### Definition via coalgebras
{#DefinitionViaCoalgebras}
In ([BenZvi-Nadler 13, remark 1.17](#BenZviNadler13)) it is observed that
$$
Corr_n(\mathbf{H}) \simeq E_n Alg_b(\mathbf{H}^{op})
$$
is equivalently the [[(∞,n)-category]] of [[En-algebras]] and [[(∞,1)-bimodules]] between them in the [[opposite (∞,1)-category]] of $\mathbf{H}$ (since every object in a cartesian category is uniquely a [[coalgebra]] by its [[diagonal]] map).
(This immediately implies that every object in $Corr_n(\mathbf{H})$ is a self-[[fully dualizable object]].)
To see how this works, consider $X \in \mathbf{H}$ any object regarded as a coalgebra in $\mathbf{H}$ via its [[diagonal map]] ([here](cartesian+monoidal+infinity%2C1-category#CoalgebraObjects)). Then a [[comodule]] $E$ over it is a [[co-action]]
$$
E \to E \times X
$$
and hence is canonically given by just a map $E \to X$.
Then for
$$
\array{
&& E_1 &&&& E_2
\\
& \swarrow && \searrow && \swarrow && \searrow
\\
X && && Y && && Z
}
$$
two consecutive correspondences, now interpreted as two bi-comodules, their [[tensor product]] of comodules over $Y$ as a coalgebra is the limit over
$$
E_1 \times E_2
\stackrel{\to}{\to}
E_1 \times Y \times E_2
\stackrel{\to}{\stackrel{\to}{\to}}
...
$$
This is indeed the fiber product
$$
E_1 \underset{Y}{\times} E_2 \stackrel{(p_1, p_2)}{\to} E_1 \times E_2
$$
as it should be for the composition of [[correspondences]].
### With the phased tensor product
{#PhasedTensorProduct}
+-- {: .num_defn #InSliceWithPhasedTensorProduct}
###### Proposition
For $\mathbf{H}$ an [[(∞,1)-topos]] and $\mathcal{C} \in Cat_{(\infty,n)}(\mathbf{H})$ a [[symmetric monoidal (∞,n)-category|symmetric monoidal]] [[internal (∞,n)-category]] then there is a [[symmetric monoidal (∞,n)-category]]
$$
Corr_n(\mathbf{H}_{/\mathcal{C}})^\otimes \in SymmMon (\infty,n)Cat
$$
whose [[k-morphisms]] are $k$-fold correspondence in $\mathbf{H}$ over $k$-fold correspondences in $\mathcal{C}$, and whose monoidal structure is given by
$$
\left[
\array{
X_1
\\
\downarrow^{\mathrlap{\mathbf{L}_1}}
\\
\mathcal{C}_0
}
\right]
\otimes
\left[
\array{
X_2
\\
\downarrow^{\mathrlap{\mathbf{L}_2}}
\\
\mathcal{C}_0
}
\right]
\coloneqq
\left[
\array{
X_1 \times X_2
\\
\downarrow^{\mathrlap{(\mathbf{L}_1, \mathbf{L}_2)}}
\\
\mathcal{C}_0 \times \mathcal{C}_0
\\
\downarrow^{\mathrlap{\otimes_{\mathcal{C}}}}
\\
\mathcal{C}_0
}
\right]
\,.
$$
=--
This is ([Haugseng 14, def. 4.6, corollary 7.5](#Haugseng14))
+-- {: .num_remark}
###### Remark
If $\mathcal{C}_0$ is (or is regarded as) a [[moduli stack]] for some kind of bundles forming a [[linear homotopy type theory]] over $\mathbf{H}$, then the phased tensor product is what is also called the _[[external tensor product]]_.
=--
+-- {: .num_example}
###### Example
Examples of phased tensor products include
* the [Cauchy product of species](species#HoTTCauchyProduct);
* some [[external tensor products]] in [[indexed monoidal categories]];
=--
## Properties
{#Properties}
### Full dualizability
{#FullDualizability}
+-- {: .num_prop #CorrnGrpdHasDuals}
###### Proposition
$Corr_n(\infty Grpd)$ is a [[symmetric monoidal (∞,n)-category|symmetric monoidal]] [[(∞,n)-category with duals]].
More generally, if $\mathcal{C}$ is a symmetric monoidal $(\infty,n)$-category with
duals, then so is $Corr_n(\infty Grpd,\mathcal{C})^\otimes$ equipped with the
phased tensor product of prop. \ref{InSliceWithPhasedTensorProduct}.
In particular every object in these is a [[fully dualizable object]].
=--
This appears as ([Lurie, remark 3.2.3](#Lurie)). A proof is written down in ([Haugseng 14, corollary 6.6](#Haugseng14)).
