Universal algebra uniformly captures various algebraic structures, by expressing them as equational theories or abstract clones. The ubiquity of algebraic structures in mathematics and related fields has given rise to several variants of universal algebra, such as theories of symmetric operads, non-symmetric operads, generalised operads, PROPs, PROs, and monads. These variants of universal algebra are called notions of algebraic theory. In this paper, we develop a unified framework for them. The key observation is that each notion of algebraic theory can be identified with a monoidal category, in such a way that algebraic theories correspond to monoid objects therein. To incorporate semantics, we introduce a categorical structure called metamodel, which formalises a definition of models of algebraic theories. We also define morphisms between notions of algebraic theory, which are a monoidal version of profunctors. Every strong monoidal functor gives rise to an adjoint pair of such morphisms, and provides a uniform method to establish isomorphisms between categories of models in different notions of algebraic theory. A general structure-semantics adjointness result and a double categorical universal property of categories of models are also shown.
The framework by Avery @cite_3 is relative to a well-behaved 2-category (which he calls a ). In the basic setting of @math he identifies algebraic theories with identity-on-objects functor from a certain category @math of arities, calling them . In this case, the relationship to our work may be established by the fact that (putting size issues aside) identity-on-objects functors from @math correspond to monoid objects in @math (with the profunctor composition as the monoidal structure). This way we may understand Avery's framework (with respect to the setting @math ) within ours, although for general setting probably we cannot do so. However we remark that for specific examples of settings treated in @cite_3 , it seems that proto-theories therein can be identified with monoid objects in the category of a suitable variant of profunctors.
@cite_3
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Whereas formal category theory is classically considered within a @math -category, in this paper a double-dimensional approach is taken. More precisely we develop such theory within the setting of hypervirtual double categories, a notion extending that of virtual double category by adding cells with nullary target. [...] After this the notion of weak' Kan extension within a hypervirtual double category is considered, together with three strengthenings. [...] The notion of yoneda embedding is then considered in a hypervirtual double category, and compared to that of a good yoneda structure on a @math -category; the latter in the sense of Street-Walters and Weber. Conditions are given ensuring that a yoneda embedding @math defines @math as the free small cocompletion of @math , in a suitable sense. In the second half we consider formal category theory in the presence of algebraic structures. In detail: to a monad @math on a hypervirtual double category @math several hypervirtual double categories @math of @math -algebras are associated, [...]. This is followed by the study of the creation of, amongst others, left Kan extensions by the forgetful functors @math . The main motivation of this paper is the description of conditions ensuring that yoneda embeddings in @math lift along these forgetful functors, as well as ensuring that such lifted algebraic yoneda embeddings again define free small cocompletions, now in @math . As a first example we apply the previous to monoidal structures on categories, hence recovering Day convolution of presheaves and Im-Kelly's result on free monoidal cocompletion, as well as obtaining a "monoidal Yoneda lemma". Let @math be a smooth scheme over an algebraically closed field @math of characteristic zero and @math a regular function, and write @math Crit @math , as a closed subscheme of @math . The motivic vanishing cycle @math is an element of the @math -equivariant motivic Grothendieck ring @math defined by Denef and Loeser math.AG 0006050 and Looijenga math.AG 0006220, and used in Kontsevich and Soibelman's theory of motivic Donaldson-Thomas invariants, arXiv:0811.2435. We prove three main results: (a) @math depends only on the third-order thickenings @math of @math . (b) If @math is another smooth scheme, @math is regular, @math Crit @math , and @math is an embedding with @math and @math an isomorphism, then @math equals @math "twisted" by a motive associated to a principal @math -bundle defined using @math , where now we work in a quotient ring @math of @math . (c) If @math is an "oriented algebraic d-critical locus" in the sense of Joyce arXiv:1304.4508, there is a natural motive @math , such that if @math is locally modelled on Crit @math , then @math is locally modelled on @math . Using results from arXiv:1305.6302, these imply the existence of natural motives on moduli schemes of coherent sheaves on a Calabi-Yau 3-fold equipped with "orientation data", as required in Kontsevich and Soibelman's motivic Donaldson-Thomas theory arXiv:0811.2435, and on intersections of oriented Lagrangians in an algebraic symplectic manifold. This paper is an analogue for motives of results on perverse sheaves of vanishing cycles proved in arXiv:1211.3259. We extend this paper to Artin stacks in arXiv:1312.0090. There are many category-theoretic notions of algebraic theory, including Lawvere theories, monads, PROPs and operads. The first central notion of this thesis is a common generalisation of these, which we call a proto-theory. In order to define models of a proto-theory in a category, we need a way of relating the arities of the proto-theory