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mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, a simplicial set is an object composed of ''simplices'' in a specific way. Simplicial sets are higher-dimensional generalizations of
directed graph In mathematics, and more specifically in graph theory, a directed graph (or digraph) is a graph that is made up of a set of vertices connected by directed edges, often called arcs. Definition In formal terms, a directed graph is an ordered pa ...
s,
partially ordered set In mathematics, especially order theory, a partially ordered set (also poset) formalizes and generalizes the intuitive concept of an ordering, sequencing, or arrangement of the elements of a Set (mathematics), set. A poset consists of a set toget ...
s and
categories Category, plural categories, may refer to: Philosophy and general uses *Categorization, categories in cognitive science, information science and generally *Category of being * ''Categories'' (Aristotle) *Category (Kant) * Categories (Peirce) * ...
. Formally, a simplicial set may be defined as a
contravariant functor In mathematics, specifically category theory, a functor is a mapping between categories. Functors were first considered in algebraic topology, where algebraic objects (such as the fundamental group) are associated to topological spaces, and ...
from the
simplex category In mathematics, the simplex category (or simplicial category or nonempty finite ordinal category) is the category of non-empty finite ordinals and order-preserving maps. It is used to define simplicial and cosimplicial objects. Formal definition ...
to the
category of sets In the mathematical field of category theory, the category of sets, denoted as Set, is the category whose objects are sets. The arrows or morphisms between sets ''A'' and ''B'' are the total functions from ''A'' to ''B'', and the composition o ...
. Simplicial sets were introduced in 1950 by
Samuel Eilenberg Samuel Eilenberg (September 30, 1913 – January 30, 1998) was a Polish-American mathematician who co-founded category theory (with Saunders Mac Lane) and homological algebra. Early life and education He was born in Warsaw, Kingdom of Poland to ...
and Joseph A. Zilber. Every simplicial set gives rise to a "nice"
topological space In mathematics, a topological space is, roughly speaking, a geometrical space in which closeness is defined but cannot necessarily be measured by a numeric distance. More specifically, a topological space is a set whose elements are called points ...
, 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 "
well-behaved In mathematics, when a mathematical phenomenon runs counter to some intuition, then the phenomenon is sometimes called pathological. On the other hand, if a phenomenon does not run counter to intuition, it is sometimes called well-behaved. Th ...
" topological space for the purposes of homotopy theory. Specifically, the category of simplicial sets carries a natural
model structure In mathematics, particularly in homotopy theory, a model category is a category with distinguished classes of morphisms ('arrows') called ' weak equivalences', ' fibrations' and 'cofibrations' satisfying certain axioms relating them. These abstr ...
, and the corresponding
homotopy category In mathematics, the homotopy category is a category built from the category of topological spaces which in a sense identifies two spaces that have the same shape. The phrase is in fact used for two different (but related) categories, as discussed be ...
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 In mathematics, higher category theory is the part of category theory at a ''higher order'', which means that some equalities are replaced by explicit arrows in order to be able to explicitly study the structure behind those equalities. Higher cate ...
. 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 In geometry, a simplex (plural: simplexes or simplices) is a generalization of the notion of a triangle or tetrahedron to arbitrary dimensions. The simplex is so-named because it represents the simplest possible polytope in any given dimension. ...
and their incidence relations. This is similar to the approach of
CW complex A CW complex (also called cellular complex or cell complex) is a kind of a topological space that is particularly important in algebraic topology. It was introduced by J. H. C. Whitehead (open access) to meet the needs of homotopy theory. This cl ...
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 space In topology, a compactly generated space is a topological space whose topology is coherent with the family of all compact subspaces. Specifically, a topological space ''X'' is compactly generated if it satisfies the following condition: :A subsp ...
s. 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, classically studying zeros of multivariate polynomials. Modern algebraic geometry is based on the use of abstract algebraic techniques, mainly from commutative algebra, for solving geometrical ...
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'' â†’ ''C'', ''A'' â†’ ''C'' and ''A'' â†’ ''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'' âˆ’ 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 complex In combinatorics, an abstract simplicial complex (ASC), often called an abstract complex or just a complex, is a family of sets that is closed under taking subsets, i.e., every subset of a set in the family is also in the family. It is a purely c ...
es, 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'' â†’ ''X''''n''−1 (''n'' = 1, 2, 3, ... and 0 â‰¤ ''i'' â‰¤ ''n'') and ''degeneracy maps'' ''s''''n'',''i'' : ''X''''n''→''X''''n''+1 (''n'' = 0, 1, 2, ... and 0 â‰¤ ''i'' â‰¤ ''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 Category theory is a general theory of mathematical structures and their relations that was introduced by Samuel Eilenberg and Saunders Mac Lane in the middle of the 20th century in their foundational work on algebraic topology. Nowadays, cate ...