+-- {: .num_conjecture }
###### Conjecture
The canonical $O(n)$-[[∞-action]] on $Corr_n(\infty Grpd)$ induced via prop. \ref{CorrnGrpdHasDuals} by the [[cobordism hypothesis]] (see there at _[the canonical O(n)-action](cobordism+hypothesis#TheCanonicalOnAction)_) is trivial.
=--
This statement appears in ([Lurie, below remark 3.2.3](#Lurie)) without formal proof. For more see ([Haugseng 14, remark 9.7](#Haugseng14)).
More generally:
+-- {: .num_prop }
###### Proposition
For $\mathbf{H}$ an [[(∞,1)-topos]], then $Corr_n(\mathbf{H})$
is an [[(∞,n)-category with duals]].
And generally, for $\mathcal{C} \in SymmMon (\infty,n)Cat(\mathbf{H})$ a [[symmetric monoidal (∞,n)-category]] [[internal (∞,n)-category|internal]] to $\mathbf{C}$, then $Corr_n(\mathbf{H}_{/\mathbf{C}})$ equipped with the phased tensor product of prop. \ref{InSliceWithPhasedTensorProduct} is an [[(∞,n)-category with duals]]
=--
([Haugseng 14, cor. 7.8](#Haugseng14))
Let $Bord_n$ be the [[(∞,n)-category of cobordisms]].
+-- {: .num_prop #HomsFromBordIntoSpan}
###### Claim
The following data are equivalent
1. Symmetric monoidal $(\infty,n)$-functors
$$
Bord_n \to Corr_n(\infty Grpd)
$$
1. Pairs $(X,V)$, where $X$ is a [[topological space]] and $V \to X$ a [[vector bundle]] of [[rank]] $n$.
=--
This appears as ([Lurie, claim 3.2.4](#Lurie)).
## Related concepts
* [[symplectic category]]
* [[span trace]]
* [[relations]], [[bicategory of relations]]
* [[sheaf with transfer]], [[Mackey functor]]
## References
For references on 1- and 2-categories of spans see at _[[correspondences]]_.
An explicit definition of the [[(∞,2)-category]] of spans in [[∞Grpd]] is in section 10 of
* {#DyckerhoffKapranov12} Tobias Dyckerhoff, [[Mikhail Kapranov]], _Higher Segal spaces I_, ([arxiv:1212.3563](http://arxiv.org/abs/1212.3563))
An inductive definition of the [[symmetric monoidal (∞,n)-category]] $Span_n(\infty Grpd)/C$ of spans of [[∞-groupoid]] over a symmetric monoidal $(\infty,n)$-category $C$ is sketched in section 3.2 of
* {#Lurie} [[Jacob Lurie]], _[[On the Classification of Topological Field Theories]]_
there denoted $Fam_n(C)$. Notice the heuristic discussion on page 59.
More detailed discussion is given in
* {#Haugseng14} [[Rune Haugseng]], _Iterated spans and "classical" topological field theories_ ([arXiv:1409.0837](http://arxiv.org/abs/1409.0837))
* [[Yonatan Harpaz]], _Ambidexterity and the universality of finite spans_ ([arXiv:1703.09764](https://arxiv.org/abs/1703.09764))
Both articles comment on the relation to [[schreiber:Local prequantum field theory]].
The generalization to an $(\infty,n)$-category $Span_n((\infty,1)Cat^Adj)$ of spans between [[cobordism hypothesis|(∞,n)-categories with duals]] is discussed on p. 107 and 108.
