Simplicial set

In mathematics, a simplicial set is an object made up of "simplices" in a specific way. Simplicial sets are higher-dimensional generalizations of directed graphs, partially ordered sets and categories. Formally, a simplicial set may be defined as a contravariant functor from the simplex category to the category of sets. Simplicial sets were introduced in 1950 by Samuel Eilenberg and J. A. Zilber.
One may view a simplicial set as a purely combinatorial construction designed to capture the notion of a "well-behaved" topological space for the purposes of homotopy theory. Specifically, the category of simplicial sets carries a natural model structure, and the corresponding homotopy category is equivalent to the familiar homotopy category of topological spaces.
Simplicial sets are used to define quasi-categories, a basic notion of higher category theory. A construction analogous to that of simplicial sets can be carried out in any category, not just in the category of sets, yielding the notion of simplicial objects.
Motivation.
A simplicial set is a categorical (that is, purely algebraic) model capturing those topological spaces that can be built up (or faithfully represented up to homotopy) from simplices and their incidence relations. This is similar to the approach of CW complexes to modeling topological spaces, with the crucial difference that simplicial sets are purely algebraic and do not carry any actual topology.
To get back to actual topological spaces, there is a "geometric realization" functor which turns simplicial sets into compactly generated Hausdorff spaces. Most classical results on CW complexes in homotopy theory are generalized by analogous results for simplicial sets. While algebraic topologists largely continue to prefer CW complexes, there is a growing contingent of researchers interested in using simplicial sets for applications in algebraic geometry where CW complexes do not naturally exist.
Intuition.
Simplicial sets can be viewed as a higher-dimensional generalization of directed multigraphs. A simplicial set contains vertices (known as "0-simplices" in this context) and arrows ("1-simplices") between some of these vertices. Two vertices may be connected by several arrows, and directed loops that connect a vertex to itself are also allowed. Unlike directed multigraphs, simplicial sets may also contain higher simplices. A 2-simplex, for instance, can be thought of as a two-dimensional "triangular" shape bounded by a list of three vertices "A", "B", "C" and three arrows "B" → "C", "A" → "C" and "A" → "B". In general, an "n"-simplex is an object made up from a list of "n" + 1 vertices (which are 0-simplices) and "n" + 1 faces (which are ("n" − 1)-simplices). The vertices of the "i"-th face are the vertices of the "n"-simplex minus the "i"-th vertex. The vertices of a simplex need not be distinct and a simplex is not determined by its vertices and faces: two different simplices may share the same list of faces (and therefore the same list of vertices), just like two different arrows in a multigraph may connect the same two vertices.
Simplicial sets should not be confused with abstract simplicial complexes, which generalize simple undirected graphs rather than directed multigraphs.
Formally, a simplicial set "X" is a collection of sets "X""n", "n" = 0, 1, 2, ..., together with certain maps between these sets: the "face maps" "d""n","i" : "X""n" → "X""n"−1 ("n" = 1, 2, 3, ... and 0 ≤ "i" ≤ "n") and "degeneracy maps" "s""n","i" : "X""n"→"X""n"+1 ("n" = 0, 1, 2, ... and 0 ≤ "i" ≤ "n"). We think of the elements of "X""n" as the "n"-simplices of "X". The map "d""n","i" assigns to each such "n"-simplex its "i"-th face, the face "opposite to" (i.e. not containing) the "i"-th vertex. The map "s""n","i" assigns to each "n"-simplex the degenerate ("n"+1)-simplex which arises from the given one by duplicating the "i"-th vertex. This description implicitly requires certain consistency relations among the maps "d""n","i" and "s""n","i". Rather than requiring these "simplicial identities" explicitly as part of the definition, the short and elegant modern definition uses the language of category theory.
Formal definition.
Let Δ denote the simplex category. The objects of Δ are nonempty linearly ordered sets of the form
with "n"≥0. The morphisms in Δ are (non-strictly) order-preserving functions between these sets.
A simplicial set "X" is a contravariant functor
where Set is the category of sets. (Alternatively and equivalently, one may define simplicial sets as covariant functors from the opposite category Δop to Set.) Simplicial sets are therefore nothing but presheaves on Δ. Given a simplicial set "X," we often write "Xn" instead of "X"(["n"]).
Simplicial sets form a category, usually denoted sSet, whose objects are simplicial sets and whose morphisms are natural transformations between them.
If we consider "covariant" functors "X" : Δ → Set instead of a contravariant ones, we arrive at the definition of a cosimplicial set. The corresponding category of cosimplicial sets is denoted by cSet.
Face and degeneracy maps.
The simplex category Δ is generated by two particularly important families of morphisms (maps), whose images under a given simplicial set functor are called face maps and degeneracy maps of that simplicial set.