.


Formal definition

Let Δ denote the
simplex category In mathematics, the simplex category (or simplicial category or nonempty finite ordinal category) is the category of non-empty finite ordinals and order-preserving maps. It is used to define simplicial and cosimplicial objects. Formal definition ...
. The objects of Δ are nonempty
linearly ordered In mathematics, a total 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 ( reflexive) ...
sets of the form : 'n''= with ''n''≥0. The morphisms in Δ are (non-strictly)
order-preserving function In mathematics, a monotonic function (or monotone function) is a function between ordered sets that preserves or reverses the given order. This concept first arose in calculus, and was later generalized to the more abstract setting of orde ...
s between these sets. A simplicial set ''X'' is a
contravariant functor In mathematics, specifically category theory, a functor is a mapping between categories. Functors were first considered in algebraic topology, where algebraic objects (such as the fundamental group) are associated to topological spaces, and ...
:''X'' : Δ → Set where Set is the
category of sets In the mathematical field of category theory, the category of sets, denoted as Set, is the category whose objects are sets. The arrows or morphisms between sets ''A'' and ''B'' are the total functions from ''A'' to ''B'', and the composition o ...
. (Alternatively and equivalently, one may define simplicial sets as covariant functors from the
opposite category In category theory, a branch of mathematics, the opposite category or dual category ''C''op of a given category ''C'' is formed by reversing the morphisms, i.e. interchanging the source and target of each morphism. Doing the reversal twice yields t ...
Δop to 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 In category theory, a branch of mathematics, a natural transformation provides a way of transforming one functor into another while respecting the internal structure (i.e., the composition of morphisms) of the categories involved. Hence, a natural ...
between them. This is nothing but the category of presheaves on Δ. As such, it is a
topos In mathematics, a topos (, ; plural topoi or , or toposes) is a category that behaves like the category of sheaves of sets on a topological space (or more generally: on a site). Topoi behave much like the category of sets and possess a notio ...
.


Face and degeneracy maps and simplicial identities

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 \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 partially ordered set (also poset) formalizes and generalizes the intuitive concept of an ordering, sequencing, or arrangement of the elements of a Set (mathematics), set. A poset consists of a set toget ...
(''S'',≤), we can define a simplicial set ''NS'', the
nerve A nerve is an enclosed, cable-like bundle of nerve fibers (called axons) in the peripheral nervous system. A nerve transmits electrical impulses. It is the basic unit of the peripheral nervous system. A nerve provides a common pathway for the e ...
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 â‰¤ ''a''1 â‰¤ ... ≤ ''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'' â‰¤ ''j''. Concretely, the ''n''-simplices of the nerve ''NC'' can be thought of as sequences of ''n'' composable morphisms in ''C'': ''a''0 â†’ ''a''1 â†’ ... â†’ ''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 In mathematics, the Yoneda lemma is arguably the most important result in category theory. It is an abstract result on functors of the type ''morphisms into a fixed object''. It is a vast generalisation of Cayley's theorem from group theory (view ...
, 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 In mathematics, specifically category theory, an overcategory (and undercategory) is a distinguished class of categories used in multiple contexts, such as with covering spaces (espace etale). They were introduced as a mechanism for keeping track ...
of Δ over ''X''. The following isomorphism shows that a simplicial set ''X'' is a
colimit In category theory, a branch of mathematics, the abstract notion of a limit captures the essential properties of universal constructions such as products, pullbacks and inverse limits. The dual notion of a colimit generalizes constructions such ...
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 of compactly-generated
Hausdorff topological space In topology and related branches of mathematics, a Hausdorff space ( , ), separated space or T2 space is a topological space where, for any two distinct points, there exist neighbourhoods of each which are disjoint from each other. Of the many ...
s. Intuitively, the realization of ''X'' is the topological space (in fact a
CW complex A CW complex (also called cellular complex or cell complex) is a kind of a topological space that is particularly important in algebraic topology. It was introduced by J. H. C. Whitehead (open access) to meet the needs of homotopy theory. This cl ...