with the objects of the category. This leads to our second central notion, that of an interpretation of arities, or aritation for short. We show that every aritation gives rise to a semantics functor sending proto-theories to models. In fact this functor always has an adjoint, giving a structure-semantics adjunction. Furthermore, we show that the semantics of proto-theories generalises the classical semantics of many existing notions of algebraic theory. Another aim of this thesis is to find a convenient category of monads in the following sense. Every right adjoint gives rise to a monad on its codomain, and more generally so does any functor that admits a codensity monad. However, not all functors have codensity monads. This means that the semantics functor for monads on a category, viewed as a functor into the category of all functors into the base category, does not have a left adjoint. We seek a generalisation of monads with a semantics functor that does have a left adjoint. This can be accomplished with proto-theories, but at a cost. The classical semantics functor for monads is full and faithful; this is a kind of completeness theorem, and is highly desirable in a notion of algebraic theory. However, the semantics functor for general proto-theories is not full and faithful. Thus we seek a generalisation of monads and their semantics for which the semantics functor is both full and faithful and has a left adjoint. We show that such a generalisation is given, for suitable base categories, by a certain kind of topological proto-theory. In this paper we present an algebraic construction of the category of monotonic predicate transformers from the category of relations which is similar to the standard algebraic construction of the integers from the natural numbers. The same construction yields the category of relations from the category of total functions. This provides a mechanism through which the rich type structure of the category of total functions can be promoted to successively weaker ones in the categories of relations and predicate transformers. In addition, it has exposed two complete rules for the refinement and composition of specifications in Morgan's refinement calculus. We give a general category theoretic formulation of the substitution structure underlying the category theoretic study of variable binding proposed by Fiore, Plotkin, and Turi. This general formulation provides the foundation for their work on variable binding, as well as Tanaka's linear variable binding and variable binding for other binders and for mixtures of binders as for instance in the Logic of Bunched Implications. The key structure developed by was a substitution monoidal structure, from which their formulation of binding was derived; so we give an abstract formulation of a substitution monoidal structure, then, at that level of generality, derive the various category theoretic structures they considered. The central construction we use is that of a pseudo-distributive law between 2-monads on Cat, which suffices to induce a pseudo-monad on Cat, and hence a substitution monoidal structure on the free object on 1. We routinely generalise that construction to account for types. The Distributional Compositional Categorical (DisCoCat) model is a mathematical framework that provides compositional semantics for meanings of natural language sentences. It consists of a computational procedure for constructing meanings of sentences, given their grammatical structure in terms of compositional type-logic, and given the empirically derived meanings of their words. For the particular case that the meaning of words is modelled within a distributional vector space model, its experimental predictions, derived from real large scale data, have outperformed other empirically validated methods that could build vectors for a full sentence. This success can be attributed to a conceptually motivated mathematical underpinning, something which the other methods lack, by integrating qualitative compositional type-logic and quantitative modelling of meaning within a category-theoretic mathematical framework. The type-logic used in the DisCoCat model is Lambekʼs pregroup grammar. Pregroup types form a poset al compact closed category, which can be passed, in a functorial manner, on to the compact closed structure of vector spaces, linear maps and tensor product. The diagrammatic versions of the equational reasoning in compact closed categories can be interpreted as the flow of word meanings within sentences. Pregroups simplify Lambekʼs previous type-logic, the Lambek calculus. The latter and its extensions have been extensively used to formalise and reason about various linguistic phenomena. Hence, the apparent reliance of the DisCoCat on pregroups has been seen as a shortcoming. This paper addresses this concern, by pointing out that one may as well realise a functorial passage from the original type-logic of Lambek, a monoidal bi-closed category, to vector spaces, or to any other model of meaning organised within a monoidal bi-closed category. The corresponding string diagram calculus, due to Baez and Stay, now depicts the flow of word meanings, and also reflects the structure of the parse trees of the Lambek calculus. In our LICS 2004 paper we introduced an approach to the study of the local structure of finite algebras and relational structures that aims at applications in the Constraint Satisfaction Problem (CSP). This approach involves a graph associated with an algebra @math or a relational structure A, whose vertices are the elements of @math (or A), the edges represent subsets of @math such that the restriction of some term operation of @math is ‘good’ on the subset, that is, act as an operation of one of the 3 types: semilattice, majority, or affine. In this paper we significantly refine and advance this approach. In particular, we prove certain connectivity and rectangularity properties of relations over algebras related to components of the graph connected by semilattice and affine edges. We also prove a result similar to 2-decomposition of relations invariant under a majority operation, only here we do not impose any restrictions on the relation. These results allow us to give a new, somewhat more intuitive proof of the bounded width theorem: the CSP over algebra @math has bounded width if and only if @math does not contain affine edges. Actually, this result shows that bounded width implies width (2,3). We also consider algebras with edges from a restricted set of types. In particular, it can be proved that type restrictions are preserved under the standard algebraic constructions. Finally, we prove that algebras without semilattice edges have few subalgebras of powers, that is, the CSP over such algebras is also polynomial time. Let ( O ) k be the ring of integers of a finite extension k of the field ( Q ) p of p-adic numbers. The endomorphisms of a formal group law defined over ( O ) k provide nontrivial examples of commuting formal series with coefficients in ( O ) k . This article deals with the inverse problem formulated by Jonathan Lubin within the context of non-Archimedean dynamical systems. We present a large family of series, with coefficients in ( Z ) p , which satisfy Lubin's conjecture. These series are constructed with the help of Lubin–Tate formal group laws over ( Q ) p . We introduce the notion of minimally ramified series which turn out to be modulo p reductions of some series of this family. The commutant monoids of these minimally ramified series are determined by using the Fontaine–Wintenberger theory of the field of norms which allows an interpretation of them as automorphisms of ( Z ) p -extensions of local fields of characteristic zero. A particularly effective example illustrating the paper is given by a family of series generalizing Cebysev polynomials For a simply-connected simple algebraic group @math over @math , we exhibit a subvariety of its affine Grassmannian that is closely related to the nilpotent cone of @math , generalizing a well-known fact about @math . Using this variety, we construct a sheaf-theoretic functor that, when combined with the geometric Satake equivalence and the Springer correspondence, leads to a geometric explanation for a number of known facts (mostly due to Broer and Reeder) about small representations of the dual group. The order-enriched category of monotonic predicate transformers over posets is a model of the refinement calculus of higher order imperative programs and pre-post specifications. This category is shown to be equivalent to the category of spans over ideal relations, and ideal relations are shown to be spans over monotonic functions between posets. To do this we use a skew span construction because the standard categorical span constructions are inapplicable. Axioms are given for products and coproducts of underlying posets as well as the homset as a coexponent, using inequations (for various kinds of lax adjunctions) and conditional equations (for adjunctions in subcategories) that are shown to uniquely determine the structures. The model is described in elementary terms using power allegories, an axiomatic calculus of relations, which makes the proofs accessible to non-specialists and shows that the results generalize to other base categories. We prove a homological mirror symmetry equivalence between an @math -brane category for the pair of pants, computed as a wrapped microlocal sheaf category, and a @math -brane category for a mirror LG model, understood as a category of matrix factorizations. The equivalence improves upon prior results in two ways: it intertwines evident affine Weyl group symmetries on both sides, and it exhibits the relation of wrapped microlocal sheaves along different types of Lagrangian skeleta for the same hypersurface. The equivalence proceeds through the construction of a combinatorial realization of the @math -model via arboreal singularities. The constructions here represent the start of a program to generalize to higher dimensions many of the structures which have appeared in topological approaches to Fukaya categories of surfaces. A category N of labeled (oriented) trivalent graphs (nets) or ribbon graphs is extended by new generators called fusing, braiding, twist and switch with relations which can be called Moore-Seiberg relations. A functor to N is constructed from the category Surf of oriented surfaces with labeled boundary and their homeomorphisms. Given a (eventually non-semisimple) k-linear abelian ribbon braided category @math with some finiteness conditions we construct a functor from a central extension of N with the set of labels @math ObC to k-vector spaces. Composing the functors we get a modular functor from a central extension of Surfto k-vector spaces. In this paper we construct new categorical models for the identity types of Martin-Lof type theory, in the categories Top of topological spaces and SSet of simplicial sets. We do so building on earlier work of Awodey and Warren [2009], which has suggested that a suitable environment for the interpretation of identity types should be a category equipped with a weak factorization system in the sense of Bousfield--Quillen. It turns out that this is not quite enough for a sound model, due to some subtle coherence issues concerned with stability under substitution; and so our first task is to introduce a slightly richer structure, which we call a homotopy-theoretic model of identity types, and to prove that this is sufficient for a sound interpretation. Now, although