Editing Simplicial set

{{Short description|Mathematical construction used in homotopy theory}}
In [[mathematics]], a '''simplicial set''' is a sequence of sets with internal order structure ([[Abstract simplicial complex|abstract simplices]]) and maps between them.  Simplicial sets are higher-dimensional generalizations of [[directed graph]]s. 

Every simplicial set gives rise to a "nice" [[topological space]], known as its geometric realization. This realization consists of [[Simplex|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 category|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]]. Simplicial sets were introduced in 1950 by [[Samuel Eilenberg]] and Joseph A. Zilber.<ref>{{Cite journal|last=Eilenberg|first=Samuel|last2=Zilber|first2=J. A.|date=1950|title=Semi-Simplicial Complexes and Singular Homology|journal=Annals of Mathematics|volume=51|issue=3|pages=499–513|doi=10.2307/1969364|jstor=1969364}}</ref>

Simplicial sets are used to define [[quasi-category|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 [[simplex|simplices]] and their incidence relations.  This is similar to the approach of [[CW complex]]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]] which turns simplicial sets into [[compactly generated Hausdorff space]]s. Most classical results on CW complexes in [[homotopy theory]] are generalized by analogous results for simplicial sets.  While [[algebraic topology|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 [[Graph (discrete mathematics)#Directed graph|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''&nbsp;→&nbsp;''C'', ''A''&nbsp;→&nbsp;''C'' and ''A''&nbsp;→&nbsp;''B''. In general, an ''n''-simplex is an object made up from a list of ''n''&nbsp;+&nbsp;1 vertices (which are 0-simplices) and ''n''&nbsp;+&nbsp;1 faces (which are (''n''&nbsp;−&nbsp;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]]es, which generalize [[graph (discrete mathematics)|simple undirected graphs]] rather than directed multigraphs.

Formally, a simplicial set ''X'' is a collection of sets ''X''<sub>''n''</sub>, ''n''&nbsp;=&nbsp;0,&nbsp;1,&nbsp;2,&nbsp;..., together with certain maps between these sets: the ''face maps'' ''d''<sub>''n'',''i''</sub>&nbsp;:&nbsp;''X''<sub>''n''</sub>&nbsp;→&nbsp;''X''<sub>''n''−1</sub> (''n''&nbsp;=&nbsp;1,&nbsp;2,&nbsp;3,&nbsp;... and 0&nbsp;≤&nbsp;''i''&nbsp;≤&nbsp;''n'') and ''degeneracy maps'' ''s''<sub>''n'',''i''</sub>&nbsp;:&nbsp;''X''<sub>''n''</sub>→''X''<sub>''n''+1</sub> (''n''&nbsp;=&nbsp;0,&nbsp;1,&nbsp;2,&nbsp;... and 0&nbsp;≤&nbsp;''i''&nbsp;≤&nbsp;''n''). We think of the elements of ''X''<sub>''n''</sub> as the ''n''-simplices of ''X''. The map ''d''<sub>''n'',''i''</sub> 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''<sub>''n'',''i''</sub> 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''<sub>''n'',''i''</sub> and ''s''<sub>''n'',''i''</sub>. 

Rather than requiring these ''simplicial identities'' explicitly as part of the definition, the short modern definition uses the language of [[category theory]].

==Formal definition==

Let Δ denote the [[simplex category]]. The objects of Δ are nonempty [[total order|totally ordered]] sets. Each object is uniquely order isomorphic to an object of the form
:[''n''] = {0, 1, ..., ''n''}
with ''n'' ≥ 0. The morphisms in Δ are (non-strictly) [[order-preserving function]]s between these sets.