The extension to the case when the ambient $\infty$-topos is varied is in
* {#LiBland15} [[David Li-Bland]], _The stack of higher internal categories and stacks of iterated spans_, ([arXiv:1506.08870](http://arxiv.org/abs/1506.08870))
The application of $Span_n(\infty Grpd/C)$ to the construction of [[FQFT]]s is further discussed in section 3 of
* [[Dan Freed]], [[Mike Hopkins]], [[Jacob Lurie]], [[Constantin Teleman]], _[[Topological Quantum Field Theories from Compact Lie Groups]]_
Discussion of $Span_n(\mathbf{H}) \simeq Alg_{E_n}(\mathbf{H}^{op})$ is around remark 1.17 of
* {#BenZviNadler13} [[David Ben-Zvi]], [[David Nadler]], _Nonlinear traces_ ([arXiv:1305.7175](http://arxiv.org/abs/1305.7175))
A discussion of a version $Span(B)$for $B$ a [[2-category]] with $Span(B)$ regarded as a [[tricategory]] and then as a 1-object [[tetracategory]] is in
* {#Hoffnung} [[Alex Hoffnung]], _Spans in 2-Categories: A monoidal tricategory_ ([arXiv:1112.0560](http://arxiv.org/abs/1112.0560))
A discussion that $Span_2(-)$ in a [[2-category]] with weak [[finite limits]] is a [[compact closed 2-category]]:
* {#Stay13} [[Mike Stay]], _Compact Closed Bicategories_ ([arXiv:1301.1053](http://arxiv.org/abs/1301.1053))
See also
* {#ayalaFrancis17} [[David Ayala]], [[John Francis]], _Fibrations of $\infty$-Categories_ ([arXiv:1702.02681](https://arxiv.org/abs/1702.02681))
Coisotropic orrespondences for derived Poisson stacks:
* {#HMS19} [[Rune Haugseng]], Valerio Melani, [[Pavel Safronov]], _Shifted Coisotropic Correspondences_ ([arXiv:1904.11312](https://arxiv.org/abs/1904.11312))
[[!redirects (∞,n)-category of spans]]
[[!redirects (∞,n)-categories of spans]]
[[!redirects (infinity,n)-categories of spans]]
[[!redirects (∞,n)-category of correspondences]]
[[!redirects (∞,n)-categories of correspondences]]
[[!redirects (infinity,n)-category of correspondences]]
[[!redirects (infinity,n)-categories of correspondences]]
[[!redirects (∞,1)-category of correspondences]]
[[!redirects (∞,1)-categories of corespondences]]
[[!redirects (infinity,1)-category of correspondences]]
[[!redirects (infinity,1)-categories of correspondences]]
[[!redirects (2,1)-category of correspondences]]
[[!redirects (2,1)-categories of correspondences]]
[[!redirects higher correspondence]]
[[!redirects higher correspondences]]
[[!redirects higher category of correspondences]]
[[!redirects higher categories of correspondences]]
[[!redirects n-fold corespondence]]
[[!redirects n-fold corespondences]]
[[!redirects n-fold correspondence]]
[[!redirects n-fold correspondences]]
[[!redirects (∞,n)-category of n-fold correspondences]]
[[!redirects (infinity,n)-category of n-fold correspondences]]
[[!redirects phased tensor product]]
[[!redirects phased tensor products]]
|
(infinity,n)-category with adjoints | https://ncatlab.org/nlab/source/%28infinity%2Cn%29-category+with+adjoints |
+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Higher category theory
+--{: .hide}
[[!include higher category theory - contents]]
=--
=--
=--
#Contents#
* table of contents
{:toc}
## Idea
An [[(∞,n)-category]] $\mathcal{C}$ is said to _have 1-adjoints_ if in its [[homotopy 2-category]] $Ho_2(\mathcal{C})$ every [[1-morphism]] is part of an [[adjunction]]. By [[recursion]], for $n \geq 3$ and $k \geq 2$ an [[(∞,n)-category]] _has $k$-adjoints_ if for every pair $X, Y$ of [[objects]] the [[hom object|hom (∞,n-1)-category]] $\mathcal{C}(X,Y)$ has adjoints for $(k-1)$-morphisms.
An $(\infty,n)$-category _has all adjoints_ (or just _has adjoints_, for short) if it has adjoints for $k$-morphisms for $0 \lt k \lt n$.
If in addtition every [[object]] in $\mathcal{C}$ is a [[fully dualizable object]], then $\mathcal{C}$ is called an _[[(∞,n)-category with duals]]_.
## Properties
### Internal language
The [[internal language]] of $(\infty,n)$-categories with duals seems plausibly to be axiomatizable in [[opetopic type theory]].
## Examples
* An [[(∞,n)-category of cobordisms]] has all adjoints.
## Related concepts
* [[(∞,n)-category with duals]]
* [[fully dualizable object]]
## References
{#References}
The notion appears first in section 2.3 of
* [[Jacob Lurie]], _[[On the Classification of Topological Field Theories]]_
A model for $(\infty,n)$-categories with all adjoints in terms of [[(∞,1)-sheaves]] on a [[site]] of a variant of $n$-[[dimension|dimensional]] [[manifolds]] with [[embeddings]] between them is discussed in
* [[David Ayala]], [[Nick Rozenblyum]], _Weak $n$-categories are sheaves on iterated submersions of $\leq n$-manifolds_ (in preparation)
* [[David Ayala]], [[Nick Rozenblyum]], _Weak $n$-categories with adjoints are sheaves on $n$-manifolds_ (in preparation)
previewed in
* {#Ayala12} [[David Ayala]], _Higher categories are sheaves on manifolds_, talk at _[FRG Conference on Topology and Field Theories](http://www.nd.edu/~cmnd/conferences/topology/)_, U. Notre Dame (2012) ([video](http://www.youtube.com/watch?v=8nm2ByS5NnY))
**Abstract** [[topological chiral homology|Chiral]]/[[factorization homology]] gives a procedure for constructing a [[topological field theory]] from the data of an [[En-algebra]]. I'll explain a multi-object version of this construction which produces a topological field theory from the data of an $n$-category with adjoints. This construction is a consequence of a more primitive result which asserts an equivalence between [[(infinity,n)-category|n-categories]] with adjoints and "transversality sheaves" on [[framed manifold|framed]] $n$-[[manifolds]] - of which there is an abundance of examples.