The "face maps" of a simplicial set "X" are the images in that simplicial set of the morphisms formula_1, where formula_2 is the only (order-preserving) injection formula_3 that "misses" formula_4.
Let us denote these face maps by formula_5 respectively, so that formula_6 is a map formula_7. If the first index is clear, we write formula_8 instead of formula_6.
The "degeneracy maps" of the simplicial set "X" are the images in that simplicial set of the morphisms formula_10, where formula_11 is the only (order-preserving) surjection formula_12 that "hits" formula_4 twice.
Let us denote these degeneracy maps by formula_14 respectively, so that formula_15 is a map formula_16. If the first index is clear, we write formula_17 instead of formula_15.
The defined maps satisfy the following simplicial identities:
Conversely, given a sequence of sets "Xn" together with maps formula_25 and formula_26 that satisfy the simplicial identities, there is a unique simplicial set "X" that has these face and degeneracy maps. So the identities provide an alternative way to define simplicial sets.
Examples.
Given a partially ordered set ("S",≤), we can define a simplicial set "NS", the nerve of "S", as follows: for every object ["n"] of Δ we set "NS"(["n"]) = hompo-set( ["n"] , "S"), the order-preserving maps from ["n"] to "S". Every morphism φ:["n"]→["m"] in Δ is an order preserving map, and via composition induces a map "NS"(φ) : "NS"(["m"]) → "NS"(["n"]). It is straightforward to check that "NS" is a contravariant functor from Δ to Set: a simplicial set.
Concretely, the "n"-simplices of the nerve "NS", i.e. the elements of "NS""n"="NS"(["n"]), can be thought of as ordered length-("n"+1) sequences of elements from "S": ("a"0 ≤ "a"1 ≤ ... ≤ "a""n"). The face map "d""i" drops the "i"-th element from such a list, and the degeneracy maps "s""i" duplicates the "i"-th element.
A similar construction can be performed for every category "C", to obtain the nerve "NC" of "C". Here, "NC"(["n"]) is the set of all functors from ["n"] to "C", where we consider ["n"] as a category with objects 0,1...,"n" and a single morphism from "i" to "j" whenever "i" ≤ "j".
Concretely, the "n"-simplices of the nerve "NC" can be thought of as sequences of "n" composable morphisms in "C": "a"0 → "a"1 → ... → "a""n". (In particular, the 0-simplices are the objects of "C" and the 1-simplices are the morphisms of "C".) The face map "d"0 drops the first morphism from such a list, the face map "d""n" drops the last, and the face map "d""i" for 0 < "i" < "n" drops "ai" and composes the "i"th and ("i" + 1)th morphisms. The degeneracy maps "s""i" lengthen the sequence by inserting an identity morphism at position "i".
We can recover the poset "S" from the nerve "NS" and the category "C" from the nerve "NC"; in this sense simplicial sets generalize posets and categories.
Another important class of examples of simplicial sets is given by the singular set "SY" of a topological space "Y". Here "SY""n" consists of all the continuous maps from the standard topological "n"-simplex to "Y". The singular set is further explained below.
The standard "n"-simplex and the category of simplices.
The standard "n"-simplex, denoted Δ"n", is a simplicial set defined as the functor homΔ(-, ["n"]) where ["n"] denotes the ordered set {0, 1, ... ,"n"} of the first ("n" + 1) nonnegative integers. (In many texts, it is written instead as hom(["n"],-) where the homset is understood to be in the opposite category Δop.)
By the Yoneda lemma, the "n"-simplices of a simplicial set "X" stand in 1–1 correspondence with the natural transformations from Δ"n" to "X," i.e. formula_27.
Furthermore, "X" gives rise to a category of simplices, denoted by formula_28 , whose objects are maps ("i.e." natural transformations) Δ"n" → "X" and whose morphisms are natural transformations Δ"n" → Δ"m" over "X" arising from maps ["n"] "→" ["m"] in Δ. That is, formula_28 is a slice category of Δ over "X". The following isomorphism shows that a simplicial set "X" is a colimit of its simplices:
where the colimit is taken over the category of simplices of "X".
Geometric realization.
There is a functor |•|: sSet "→" CGHaus called the geometric realization taking a simplicial set "X" to its corresponding realization in the category of compactly-generated Hausdorff topological spaces. Intuitively, the realization of "X" is the topological space (in fact a CW complex) obtained if every "n-"simplex of "X" is replaced by a topological "n-"simplex (a certain "n-"dimensional subset of ("n" + 1)-dimensional Euclidean space defined below) and these topological simplices are glued together in the fashion the simplices of "X" hang together. In this process the orientation of the simplices of "X" is lost.
To define the realization functor, we first define it on standard n-simplices "Δn" as follows: the geometric realization |Δ"n"| is the standard topological "n"-simplex in general position given by
The definition then naturally extends to any simplicial set "X" by setting
where the colimit is taken over the n-simplex category of "X". The geometric realization is functorial on sSet.