A '''simplicial set''' ''X'' is a [[functor#Covariance and contravariance|contravariant functor]]

:''X'' : Δ → '''Set'''

where '''Set''' is the [[category of sets]]. (Alternatively and equivalently, one may define simplicial sets as [[functor#Covariance and contravariance|covariant functors]] from the [[Dual (category theory)|opposite category]] Δ<sup>op</sup> ''→''f '''Set'''.) Given a simplicial set ''X,'' we often write ''X<sub>n</sub>'' 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 [[presheaf (category theory)|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 <math>\delta^{n,0},\dotsc,\delta^{n,n}\colon[n-1]\to[n]</math>, where <math>\delta^{n,i}</math> is the only (order-preserving) injection <math>[n-1]\to[n]</math> that "misses" <math>i</math>.
Let us denote these face maps by <math>d_{n,0},\dotsc,d_{n,n}</math> respectively, so that <math>d_{n,i}</math> is a map <math>X_n \to X_{n-1}</math>. If the first index is clear, we write <math>d_i</math> instead of <math>d_{n,i}</math>.

The ''degeneracy maps'' of the simplicial set ''X'' are the images in that simplicial set of the morphisms <math>\sigma^{n,0},\dotsc,\sigma^{n,n}\colon[n+1]\to[n]</math>, where <math>\sigma^{n,i}</math> is the only (order-preserving) surjection <math>[n+1]\to[n]</math> that "hits" <math>i</math> twice.
Let us denote these degeneracy maps by <math>s_{n,0},\dotsc,s_{n,n}</math> respectively, so that <math>s_{n,i}</math> is a map <math>X_n \to X_{n+1}</math>. If the first index is clear, we write <math>s_i</math> instead of <math>s_{n,i}</math>.

The defined maps satisfy the following '''simplicial identities''':

#<math>d_i d_j = d_{j-1} d_i</math>  if ''i'' < ''j''. (This is short for <math>d_{n-1,i} d_{n,j} = d_{n-1,j-1} d_{n,i}</math> if 0 ≤ ''i'' < ''j'' ≤ ''n''.)
#<math>d_i s_j = s_{j-1}d_i</math>  if ''i'' < ''j''.
#<math>d_i s_j = \text{id}</math> if ''i'' = ''j'' or ''i'' = ''j''&nbsp;+&nbsp;1.
#<math>d_i s_j = s_j d_{i-1}</math>  if ''i'' > ''j''&nbsp;+&nbsp;1.
#<math>s_i s_j = s_{j+1} s_i</math>  if ''i'' ≤ ''j''.

Conversely, given a sequence of sets ''X<sub>n</sub>'' together with maps <math>d_{n,i} : X_n \to X_{n-1}</math> and <math>s_{n,i} : X_n \to X_{n+1}</math> 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'', called the [[nerve (category theory)|nerve]] of ''S'', as follows: for every object [''n''] of Δ we set ''NS''([''n'']) = hom<sub>'''poset'''</sub>( [''n''] , ''S''), the set of order-preserving maps from [''n''] to ''S''. Every morphism &phi;: [''n''] → [''m''] in Δ is an order preserving map, and via composition induces a map ''NS''(&phi;) : ''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''<sub>''n''</sub> = ''NS''([''n'']), can be thought of as ordered length-(''n''+1) sequences of elements from ''S'':  (''a''<sub>0</sub>&nbsp;≤&nbsp;''a''<sub>1</sub>&nbsp;≤ ... ≤&nbsp;''a''<sub>''n''</sub>). The face map ''d''<sub>''i''</sub> drops the ''i''-th element from such a list, and the degeneracy maps ''s''<sub>''i''</sub> 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''&nbsp;≤&nbsp;''j''.

