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In
mathematics Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...
, a simplicial set is a sequence of sets with internal order structure ( abstract simplices) and maps between them. Simplicial sets are higher-dimensional generalizations of directed graphs. Every simplicial set gives rise to a "nice"
topological space In mathematics, a topological space is, roughly speaking, a Geometry, geometrical space in which Closeness (mathematics), closeness is defined but cannot necessarily be measured by a numeric Distance (mathematics), distance. More specifically, a to ...
, known as its geometric realization. This realization consists of geometric simplices, glued together according to the rules of the simplicial set. Indeed, one may view a simplicial set as a purely combinatorial construction designed to capture the essence of a 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. Formally, a simplicial set may be defined as a contravariant functor from the simplex category to the
category of sets In the mathematical field of category theory, the category of sets, denoted by Set, is the category whose objects are sets. The arrows or morphisms between sets ''A'' and ''B'' are the functions from ''A'' to ''B'', and the composition of mor ...
. Simplicial sets were introduced in 1950 by Samuel Eilenberg and Joseph A. Zilber. Simplicial sets are used to define quasi-categories, a basic notion of
higher category theory In mathematics, higher category theory is the part of category theory at a ''higher order'', which means that some equalities are replaced by explicit morphism, arrows in order to be able to explicitly study the structure behind those equalities. H ...
. 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 complex In mathematics, and specifically in topology, a CW complex (also cellular complex or cell complex) is a topological space that is built by gluing together topological balls (so-called ''cells'') of different dimensions in specific ways. It generali ...
es 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 In mathematics, specifically category theory, a functor is a Map (mathematics), mapping between Category (mathematics), categories. Functors were first considered in algebraic topology, where algebraic objects (such as the fundamental group) ar ...
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 Algebraic geometry is a branch of mathematics which uses abstract algebraic techniques, mainly from commutative algebra, to solve geometry, geometrical problems. Classically, it studies zero of a function, zeros of multivariate polynomials; th ...
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 modern definition uses the language of
category theory Category theory is a general theory of mathematical structures and their relations. It was introduced by Samuel Eilenberg and Saunders Mac Lane in the middle of the 20th century in their foundational work on algebraic topology. Category theory ...
.


Formal definition

Let Δ denote the simplex category. The objects of Δ are nonempty
totally ordered In mathematics, a total order or linear order is a partial order in which any two elements are comparable. That is, a total order is a binary relation \leq on some set X, which satisfies the following for all a, b and c in X: # a \leq a ( r ...
sets. Each object is uniquely order isomorphic to an object of the form : 'n''= with ''n'' ≥ 0. The morphisms in Δ are (non-strictly) order-preserving functions between these sets. A simplicial set ''X'' is a contravariant functor :''X'' : Δ → Set where Set is the
category of sets In the mathematical field of category theory, the category of sets, denoted by Set, is the category whose objects are sets. The arrows or morphisms between sets ''A'' and ''B'' are the functions from ''A'' to ''B'', and the composition of mor ...
. (Alternatively and equivalently, one may define simplicial sets as covariant functors from the opposite category Δop ''→''f Set.) 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. This is the category of presheaves on Δ. As such, it is a topos.


