HOME

TheInfoList



OR:

In 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 pai ...
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. A poset consists of a set together with a binar ...
s and categories. 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 definit ...
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 ...
. Simplicial sets were introduced in 1950 by Samuel Eilenberg 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 po ...
, 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. T ...
" topological space for the purposes of
homotopy theory In mathematics, homotopy theory is a systematic study of situations in which maps can come with homotopies between them. It originated as a topic in algebraic topology but nowadays is studied as an independent discipline. Besides algebraic topol ...
. 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 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 cla ...
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 mapping between categories. Functors were first considered in algebraic topology, where algebraic objects (such as the fundamental group) are associated to topological spaces, an ...
which turns simplicial sets into compactly generated Hausdorff spaces. Most classical results on CW complexes in
homotopy theory In mathematics, homotopy theory is a systematic study of situations in which maps can come with homotopies between them. It originated as a topic in algebraic topology but nowadays is studied as an independent discipline. Besides algebraic topol ...
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'' 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 ...
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.


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 definit ...
. The objects of Δ are nonempty linearly ordered sets 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 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 ...
. (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 yield ...
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 between them. This is nothing but the category of
presheaves In mathematics, a sheaf is a tool for systematically tracking data (such as sets, abelian groups, rings) attached to the open sets of a topological space and defined locally with regard to them. For example, for each open set, the data could ...
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. A poset consists of a set together with a binar ...
(''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 ...
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, 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 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 su ...
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 spaces. 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 cla ...
) 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 su ...
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 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 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 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 com ...
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 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 definit ...
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 yield ...
. 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 ...
, 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 T ...
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 of ...
, we obtain the categories sGrp of simplicial groups 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 com ...
s, 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.


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 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 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 topo ...
. 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 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 In mathematics, a Δ-set ''S'', often called a semi-simplicial set, is a combinatorial object that is useful in the construction and triangulation of topological spaces, and also in the computation of related algebraic invariants of such spaces. ...
* Dendroidal set, a generalization of simplicial set * Simplicial presheaf * Quasi-category * Kan complex * Dold–Kan correspondence *
Simplicial homotopy In algebraic topology, a simplicial homotopypg 23 is an analog of a homotopy between topological spaces for simplicial sets. If :f, g: X \to Y are maps between simplicial sets, a simplicial homotopy from ''f'' to ''g'' is a map :h: X \times \Delta^ ...
* Simplicial sphere *
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 ...


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