Concretely, the ''n''-simplices of the nerve ''NC'' can be thought of as sequences of ''n'' composable morphisms in ''C'': ''a''<sub>0</sub>&nbsp;→&nbsp;''a''<sub>1</sub>&nbsp;→&nbsp;...&nbsp;→&nbsp;''a''<sub>''n''</sub>. (In particular, the 0-simplices are the objects of ''C'' and the 1-simplices are the morphisms of ''C''.) The face map ''d''<sub>0</sub> drops the first morphism from such a list, the face map ''d''<sub>''n''</sub> drops the last, and the face map ''d''<sub>''i''</sub> for 0&nbsp;<&nbsp;''i''&nbsp;<&nbsp;''n'' drops ''a<sub>i</sub>'' and composes the ''i''-th and (''i''&nbsp;+&nbsp;1)-th morphisms. The degeneracy maps ''s''<sub>''i''</sub> lengthen the sequence by inserting an identity morphism at position&nbsp;''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''<sub>''n''</sub> 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<sup>''n''</sup>, is a simplicial set defined as the functor hom<sub>Δ</sub>(-, [''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 Δ<sup>op</sup>.<ref>{{harvnb|Gelfand|Manin|2013}}</ref>)

By the [[Yoneda lemma]], the ''n''-simplices of a simplicial set ''X'' stand in 1–1 correspondence with the natural transformations from Δ<sup>''n''</sup> to ''X,'' i.e. <math>X_n = X([n])\cong \operatorname{Nat}(\operatorname{hom}_\Delta(-,[n]),X)=
\operatorname{hom}_{\textbf{sSet}}(\Delta^n,X)</math>.

Furthermore, ''X'' gives rise to a [[category of elements|category of simplices]], denoted by <math>\Delta\downarrow{X}</math> , whose objects are maps (''i.e.'' natural transformations) Δ<sup>''n''</sup> → ''X'' and whose morphisms are natural transformations Δ<sup>''n''</sup> → Δ<sup>''m''</sup> over ''X'' arising from maps [''n''] ''→'' [''m''] in Δ.  That is, <math>\Delta\downarrow{X}</math> is a [[slice category]] of Δ over ''X''. The [[density theorem (category theory)|following isomorphism]] shows that a simplicial set ''X'' is a [[colimit]] of its simplices:<ref>{{harvnb|Goerss|Jardine|1999|p=7}}</ref>

: <math>X \cong \varinjlim_{\Delta^n \to X} \Delta^n</math>

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 space|compactly-generated]] [[Hausdorff topological space]]s. 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''&nbsp;+&nbsp;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<sup>''n''</sup> as follows: the geometric realization |Δ<sup>''n''</sup>| is the standard topological ''n''-[[simplex]] in general position given by

:<math>|\Delta^n| = \{(x_0, \dots, x_n) \in \mathbb{R}^{n+1}: 0\leq x_i \leq 1, \sum x_i = 1 \}.</math>

The definition then naturally extends to any simplicial set ''X'' by setting

:|X| = lim<sub>Δ<sup>''n''</sup> → ''X''</sub> | Δ<sup>''n''</sup>|

where the [[Limit (category theory)|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 category|cartesian closed]]; the [[Product (category theory)|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'']) = hom<sub>'''T''op'''''</sub>(|Δ<sup>''n''</sup>|, ''Y'') for each object [''n''] ∈ Δ.

Every order-preserving map &phi;:[''n'']→[''m''] induces a continuous map |Δ<sup>''n''</sup>|→|Δ<sup>''m''</sup>| by

:<math>(x_0,...,x_n) \in |\Delta_n| \mapsto (y_j),~~ y_j = \sum_{\phi(i) =j}x_i.</math>

Then, by composition it yields to a map ''SY''(''&phi;'') : ''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 [[adjoint functor|right adjoint]] to the geometric realization functor described above, i.e.:

:hom<sub>'''Top'''</sub>(|''X''|, ''Y'') &cong; hom<sub>'''sSet'''</sub>(''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 category|model structure]] on the category of simplicial sets, one has to define fibrations, cofibrations and weak equivalences. One can define [[Fibration of simplicial sets|fibrations]] to be [[Kan fibration]]s.  A map of simplicial sets is defined to be a weak equivalence if its geometric realization is a [[Weak homotopy equivalence|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 model category|proper]] [[closed model category|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 adjunction|Quillen equivalence]] of [[closed model category|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'': Δ<sup>op</sup> → ''C,''

where Δ still denotes the [[simplex category]] and <sup>op</sup> the [[opposite 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 [[group (mathematics)|group]]s and '''sAb''' of simplicial [[abelian group]]s, respectively.