both Top and SSet are categories endowed with a weak factorization system---and indeed, an entire Quillen model structure---exhibiting the additional structure required for a homotopy-theoretic model is quite hard to do. However, the categories we are interested in share a number of common features, and abstracting these leads us to introduce the notion of a path object category. This is a relatively simple axiomatic framework, which is nonetheless sufficiently strong to allow the construction of homotopy-theoretic models. Now by exhibiting suitable path object structures on Top and SSet, we endow those categories with the structure of a homotopy-theoretic model and, in this way, obtain the desired topological and simplicial models of identity types. Author(s): Johnson, William Andrew | Advisor(s): Scanlon, Thomas | Abstract: This dissertation is a collection of results in model theory, related in one way or another to fields, NIP theories, and elimination of imaginaries. The most important result is a classification of dp-minimal fields, presented in Chapter 9. We construct in a canonical fashion a non-trivial Hausdorff definable field topology on any unstable dp-minimal field. Usingthis we classify the dp-minimal pure fields and valued fields up to elementary equivalence. Furthermore we prove that every VC-minimal field is real closed or algebraically closed.In Chapter 11, we analyze the theories of existentially closed fields with several valuations and orderings, as studied by van den Dries. We show that these model complete theories are NTP2, and analyze forking, dividing, and burden in these theories. The theory of algebraically closed fields with n independent valuation rings turns out to be an example of such a theory. This provides a new and natural example of an NTP2 theory which is neithersimple nor NIP, nor even a conceptual hybrid of something simple and something NIP.In Chapter 8, we exhibit a bad failure of elimination of imaginaries in a dense o-minimal structure. We produce an exotic interpretable set which cannot be put in definable bijection with a definable set, after naming any amount of parameters. However, we show that these exotic interpretable sets are still amenable to some of the tools of tame topology: they mustadmit nice definable topologies locally homeomorphic to definable sets.Chapter 12 proves the existence of Z nZ-valued definable strong Euler characteristics on pseudofinite fields, which measure the non-standard “size” of definable sets, mod n. The non-trivial result is that these “sizes” are definable in families of definable sets. This could probably be proven e cohomology, but we give a more elementary proof relying heavily on the theory of abelian varieties.We also present simplified and new proofs of several model-theoretic facts, including the definability of irreducibility and Zariski closure in ACF (Chapter 10), and elimination of imaginaries in ACVF (Chapter 6). This latter fact was originally proven by Haskell, Hrushovski, and Macpherson. We give a proof that is drastically simpler, inspired by Poizat’s proofs of elimination of imaginaries in ACF and DCF. Seven years of COMPASS.- Inductively defined relations: A brief tutorial extended abstract.- On the role of category theory in the area of algebraic specifications.- Unification of theories: A challenge for computing science.- The larch shared language: Some open problems.- The lambda calculus as an abstract data type.- Unifying theories in different institutions.- Interchange format for inter-operability of tools and translation.- Experiments with partial evaluation domains for rewrite specifications.- Class-sort polymorphism in GLIDER.- Deontic concepts in the algebraic specification of dynamic systems: The permission case.- Reification - Changing viewpoint but preserving truth.- A category-based equational logic semantics to constraint programming.- Concurrent state transformations on abstract data types.- A view on implementing processes: Categories of circuits.- Combining algebraic and set-theoretic specifications.- Minimal term rewriting systems.- InterACT: An interactive theorem and completeness prover for algebraic specifications with conditional equations.- Rewriting and reasoning with set-relations II: The non-ground case completeness.- Termination of curryfied rewrite systems.- Formal specifications and test: Correctness and oracle.- Behavioural equivalence, bisimulation, and minimal realisation.- Using limits of parchments to systematically construct institutions of partial algebras.- Behavioural specifications in type theory.- Swinging data types.- Context institutions.- Object-oriented functional programming and type reconstruction.- Moving between logical systems.- Modular algebraic specifications and the orientation of equations into rewrite rules.- A model for I O in equational languages with don't care non-determinism.