* [[Nick Rozenblyum]], _Manifolds, Higher Categories and Topological Field Theories_, talk Northwestern University (2012) ([pdf slides](http://math.northwestern.edu/~nrozen/notes/jmm12.pdf))
[[!redirects (∞,n)-category with all adjoints]]
[[!redirects (infinity,n)-category with all adjoints]]
[[!redirects (∞,n)-category with adjoints]]
[[!redirects (∞,n)-categories with adjoints]]
[[!redirects (infinity,n)-categories with adjoints]]
|
(infinity,n)-category with duals | https://ncatlab.org/nlab/source/%28infinity%2Cn%29-category+with+duals | +-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### Higher category theory
+-- {: .hide}
[[!include higher category theory - contents]]
=--
#### Monoidal categories
+-- {: .hide}
[[!include monoidal categories - contents]]
=--
=--
=--
# Contents
* table of contents
{:toc}
## Idea
An [[(∞,n)-category with adjoints]] (see there for more) and a ([[fully dualizable object|fully]]) [[dual object]] for every [[object]].
## Definition
+-- {: .num_defn }
###### Definition
Let $C$ be an [[(∞,n)-category]]. We say that
* $C$ **has adjoints for morphisms** if in its [[homotopy 2-category]] every [[morphism]] has a [[left adjoint]] and a [[right adjoint]];
* for $1 \lt k \lt n$ that $C$ **has adjoints for [[k-morphism]]s** if for every pair $X,Y \in C$ of [[object]]s, the [[hom-set|hom-(∞,n-1)-category]] $C(X,Y)$ has adjoints for $(k-1)$-morphisms.
* $C$ is an **[[(∞,n)-category with adjoints]]** if it has adjoints for [[k-morphisms]] with $0 \lt k \lt n$.
If $C$ is in addition a [[symmetric monoidal (∞,n)-category]] we say that
* $C$ **has duals for objects** if its [[homotopy category]] is a [[rigid monoidal category]].
Finally we say that
* $C$ **has duals** if it has duals for objects and has adjoints.
=--
This is ([Lurie, def. 2.3.13, def. 2.3.16](#Lurie)). See at _[[fully dualizable object]]_
## Properties
### Internal language
The [[internal language]] of $(\infty,n)$-categories with duals seems plausible to be axiomatized inside [[opetopic type theory]].
## Examples
* [[(∞,n)-category of cobordisms]]
* [[(∞,n)-category of n-fold correspondences]]
## Related concepts
* [[fundamental n-category]]
* [[rigid monoidal category]]
* [[category with duals]]
* [[dualizable object]], [[fully dualizable object]]
* [[blob n-category]]
## References
* {#Lurie} [[Jacob Lurie]], _[[On the Classification of Topological Field Theories]]_
For more see at _[[(infinity,n)-category with adjoints]]_.
[[!redirects (infinity,n)-category with duals]]
[[!redirects (infinity,n)-categories with duals]]
[[!redirects (infinity,n)-categories with duales]]
[[!redirects (∞,n)-category with duals]]
[[!redirects (∞,n)-categories with duals]]
[[!redirects (∞,n)-categories with duales]]
[[!redirects symmetric monoidal (infinity,n)-category with duals]]
[[!redirects symmetric monoidal (infinity,n)-categories with duals]]
[[!redirects symmetric monoidal (∞,n)-category with duals]]
[[!redirects symmetric monoidal (∞,n)-categories with duals]]
|
(infinity,n)-functor | https://ncatlab.org/nlab/source/%28infinity%2Cn%29-functor |
#Contents#
* table of contents
{:toc}
## Idea
... a morphism between [[(∞,n)-categories]] ...