) 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 geometry, a simplex (plural: simplexes or simplices) is a generalization of the notion of a triangle or tetrahedron to arbitrary dimensions. The simplex is so-named because it represents the simplest possible polytope in any given dimension. ...
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 In category theory, a branch of mathematics, the abstract notion of a limit captures the essential properties of universal constructions such as products, pullbacks and inverse limits. The dual notion of a colimit generalizes constructions such ...
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 In category theory, a category is Cartesian closed if, roughly speaking, any morphism defined on a product of two objects can be naturally identified with a morphism defined on one of the factors. These categories are particularly important in math ...
; the
categorical product In category theory, the product of two (or more) objects in a category is a notion designed to capture the essence behind constructions in other areas of mathematics such as the Cartesian product of sets, the direct product of groups or ring ...
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'', 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 In algebraic topology, singular homology refers to the study of a certain set of algebraic invariants of a topological space ''X'', the so-called homology groups H_n(X). Intuitively, singular homology counts, for each dimension ''n'', the ''n''- ...
of "probing" a target topological space with standard topological ''n''-simplices. Furthermore, the singular functor ''S'' is
right adjoint In mathematics, specifically category theory, adjunction is a relationship that two functors may exhibit, intuitively corresponding to a weak form of equivalence between two related categories. Two functors that stand in this relationship are kn ...
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 In mathematics, particularly in homotopy theory, a model category is a category with distinguished classes of morphisms ('arrows') called ' weak equivalences', ' fibrations' and 'cofibrations' satisfying certain axioms relating them. These abstr ...
on the category of simplicial sets, one has to define fibrations, cofibrations and weak equivalences. One can define
fibrations The notion of a fibration generalizes the notion of a fiber bundle and plays an important role in algebraic topology, a branch of mathematics. Fibrations are used, for example, in postnikov-systems or obstruction theory. In this article, all ma ...
to be
Kan fibration In mathematics, Kan complexes and Kan fibrations are part of the theory of simplicial sets. Kan fibrations are the fibrations of the standard model category structure on simplicial sets and are therefore of fundamental importance. Kan complexes are ...
s. 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 In mathematics, in particular homotopy theory, a continuous mapping :i: A \to X, where A and X are topological spaces, is a cofibration if it lets homotopy classes of maps ,S/math> be extended to homotopy classes of maps ,S/math> whenever a map ...
if it is a
monomorphism In the context of abstract algebra or universal algebra, a monomorphism is an injective homomorphism. A monomorphism from to is often denoted with the notation X\hookrightarrow Y. In the more general setting of category theory, a monomorphism ...
of simplicial sets. It is a difficult theorem of
Daniel Quillen Daniel Gray "Dan" Quillen (June 22, 1940 – April 30, 2011) was an American mathematician. He is known for being the "prime architect" of higher algebraic ''K''-theory, for which he was awarded the Cole Prize in 1975 and the Fields Medal in 197 ...
that the category of simplicial sets with these classes of morphisms becomes a model category, and indeed satisfies the axioms for a
proper Proper may refer to: Mathematics * Proper map, in topology, a property of continuous function between topological spaces, if inverse images of compact subsets are compact * Proper morphism, in algebraic geometry, an analogue of a proper map for ...
closed
simplicial model category In mathematics, particularly in homotopy theory, a model category is a category with distinguished classes of morphisms ('arrows') called ' weak equivalences', ' fibrations' and 'cofibrations' satisfying certain axioms relating them. These abstr ...
. A key turning point of the theory is that the geometric realization of a Kan fibration is a
Serre fibration The notion of a fibration generalizes the notion of a fiber bundle and plays an important role in algebraic topology, a branch of mathematics. Fibrations are used, for example, in postnikov-systems or obstruction theory. In this article, all ma ...
of spaces. With the model structure in place, a homotopy theory of simplicial sets can be developed using standard
homotopical algebra In mathematics, homotopical algebra is a collection of concepts comprising the ''nonabelian'' aspects of homological algebra as well as possibly the abelian aspects as special cases. The ''homotopical'' nomenclature stems from the fact that a ...
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 In mathematics, the homotopy category is a category built from the category of topological spaces which in a sense identifies two spaces that have the same shape. The phrase is in fact used for two different (but related) categories, as discussed be ...
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 In mathematics, the simplex category (or simplicial category or nonempty finite ordinal category) is the category of non-empty finite ordinals and order-preserving maps. It is used to define simplicial and cosimplicial objects. Formal definition ...
and op the
opposite category In category theory, a branch of mathematics, the opposite category or dual category ''C''op of a given category ''C'' is formed by reversing the morphisms, i.e. interchanging the source and target of each morphism. Doing the reversal twice yields t ...