- Tool design for structuring mechanisms for algebraic specification languages with initial semantics. This paper illustrates a hierarchical generative model for representing and recognizing compositional object categories with large intra-category variance. In this model, objects are broken into their constituent parts and the variability of configurations and relationships between these parts are modeled by stochastic attribute graph grammars, which are embedded in an And-Or graph for each compositional object category. It combines the power of a stochastic context free grammar (SCFG) to express the variability of part configurations, and a Markov random field (MRF) to represent the pictorial spatial relationships between these parts. As a generative model, different object instances of a category can be realized as a traversal through the And-Or graph to arrive at a valid configuration (like a valid sentence in language, by analogy). The inference recognition procedure is intimately tied to the structure of the model and follows a probabilistic formulation consisting of bottom-up detection steps for the parts, which in turn recursively activate the grammar rules for top-down verification and searches for missing parts. We present experiments comparing our results to state of art methods and demonstrate the potential of our proposed framework on compositional objects with cluttered backgrounds using training and testing data from the public Lotus Hill and Caltech datasets. Kleene's theorem on the coincidence of regular and rational languages in free monoids has been generalized by Schutzenberger to a description of the recognizable formal power series in noncommuting variables over arbitrary semirings and by Ochmanski to a characterization of the recognizable languages in trace monoids. We will describe the recognizable formal power series over arbitrary semirings and in partially commuting variables, i.e. over trace monoids. We prove that the recognizable series are certain rational power series, which can be constructed from the polynomials by using the operations sum, product, and a restricted star which is applied only to series for which the elements in the support all have the same connected alphabet. The converse is true if the underlying semiring is commutative. Moreover, if in addition the semiring is idempotent then the same result holds with a star restricted to series for which the elements in the support have connected (possibly different) alphabets. It is shown that these assumptions over the semiring are necessary. This provides a joint generalization of Kleene's, Schutzenberger's and Ochmanski's theorems. We inspect Vietoris-Rips complexes @math of certain metric spaces @math using a new generalization of Bestvina-Brady discrete Morse theory. Our main result is a pair of metric criteria on @math , called the Morse Criterion and Link Criterion, that allow us to deduce information about the homotopy types of certain @math . One application is to topological data analysis, specifically persistence of homotopy type for certain Vietoris-Rips complexes. For example we recover some results of Adamaszek-Adams and Hausmann regarding homotopy types of @math . Another application is to geometric group theory; we prove that any group acting geometrically on a metric space satisfying a version of the Link Criterion admits a geometric action on a contractible simplicial complex, which has implications for the finiteness properties of the group. This applies for example to asymptotically @math groups. We also prove that any group with a word metric satisfying the Link Criterion in an appropriate range has a contractible Vietoris-Rips complex, and use combings to exhibit a family of groups with this property. We define a monad on the category of complete metric spaces with short maps, which assigns to each space the space of Radon probability measures on it with finite first moment, equipped with the Kantorovich--Wasserstein distance. It is analogous to the Giry monad on the category of Polish spaces, and it extends a construction due to van Breugel for compact and for 1-bounded complete metric spaces. We prove that this Kantorovich monad arises from a colimit construction on finite powers, which formalizes the intuition that probability measures are limits of finite samples. The proof relies on a new criterion for when an ordinary left Kan extension of lax monoidal functors is a monoidal Kan extension. This colimit characterization allows for the development of integration theory and other things, such as the treatment of measures on spaces of measures, completely without measure theory. We also show that the category of algebras of the Kantorovich monad is equivalent to the category of closed convex subsets of Banach spaces with short affine maps as the morphisms. In this paper we prove the results announced in [13]. Let G be a semi- simple, simply connected algebraic group defined over an algebraically closed field k. Let T be a maximal torus, B a Bore1 subgroup, B 3 T. Let W be the Weyl group of G. Let R (resp. R+ ) be the set of roots (resp. positive roots) relative to T (resp. B). Let S be the set of simple roots in R+. Let P be a maximal parabolic subgroup in G with associated fundamental weight w. Let W, be the Weyl group of P, and Wp be the set of minimal representatives of W W,. For w E Wp, let e(w) be the point and X(w) the Schubert variety in G P associated to w. In this paper we deter- mine the multiplicity m,(w) of X(w) at e(z), where e(z) E X(w), for all minuscule P’s and also for P = Pgn, G being of type C, (here Pun denotes the maximal parabolic subgroup obtained by omitting a,). The determina- tion of m,(w) is done as follows. Let L be the ample generator of Pic(G P). A basis has been constructed for @(X(w), L”) in terms of standard monomials on X(w) (cf. [ 16, 11 I). Let U; be the unipotent subgroup of G generated by U-,, ?ET(R+ - Rp+) (here R, denotes the set of roots of P and U, denotes the unipotent subgroup of G, associated to tl E R). Then U; e(r) gives an affme neighborhood of e(z) in G P. Let A, be the affine algebra of U, e(z) and A,.. = A, &, where & is the ideal of elements of A, that vanish on X(w) n U; e(z). Let M,,, be the maximal ideal in A, H, corresponding to e(r). Then using the results of [ 16, 111, we obtain a basis of M;, &f:,+,,’ . This enables us to obtain an inductive formula for F,,,, the Hilbert polynomial of X(w) at e(T) (cf. Corollaries 3.8 and 4.11), and also express m, (w) in terms of m, (w’)‘s, X(w’)‘s being the Schubert divisors in X(w) such that e(T)E X(w’) (cf. Theorems 3.7 and 4.10). Using this we