## Related concepts
* [[function]]
* [[functor]]
* [[2-functor]] / [[pseudofunctor]]
* [[n-functor]]
* [[(∞,1)-functor]]
* **(∞,n)-functor**
* [[monoidal (∞,n)-functor]]
[[!redirects (∞,n)-functor]]
[[!redirects (∞,n)-functors]]
[[!redirects (infinity,n)-functors]]
|
(infinity,n)-module | https://ncatlab.org/nlab/source/%28infinity%2Cn%29-module |
+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Higher category theory
+--{: .hide}
[[!include higher category theory - contents]]
=--
#### Higher linear algebra
+-- {: .hide}
[[!include homotopy - contents]]
=--
=--
=--
#Contents#
* table of contents
{:toc}
## Idea
A notion of _$n$-module_ ($n$-vector space) is a [[categorification]] of the notion of [[module]] ([[vector space]]).
There are various different notions of $n$-vector spaces.
One notion is: an $n$-vector space is a [[chain complex]] of [[vector space]]s in degrees 0 to $n$. For $n=2$ this is a [[Baez-Crans 2-vector space]]. This is useful for lots of things, but tends to be too restrictive in other contexts.
Another is, recursively: an $(n-1)$-algebra object (or its $(n-1)$-category of modules) in the $n$-category of $(n-1)$-bimodules. For higher $n$ this is envisioned in ([FHLT, section 7](#FHLT)), details are in spring. It includes the previous concept as a special case.
For $n=2$ this subsumes various other definitions of [[2-vector space]] that are in the literature, such as notably the notion of [[Kapranov-Voevodsky 2-vector space]].
We sketch the iterative definition of $n$-vector spaces. More details are below.
Assume that a notion of [[n-category]] is chosen for each $n$ (for instance [[(n,1)-category]]), that a notion of [[symmetric monoidal category|symmetric monoidal]] $n$-category is fixed (for instance [[symmetric monoidal (∞,1)-category]]) and that a notion of (weak) commutative [[monoid]] objects and [[module]] and [[bimodule]] object in a symmetric monoidal $n$-category is fixed (for instance the notion of [[algebra in an (∞,1)-category]]).
Then we have the following recursive (rough) definition:
fix a ground [[field]] $k$.
* a 0-vector space over $k$ is an elemment of $k$. The [[0-category]] of 0-vector spaces is the set
$$
0 Vect_k = k
\,.
$$
* The category $1 Vect_k$ is just [[Vect]].
* For $n \gt 1$, the [[n-category]] $n Vect$ of **$n$-vector spaces** over $k$ is the $n$-category with objects algebra objects in $(n-1)Vect$ and morphisms bimodule objects in $(n-1)Vect$.
Here we think of an algebra object $A \in (n-1)Vect$ as a basis for the $n$-vector space which is the $(n-1)$-category $A Mod$.
With this definition we have that $2 Vect$ is the [[2-category]] of $k$-[[algebra]]s, [[bimodule]]s and bimodule homomorphisms.
More generally, let $k$ here be a [[ring spectrum]]. Set
* $(\infty,0)Vect_k := k$ -- a [[symmetric monoidal (infinity,1)-category|symmetric monoidal]] [[∞-groupoid]];
* $(\infty,1)Vect_k := k Mod$ the [[symmetric monoidal (∞,1)-category]] of modules over that ring spectrum;
* $(\infty,n)Vect_k := (\infty,n-1) Mod$ the [[symmetric monoidal (∞,n)-category]] of modules over $(\infty,n-1)Mod$.
## Definition
{#Definition}
Following the [above idea](IteratedModules) we have the following definition.
+-- {: .num_defn #nVectViaALgObjects}
###### Definition
Fix a [[ring]] $k$ (usually taken to be a [[field]] if one speaks of "vector spaces" instead of just [[module]]s, but this is not actually essential for the construction). This may be an [[ring spectrum|∞-ring]].
For $n \in \mathbb{N}$, define an [[symmetric monoidal (∞,n)-category]] $n Vect_k$ of **$(\infty,n)$-vector spaces** as follows (the bi-counting follows the pattern of [[(n,r)-categories]]).
An **$(\infty,0)$-vector space** is an element of $k$. If $k$ is an ordinary ring, then the [[0-category]] $0 Vect$ is the underlying set of $k$, regarded as a [[symmetric monoidal category]] using the product structure on $k$. If $k$ is more generally an [[ring spectrum|∞-ring]], then the "stabilized [[(∞,0)-category]]" (= [[spectrum]]) of $(\infty,0)$-vector spaces is $k$ itself: $(\infty,0)Vect_k \simeq k$.
An **[[(∞,1)-vector space]]** is an [[module spectrum|∞-module]] over $k$. The [[(∞,1)-category]] of $(\infty,1)$-vector spaces is
$$
(\infty,1)Vect_k := k Mod
\,,
$$
the $(\infty,1)$-category of $k$-[[module spectra]].