. When ''C'' is the
category of sets In the mathematical field of category theory, the category of sets, denoted as Set, is the category whose objects are sets. The arrows or morphisms between sets ''A'' and ''B'' are the total functions from ''A'' to ''B'', and the composition o ...
, 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 There a ...
or category of abelian groups, we obtain the categories sGrp of simplicial
group A group is a number of persons or things that are located, gathered, or classed together. Groups of people * Cultural group, a group whose members share the same cultural identity * Ethnic group, a group whose members share the same ethnic ide ...
s and sAb of simplicial
abelian group In mathematics, an abelian group, also called a commutative group, is a group in which the result of applying the group operation to two group elements does not depend on the order in which they are written. That is, the group operation is commut ...
s, respectively.
Simplicial group In mathematics, more precisely, in the theory of simplicial sets, a simplicial group is a simplicial object in the category of groups. Similarly, a simplicial abelian group is a simplicial object in the category of abelian groups. A simplicial grou ...
s 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 In mathematics, more precisely, in the theory of simplicial sets, the Dold–Kan correspondence (named after Albrecht Dold and Daniel Kan) states that there is an equivalence between the category of (nonnegatively graded) chain complexes and the ...
which yields an equivalence of categories between simplicial abelian groups and bounded
chain complex In mathematics, a chain complex is an algebraic structure that consists of a sequence of abelian groups (or modules) and a sequence of homomorphisms between consecutive groups such that the image of each homomorphism is included in the kernel of t ...
es and is given by functors :''N:'' sAb → Ch+ and : Γ: Ch+ →  sAb.


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 ac ...
s of
group A group is a number of persons or things that are located, gathered, or classed together. Groups of people * Cultural group, a group whose members share the same cultural identity * Ethnic group, a group whose members share the same ethnic ide ...
s. 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 Algebraic ''K''-theory is a subject area in mathematics with connections to geometry, topology, ring theory, and number theory. Geometric, algebraic, and arithmetic objects are assigned objects called ''K''-groups. These are groups in the sense ...
. In this work, which earned him a
Fields Medal The Fields Medal is a prize awarded to two, three, or four mathematicians under 40 years of age at the International Congress of the International Mathematical Union (IMU), a meeting that takes place every four years. The name of the award ho ...
, 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 In topology, a branch of mathematics, the loop space Ω''X'' of a pointed topological space ''X'' is the space of (based) loops in ''X'', i.e. continuous pointed maps from the pointed circle ''S''1 to ''X'', equipped with the compact-open topolo ...
. 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 arrows in order to be able to explicitly study the structure behind those equalities. Higher cate ...
and
derived algebraic geometry Derived algebraic geometry is a branch of mathematics that generalizes algebraic geometry to a situation where commutative rings, which provide local charts, are replaced by either differential graded algebras (over \mathbb), simplicial commutati ...
. 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 In mathematics, a dendroidal set is a generalization of simplicial sets introduced by . They have the same relation to (colored symmetric) operads, also called symmetric multicategories, that simplicial sets have to categories. Definition A dend ...
, a generalization of simplicial set *
Simplicial presheaf In mathematics, more specifically in homotopy theory, a simplicial presheaf is a presheaf on a site (e.g., the category of topological spaces) taking values in simplicial sets (i.e., a contravariant functor from the site to the category of simplici ...
*
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 In mathematics, Kan complexes and Kan fibrations are part of the theory of simplicial set In mathematics, a simplicial set is an object composed of ''simplices'' in a specific way. Simplicial sets are higher-dimensional generalizations of directed ...
*
Dold–Kan correspondence In mathematics, more precisely, in the theory of simplicial sets, the Dold–Kan correspondence (named after Albrecht Dold and Daniel Kan) states that there is an equivalence between the category of (nonnegatively graded) chain complexes and the ...
* Simplicial homotopy *
Simplicial sphere In geometry and combinatorics, a simplicial (or combinatorial) ''d''-sphere is a simplicial complex homeomorphic to the ''d''-dimensional sphere. Some simplicial spheres arise as the boundaries of convex polytopes, however, in higher dimensions ...
*
Abstract simplicial complex In combinatorics, an abstract simplicial complex (ASC), often called an abstract complex or just a complex, is a family of sets that is closed under taking subsets, i.e., every subset of a set in the family is also in the family. It is a purely c ...


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