Face and degeneracy maps and simplicial identities

The morphisms (maps) of the simplex category Δ are generated by two particularly important families of morphisms, whose images under a given simplicial set functor are called the 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 \delta^,\dotsc,\delta^\colon -1to /math>, where \delta^ is the only (order-preserving) injection -1to /math> that "misses" i. Let us denote these face maps by d_,\dotsc,d_ respectively, so that d_ is a map X_n \to X_. If the first index is clear, we write d_i instead of d_. The ''degeneracy maps'' of the simplicial set ''X'' are the images in that simplicial set of the morphisms \sigma^,\dotsc,\sigma^\colon +1to /math>, where \sigma^ is the only (order-preserving) surjection +1to /math> that "hits" i twice. Let us denote these degeneracy maps by s_,\dotsc,s_ respectively, so that s_ is a map X_n \to X_. If the first index is clear, we write s_i instead of s_. The defined maps satisfy the following simplicial identities: #d_i d_j = d_ d_i if ''i'' < ''j''. (This is short for d_ d_ = d_ d_ if 0 ≤ ''i'' < ''j'' ≤ ''n''.) #d_i s_j = s_d_i if ''i'' < ''j''. #d_i s_j = \text if ''i'' = ''j'' or ''i'' = ''j'' + 1. #d_i s_j = s_j d_ if ''i'' > ''j'' + 1. #s_i s_j = s_ s_i if ''i'' ≤ ''j''. Conversely, given a sequence of sets ''Xn'' together with maps d_ : X_n \to X_ and s_ : X_n \to X_ 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 In mathematics, especially order theory, a partial order on a Set (mathematics), set is an arrangement such that, for certain pairs of elements, one precedes the other. The word ''partial'' is used to indicate that not every pair of elements need ...
(''S'', ≤), we can define a simplicial set ''NS'', called the
nerve A nerve is an enclosed, cable-like bundle of nerve fibers (called axons). Nerves have historically been considered the basic units of the peripheral nervous system. A nerve provides a common pathway for the Electrochemistry, electrochemical nerv ...
of ''S'', as follows: for every object 'n''of Δ we set ''NS''( 'n'' = homposet( 'n'', ''S''), the set of 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 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. X_n = X( \cong \operatorname(\operatorname_\Delta(-, ,X)= \operatorname_(\Delta^n,X). Furthermore, ''X'' gives rise to a category of simplices, denoted by \Delta\downarrow , 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, \Delta\downarrow is a slice category of Δ over ''X''. The following isomorphism shows that a simplicial set ''X'' is a colimit of its simplices: : X \cong \varinjlim_ \Delta^n 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 CGHaus of compactly-generated
Hausdorff topological space In topology and related branches of mathematics, a Hausdorff space ( , ), T2 space or separated space, is a topological space where distinct points have disjoint neighbourhoods. Of the many separation axioms that can be imposed on a topologic ...
s. Intuitively, the realization of ''X'' is the topological space (in fact a
CW complex In mathematics, and specifically in topology, a CW complex (also cellular complex or cell complex) is a topological space that is built by gluing together topological balls (so-called ''cells'') of different dimensions in specific ways. It generali ...
) 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 Δ''n'' as follows: the geometric realization , Δ''n'', is the standard topological ''n''- simplex in general position given by :, \Delta^n, = \. The definition then naturally extends to any simplicial set ''X'' by setting :, X, = limΔ''n'' → ''X'' , Δ''n'', 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 :(''SY'')( 'n'' = homT''op''(, Δ''n'', , ''Y'') for each object 'n''∈ Δ. Every order-preserving map φ: 'n''�� 'm''induces a continuous map , Δ''n'', →, Δ''m'', by :(x_0,...,x_n) \in , \Delta_n, \mapsto (y_j),~~ y_j = \sum_x_i. Then, by composition it yields to a map ''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.: :homTop(, ''X'', , ''Y'') ≅ homsSet(''X'', ''SY'') 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 homotopy 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 becomes a model category, and indeed 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 :, •, : ''Ho''(sSet) ↔ ''Ho''(Top) 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 :''X'' : Δ → ''C'' or equivalently a covariant functor :''X'': Δop → ''C,'' where Δ still denotes the simplex category and op the opposite category. When ''C'' is the
category of sets In the mathematical field of category theory, the category of sets, denoted by Set, is the category whose objects are sets. The arrows or morphisms between sets ''A'' and ''B'' are the functions from ''A'' to ''B'', and the composition of mor ...
, we are just talking about the simplicial sets that were defined above. Letting ''C'' be the
category of groups In mathematics, the category Grp (or Gp) has the class of all groups for objects and group homomorphisms for morphisms. As such, it is a concrete category. The study of this category is known as group theory. Relation to other categories The ...
or
category of abelian groups In mathematics, the category Ab has the abelian groups as objects and group homomorphisms as morphisms. This is the prototype of an abelian category: indeed, every small abelian category can be embedded in Ab. Properties The zero object o ...
, 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 :''N:'' sAb → Ch+ and : Γ: Ch+ →  sAb. See also: simplicial diagram.


History and uses of simplicial sets

Simplicial sets were originally used to give precise and convenient descriptions of
classifying space In mathematics, specifically in homotopy theory, a classifying space ''BG'' of a topological group ''G'' is the quotient of a weakly contractible space ''EG'' (i.e., a topological space all of whose homotopy groups are trivial) by a proper free ...
s 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. 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 G is a group with classifying space BG, then G is homotopy equivalent to the loop space \Omega BG. If BG itself is a group, we can iterate the procedure, and G is homotopy equivalent to the double loop space \Omega^2 B(BG). In case G is an abelian group, we can actually iterate this infinitely many times, and obtain that G is an infinite loop space. Even if X 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 X is an infinite loop space. In this way, one can prove that the algebraic K-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 In mathematics, higher category theory is the part of category theory at a ''higher order'', which means that some equalities are replaced by explicit morphism, arrows in order to be able to explicitly study the structure behind those equalities. H ...
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.


See also

* Delta set * Dendroidal set, a generalization of simplicial set * Simplicial presheaf *
Quasi-category In mathematics, more specifically category theory, a quasi-category (also called quasicategory, weak Kan complex, inner Kan complex, infinity category, ∞-category, Boardman complex, quategory) is a generalization of the notion of a category. T ...
* Kan complex * Dold–Kan correspondence * Simplicial homotopy * Simplicial sphere * Abstract simplicial complex * Anodyne extension * Weak equivalence between simplicial sets


Notes


References

* * * ''(An elementary introduction to simplicial sets)''. * *


Further reading

* * May, J. Peter.
Simplicial Objects in Algebraic Topology
'' University of Chicago Press 1967 * {{DEFAULTSORT:Simplicial Set Algebraic topology Homotopy theory Functors