For $k$ a field ordinary [[vector space]]s over $k$ are a [[full sub-(∞,1)-category]] of this: $1Vect_k \hookrightarrow (\infty,1)Vect_k$ .
For $n \geq 2$, an **$(\infty,n)$-vector space** is an [[algebra in an (infinity,1)-category|algebra object in the symmetric monoidal (∞,1)-category]] $(\infty,n-1)Vect$. A [[morphism]] is a [[bimodule object]]. [[k-morphism|Higher morphisms]] are defined recursively.
=--
For $\infty$ replaced by $n$ this appears as ([Schreiber, appendix A](#Schreiber)) and then with allusion to more sophisticated [[higher category theory|higher categorical]] tools in ([FHLT, def. 7.1](#FHLT)).
Notice that FHLT say "$(n-1)$-algebra" instead of "$n$-vector space", but only for the reason (p. 29) that
> The discrepancy between $m$ (the algebra level) and $n$ [the algebra level] -- for which we apologize -- is caused by the fact that the term "$n$-vector space" has been used for a much more restrictive notion than our $(n-1)$-algebras.
## Examples
### $(\infty,1)$-vector spaces
See [[(∞,1)-vector space]] for more.
### 2-Modules
+-- {: .num_remark}
###### Remark
The symmetric monoidal 3-category $Alg_k^b = 2 Mod_k$ of [[2-modules]] over $k$ is:
* [[objects]] are [[associative algebras]] over $k$;
* [[morphisms]] are [[bimodules]] of associative algebras; [[composition]] is the [[tensor product]] of bimodules;
* [[2-morphisms]] are bimodule homomorphisms.
We think of this equivalently as its essential image in $Vect_k Mod$, where
* an algebra $A$ is a placeholder for its [[module category]] $Mod_A$;
* an $A$-$B$ [[bimodule]] $N$ is a placeholder for the [[functor]]
$$
Mod_A \stackrel{(-) \otimes_A N }{\to} Mod_B
$$
* a bimodule homomorphism is a placeholder for a [[natural transformation]] of two such functors.
If we think of an algebra $A$ in terms of its [[delooping]] [[Vect]]-[[enriched category]] $B A$, then we have an [[equivalence of categories]]
$$
Mod_A \simeq Vect Cat(B A, Vect)
\,.
$$
Comparing this for the formula
$$
V \simeq Set(S,k)
$$
for a $k$-vector space $V$ with [[basis]] $S$, we see that we may
* think of the algebra objects appearing in the above as being _bases_ for a higher vector space;
* think of the bimodules as being higher [[matrix|matrices]].
=--
### 3-Modules
A _3-vector space_ according to def. \ref{nVectViaALgObjects} is
* a $k$-algebra $A$;
* equipped with an $A$-$A\otimes A$-[[bimodule]] defining the 2-multiplication, and a left $A$-[[module]] defining the unit.
Equivalently this is a [[sesquiunital sesquialgebra]].
Classes of examples come from the following construction:
* Every _commutative_ [[associative algebra]] $A$ becomes a 3-vector space.
* Every [[Hopf algebra]] canonically becomes a 3-vector space (amplified in [FHLT, p. 27](#FHLT)).
More generally: every [[hopfish algebra]].
### 4-Modules
Next, an algebra object internal to $2 Alg_k^b = 3Mod_3$, is an algebra equipped with three compatible algebra structures, a [[trialgebra]].
Its [[category of modules]] is a [[monoidal category]] equipped with two compatible product structures a [[Hopf category]].
The 2-category of 2-modules of that is a [[monoidal 2-category]].
For a review see ([Baez-Lauda 09, p. 98](BaezLauda)).
## $n$-Representations
See [[infinity-representation]].