[[Simplicial group]]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]] which yields an equivalence of categories between simplicial abelian groups and bounded [[chain complex]]es and is given by functors

:''N:'' '''sAb''' → Ch<sub>+</sub>

and

: &Gamma;: Ch<sub>+</sub> → &nbsp;'''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]]s of [[group (mathematics)|group]]s. This idea was vastly extended by [[Grothendieck]]'s idea of
considering classifying spaces of categories, and in particular by [[Daniel Quillen|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 cohomology|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 <math>G</math> is a group with classifying space <math>BG</math>, then <math>G</math> is homotopy equivalent to the loop space <math>\Omega BG</math>. If <math>BG</math> itself is a group, we can iterate the procedure, and <math>G</math> is homotopy equivalent to the double loop space  <math>\Omega^2 B(BG)</math>. In case  <math>G</math> is an abelian group, we can actually iterate this infinitely many times, and obtain that <math>G</math> is an infinite loop space.

Even if <math>X</math> 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 <math>X</math> is an infinite loop space. In this way, one can prove that the algebraic <math>K</math>-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-category|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]]
*[[Kan complex]]
*[[Dold–Kan correspondence]]
*[[Simplicial homotopy]]
*[[Simplicial sphere]]
*[[Abstract simplicial complex]]
*[[Anodyne extension]]
*[[Weak equivalence between simplicial sets]]

== Notes ==
{{reflist}}

== References ==
{{refbegin}}
* {{Cite book | last1=Goerss | first1=Paul G. | last2=Jardine | first2=John F. |author2-link=Rick Jardine| title=Simplicial Homotopy Theory | publisher=Birkhäuser | series=Progress in Mathematics | isbn=978-3-7643-6064-1 | year=1999 | volume=174 |doi=10.1007/978-3-0348-8707-6| mr=1711612}}
* {{cite book |first=Sergei I. |last=Gelfand |first2=Yuri I. |last2=Manin |author2-link=Yuri Manin| title=Methods of Homological Algebra |url=https://books.google.com/books?id=MIzqCAAAQBAJ |date=2013 |publisher=Springer |isbn=978-3-662-12492-5 }}
* {{cite web |first=Dylan G.L. |last=Allegretti |title=Simplicial Sets and van Kampen's Theorem |url=http://www.math.uchicago.edu/~may/VIGRE/VIGRE2008/REUPapers/Allegretti.pdf |citeseerx=10.1.1.539.7411 }} ''(An elementary introduction to simplicial sets)''.
* {{cite book |first=Daniel |last=Quillen |author-link=Daniel Quillen| chapter=Higher algebraic K-theory: I |editor-first=Hyman |editor-last=Bass |editor-link=Hyman Bass| title=Higher K-Theories |publisher=Springer-Verlag |series=Lecture Notes in Mathematics |volume=341 |year=1973 |pages=85–147 |isbn=3-540-06434-6 |url=https://www.springer.com/gp/book/9783540064343}}
* {{cite journal |first=Graeme B. |last=Segal |author-link=Graeme Segal|title=Categories and cohomology theories |journal=[[Topology (journal)|Topology]] |volume=13 |issue=3 |pages=293–312 |year=1974 |doi=10.1016/0040-9383(74)90022-6 |doi-access= }}
{{refend}}

== Further reading ==
*{{cite web |first=Emily |last=Riehl |author-link=Emily Riehl|title=A leisurely introduction to simplicial sets |url=http://www.math.jhu.edu/~eriehl/ssets.pdf }}
*[[J. Peter May|May, J. Peter]]. ''[http://www.math.uchicago.edu/~may/BOOKS/Simp.djvu Simplicial Objects in Algebraic Topology],'' University of Chicago Press 1967
*{{nlab|id=simplicial+set|title=simplicial set}}

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[[Category:Algebraic topology]]
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[[Category:Simplicial sets| ]]
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