It is significant that we use the category CGHaus of compactly-generated Hausdorff spaces, rather than the category Top of topological spaces, as the target category of geometric realization: like sSet and unlike Top, the category CGHaus is cartesian closed; the categorical product is defined differently in the categories Top and CGHaus, and the one in CGHaus corresponds to the one in sSet via geometric realization.
Singular set for a space.
The singular set of a topological space "Y" is the simplicial set "SY" defined by 
Every order-preserving map φ:["n"]→["m"] induces a continuous map |Δ"n"|→|Δ"m"| in a natural way, which by composition yields "SY"("φ") : "SY"(["m"]) → "SY"(["n"]). This definition is analogous to a standard idea in singular homology of "probing" a target topological space with standard topological "n"-simplices. Furthermore, the singular functor "S" is right adjoint to the geometric realization functor described above, i.e.:
for any simplicial set "X" and any topological space "Y". Intuitively, this adjunction can be understood as follows: a continuous map from the geometric realization of "X" to a space "Y" is uniquely specified if we associate to every simplex of "X" a continuous map from the corresponding standard topological simplex to "Y," in such a fashion that these maps are compatible with the way the simplices in "X" hang together.
Homotopy theory of simplicial sets.
In order to define a model structure on the category of simplicial sets, one has to define fibrations, cofibrations and weak equivalences. One can define fibrations to be Kan fibrations. A map of simplicial sets is defined to be a weak equivalence if its geometric realization is a weak equivalence of spaces. A map of simplicial sets is defined to be a cofibration if it is a monomorphism of simplicial sets. It is a difficult theorem of Daniel Quillen that the category of simplicial sets with these classes of morphisms satisfies the axioms for a proper closed simplicial model category.
A key turning point of the theory is that the geometric realization of a Kan fibration is a Serre fibration of spaces. With the model structure in place, a homotopy theory of simplicial sets can be developed using standard homotopical algebra methods. Furthermore, the geometric realization and singular functors give a Quillen equivalence of closed model categories inducing an equivalence
between the homotopy category for simplicial sets and the usual homotopy category of CW complexes with homotopy classes of continuous maps between them. It is part of the general
definition of a Quillen adjunction that the right adjoint functor (in this case, the singular set functor) carries fibrations (resp. trivial fibrations) to fibrations (resp. trivial fibrations).
Simplicial objects.
A simplicial object "X" in a category "C" is a contravariant functor
or equivalently a covariant functor
where "Δ" still denotes the simplex category. When "C" is the category of sets, we are just talking about the simplicial sets that were defined above. Letting "C" be the category of groups or category of abelian groups, we obtain the categories sGrp of simplicial groups and sAb of simplicial abelian groups, respectively.
Simplicial groups and simplicial abelian groups also carry closed model structures induced by that of the underlying simplicial sets.
The homotopy groups of simplicial abelian groups can be computed by making use of the Dold–Kan correspondence which yields an equivalence of categories between simplicial abelian groups and bounded chain complexes and is given by functors
and
History and uses of simplicial sets.
Simplicial sets were originally used to give precise and convenient descriptions of classifying spaces of groups. This idea was vastly extended by Grothendieck's idea of
considering classifying spaces of categories, and in particular by Quillen's work of algebraic K-theory. In this work, which earned him a Fields Medal, Quillen
developed surprisingly efficient methods for manipulating
infinite simplicial sets. Later these methods were used in other areas on the border between algebraic geometry and topology. For instance, the André–Quillen homology of a ring is a "non-abelian homology", defined and studied in this way.
Both the algebraic K-theory and the André–Quillen homology are defined using algebraic data to write down a simplicial set, and then taking the homotopy groups of this simplicial set.
Simplicial methods are often useful when one wants to prove that a space is a loop space. The basic idea is that if formula_32 is a group with classifying space formula_33, then formula_32 is homotopy equivalent to the loop space formula_35. If formula_33 itself is a group, we can iterate the procedure, and formula_32 is homotopy equivalent to the double loop space formula_38. In case formula_32 is an abelian group, we can actually iterate this infinitely many times, and obtain that formula_32 is an infinite loop space.
Even if formula_41 is not an abelian group, it can happen that it has a composition which is sufficiently commutative so that one can use the above idea to prove that formula_41 is an infinite loop space. In this way, one can prove that the algebraic formula_43-theory of a ring, considered as a topological space, is an infinite loop space.
In recent years, simplicial sets have been used in higher category theory and derived algebraic geometry. Quasi-categories can be thought of as categories in which the composition of morphisms is defined only up to homotopy, and information about the composition of higher homotopies is also retained. Quasi-categories are defined as simplicial sets satisfying one additional condition, the weak Kan condition.