## Related concepts
* [[vector space]],
* [[(∞,1)-module]], [[(∞,1)-module bundle]], [[(∞,1)-category of (∞,1)-modules]]
* [[2-ring]], [[2-module]]
* [[2-vector space]]
* _[[TwoVect]]_ is a Mathematica software package for computer algebra with 2-vector spaces
* **$n$-vector space**, [[n-vector bundle]],
[[!include structure on algebras and their module categories - table]]
## References
The notion of $n$-vector spaces is (defined for $n = 2$ and sketched recursively for greater $n$) in
appendix A of
* {#Schreiber} [[Urs Schreiber]], _AQFT from $n$-functorial QFT_ Communications in Mathematical Physics, Volume 291, Issue 2, pp.357-401 (2008) ([pdf](http://ncatlab.org/schreiber/files/AQFTfromFQFT.pdf))
section 7 of
* {#FHLT} [[Dan Freed]], [[Mike Hopkins]], [[Jacob Lurie]], [[Constantin Teleman]], _[[Topological Quantum Field Theories from Compact Lie Groups]]_ (2009)
Full details are in
* {#Haugseng14} [[Rune Haugseng]], _The higher Morita category of $E_n$-algebras_ ([arXiv:1412.8459](http://arxiv.org/abs/1412.8459))
Review of work on 4-modules (implicitly) as [[trialgebras]]/[[Hopf monoidal categories]] is around p. 98 of
* {#BaezLauda} [[John Baez]], [[Aaron Lauda]], _A prehistory of $n$-categorical physics_, in _Deep beauty_, 13-128, Cambridge Univ. Press, Cambridge, 2011 ([arXiv:0908.2469](http://arxiv.org/abs/0908.2469))
[[!redirects n-vector spaces]]
[[!redirects n-vector spaces]]
[[!redirects nMod]]
[[!redirects (∞,n)-vector space]]
[[!redirects (infinity,n)-vector space]]
[[!redirects (∞,n)-vector spaces]]
[[!redirects (infinity,n)-vector spaces]]
[[!redirects (infinity,n)-modules]]
[[!redirects (∞,n)-module]]
[[!redirects (∞,n)-modules]]
[[!redirects (infinity,2)-module]]
[[!redirects (infinity,2)-modules]]
[[!redirects (∞,2)-module]]
[[!redirects (∞,2)-modules]]
[[!redirects (∞, n)-vector space]]
[[!redirects (infinity, n)-vector space]]
[[!redirects (∞, n)-vector spaces]]
[[!redirects (infinity, n)-vector spaces]]
[[!redirects (infinity, n)-modules]]
[[!redirects (∞, n)-module]]
[[!redirects (∞, n)-modules]]
[[!redirects (infinity, 2)-module]]
[[!redirects (infinity, 2)-modules]]
[[!redirects (∞, 2)-module]]
[[!redirects (∞, 2)-modules]]
[[!redirects 3-vector space]]
[[!redirects 3-vector spaces]]
[[!redirects 3-module]]
[[!redirects 3-modules]]
[[!redirects 4-module]]
[[!redirects 4-modules]]
[[!redirects 4-vector space]]
[[!redirects 4-vector spaces]]
[[!redirects 2-algebra]]
[[!redirects 2-algebras]]
[[!redirects 3-algebra]]
[[!redirects 3-algebras]]
[[!redirects n-algebra]]
[[!redirects n-algebras]]
[[!redirects n-module]]
[[!redirects n-modules]]
[[!redirects (infinity,n)-vector space]]
[[!redirects (infinity, n)-vector space]]
[[!redirects n-vector space]] |
(infinity,n)-natural transformation | https://ncatlab.org/nlab/source/%28infinity%2Cn%29-natural+transformation | # $(\infty,n)$-natural transformation
## Idea
The analogue of a [[natural transformation]] for [[(∞,n)-categories]]. A special case of a [[transfor]].
## References
* [[Theo Johnson-Freyd]] and [[Claudia Scheimbauer]], *(Op)lax natural transformations, relative field theories, and the "even higher" Morita category of $E_d$-algebras. [arXiv](http://arxiv.org/abs/1502.06526)
[[!redirects (infinity,n)-natural transformation]]
[[!redirects (∞,n)-natural transformation]]
[[!redirects (infinity,n)-transformation]]
[[!redirects (∞,n)-transformation]]
|
(infinity,n)-presheaf | https://ncatlab.org/nlab/source/%28infinity%2Cn%29-presheaf |
+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Higher category theory
+--{: .hide}
[[!include higher category theory - contents]]
=--
=--
=--
#Contents#
* table of contents
{:toc}
## Idea
An **$(\infty,n)$-presheaf** on an [[(∞,n)-category]] $C$ is an [[(∞,n)-functor]] $F : C^{op} \to $ [[(∞,n)Cat]].
## Related concepts
* [[(0,1)-presheaf]]
* [[presheaf]]
* [[(2,1)-presheaf]]
* [[(∞,1)-presheaf]]
* **$(\infty,n)$-presheaf**
[[!redirects (infinity,2)-presheaf]]
[[!redirects (∞,2)-presheaf]]
[[!redirects (infinity,2)-presheaves]]
[[!redirects (∞,2)-presheaves]]
[[!redirects (∞,n)-presheaf]]
[[!redirects (infinity,n)-presheaves]]
[[!redirects (∞,n)-presheaves]]
|
(infinity,n)-sheaf | https://ncatlab.org/nlab/source/%28infinity%2Cn%29-sheaf |
+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Higher category theory
+--{: .hide}
[[!include higher category theory - contents]]
=--
#### Higher topos Theory
+--{: .hide}
[[!include (infinity,1)-topos - contents]]
=--
=--
=--
#Contents#
* table of contents
{:toc}
## Definition
For $\mathbf{H}$ an [[(∞,1)-topos]] and $n \in \mathbb{N}$, an **$(\infty,n+1)$-sheaf** on (an [[(∞,1)-site]] of definition of) $\mathbf{H}$ is an [[n-category object in an (∞,1)-category|n-fold category object in]] $\mathbf{H}$, $X \in n Cat(\mathbf{H})$.
See at _[Internal category object in an (∞,1)-category -- Iterated internalization](category+object+in+an+%28infinity%2C1%29-category#InternalnCategories)_.
The collection of all $(\infty,n)$-sheaves is an [[(∞,n)-topos]].
## Related concepts
* [[sheaf]]
* [[2-sheaf]], [[(2,1)-sheaf]]
* [[(∞,1)-sheaf]] / [[∞-stack]]
* [[(∞,2)-sheaf]]
* [[directed homotopy type theory]]
[[!redirects (∞,n)-sheaf]]
[[!redirects (∞,n)-sheaves]]
[[!redirects (infinity,n)-sheaves]]
|
(infinity,n)-topos | https://ncatlab.org/nlab/source/%28infinity%2Cn%29-topos | [[!redirects (∞,n)-topos]]
+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Higher category theory
+--{: .hide}
[[!include higher category theory - contents]]
=--
#### Higher topos Theory
+--{: .hide}
[[!include (infinity,1)-topos - contents]]
=--
=--
=--
#Contents#
* table of contents
{:toc}
## Definition
For $\mathbf{H}$ an [[(∞,1)-topos]] and $n \in \mathbb{N}$, the collection $n Cat(\mathbf{H})$ of [[n-category object in an (∞,1)-category|n-category object in]] $\mathbf{H}$, hence of [[(∞,n)-sheaves|(∞,n+1)-sheaves]] on $\mathbf{H}$ is an $(\infty,1)$-localic $(\infty,n+1)$-topos.
See at _[Internal category object in an (∞,1)-category -- Iterated internalization](category+object+in+an+%28infinity%2C1%29-category#InternalnCategories)_.
## Related concepts
* [[directed homotopy type theory]]
[[!include flavors of higher toposes -- list]]
[[!redirects (∞,n)-toposes]]
|
(infinity,n)Cat | https://ncatlab.org/nlab/source/%28infinity%2Cn%29Cat | [[!redirects (∞,n)Cat]]
+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Higher category theory
+--{: .hide}
[[!include higher category theory - contents]]
=--
#### [[categories of categories - contents|categories of categories]]
+-- {: .hide}
[[!include categories of categories - contents]]
=--
=--
=--
#Contents#
* table of contents
{:toc}
## Idea
The (large) $(\infty,n+1)$-category $(\infty,n)Cat$ is the collection of all (small) [[(∞,n)-categories]]:
* [[object]]s are [[(∞,n)-categories]];
* [[morphisms]] are [[(∞,n)-functor]]s;
* [[k-morphism]]s for $k \geq 2$ are $(k-1)$-[[transfor]]s.
There are various presentations for this. For general $n$ see for instance [this section](http://ncatlab.org/nlab/show/Theta-space#OfAll) at _[[Theta-space]]_. For low $n$ see the discussion at _[[(∞,1)Cat]]_ and _[[(∞,2)Cat]]_.
Often it is useful to consider just the maximal [[(∞,1)-category]] inside $(\infty,n)Cat$. This is what is presented by various [[model category]] structures on models for $(\infty,n)$-categories.
The discussion in ([BarwickSchommer-Pries](#BarwickSchommer-Pries)) shows that essentially all proposed models for $(\infty,n)Cat$ are in fact equivalent.
## Properties
### Automorphisms
The [[automorphism ∞-group]] of $(\infty,n)Cat$ is equivalent to $(\mathbb{Z}_2)^n$.
This is due to ([Barwick & Schommer-Pries](#BarwickSchommer-Pries)).
(See also at _[[duality]]_.)
## Related concepts
* [[Pos]]
* [[Set]]
* [[Grpd]], [[∞Grpd]]
* [[Cat]], [[Operad]]
* [[2Cat]]
* [[(∞,1)Cat]], [[(∞,1)Operad]]
* **$(\infty,n)Cat$**
## References
* [[Clark Barwick]], [[Chris Schommer-Pries]], _On the unicity of the homotopy theory of higher categories_ ([pdf](https://arxiv.org/pdf/1112.0040.pdf))
{#BarwickSchommer-Pries}
[[!redirects InfinityCommaNCat]]
category: category |