Simplicial set


In mathematics, a simplicial set is an object made up of "simplices" in a specific way. Simplicial sets are higher-dimensional generalizations of directed graphs, partially ordered sets and categories. Formally, a simplicial set may be defined as a contravariant functor from the simplex category to the category of sets. Simplicial sets were introduced in 1950 by Samuel Eilenberg and J. A. Zilber.
One may view a simplicial set as a purely combinatorial construction designed to capture the notion of a "well-behaved" topological space for the purposes of homotopy theory. Specifically, the category of simplicial sets carries a natural model structure, and the corresponding homotopy category is equivalent to the familiar homotopy category of topological spaces.
Simplicial sets are used to define quasi-categories, a basic notion of higher category theory. A construction analogous to that of simplicial sets can be carried out in any category, not just in the category of sets, yielding the notion of simplicial objects.

Motivation

A simplicial set is a categorical model capturing those topological spaces that can be built up from simplices and their incidence relations. This is similar to the approach of CW complexes to modeling topological spaces, with the crucial difference that simplicial sets are purely algebraic and do not carry any actual topology.
To get back to actual topological spaces, there is a geometric realization functor which turns simplicial sets into compactly generated Hausdorff spaces. Most classical results on CW complexes in homotopy theory are generalized by analogous results for simplicial sets. While algebraic topologists largely continue to prefer CW complexes, there is a growing contingent of researchers interested in using simplicial sets for applications in algebraic geometry where CW complexes do not naturally exist.

Intuition

Simplicial sets can be viewed as a higher-dimensional generalization of directed multigraphs. A simplicial set contains vertices and arrows 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 and n + 1 faces. 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, 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 Xn, n = 0, 1, 2, ..., together with certain maps between these sets: the face maps dn,i : Xn → Xn−1 and degeneracy maps sn,i : Xn→Xn+1. We think of the elements of Xn as the n-simplices of X. The map dn,i assigns to each such n-simplex its i-th face, the face "opposite to" the i-th vertex. The map sn,i assigns to each n-simplex the degenerate -simplex which arises from the given one by duplicating the i-th vertex. This description implicitly requires certain consistency relations among the maps dn,i and sn,i. Rather than requiring these simplicial identities explicitly as part of the definition, the short and elegant modern definition uses the language of category theory.

Formal definition

Let Δ denote the simplex category. The objects of Δ are nonempty linearly ordered sets of the form
with n≥0. The morphisms in Δ are order-preserving functions between these sets.
A simplicial set X is a contravariant functor
where Set is the category of sets. Simplicial sets are therefore nothing but presheaves on Δ. Given a simplicial set X, we often write Xn instead of X.
Simplicial sets form a category, usually denoted sSet, whose objects are simplicial sets and whose morphisms are natural transformations between them.
If we consider covariant functors X : Δ → Set instead of a contravariant ones, we arrive at the definition of a cosimplicial set. The corresponding category of cosimplicial sets is denoted by cSet.

Face and degeneracy maps

The simplex category Δ is generated by two particularly important families of morphisms, 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, where is the only injection that "misses".
Let us denote these face maps by respectively, so that is a map. If the first index is clear, we write instead of.
The degeneracy maps of the simplicial set X are the images in that simplicial set of the morphisms, where is the only surjection that "hits" twice.
Let us denote these degeneracy maps by respectively, so that is a map. If the first index is clear, we write instead of.
The defined maps satisfy the following simplicial identities:
  1. if i < j.
  2. if i < j.
  3. if i = j or i = j + 1.
  4. if i > j + 1.
  5. if i ≤ j.
Conversely, given a sequence of sets Xn together with maps and 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, we can define a simplicial set NS, the nerve of S, as follows: for every object of Δ we set NS = hompo-set, the order-preserving maps from to S. Every morphism φ:→ in Δ is an order preserving map, and via composition induces a map NS : NS → NS. 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 NSn=NS, can be thought of as ordered length- sequences of elements from S: . The face map di drops the i-th element from such a list, and the degeneracy maps si duplicates the i-th element.
A similar construction can be performed for every category C, to obtain the nerve NC of C. Here, NC is the set of all functors from to C, where we consider 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: a0 → a1 → ... → an. The face map d0 drops the first morphism from such a list, the face map dn drops the last, and the face map di for 0 < i < n drops ai and composes the ith and th morphisms. The degeneracy maps si 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 SYn 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 Δn, is a simplicial set defined as the functor homΔ where denotes the ordered set of the first nonnegative integers.
By the Yoneda lemma, the n-simplices of a simplicial set X stand in 1–1 correspondence with the natural transformations from Δn to X, i.e..
Furthermore, X gives rise to a category of simplices, denoted by , whose objects are maps Δn → X and whose morphisms are natural transformations Δn → Δm over X arising from maps → in Δ. That is, is a slice category of Δ over X. The following isomorphism shows that a simplicial set X is a colimit of its simplices:
where the colimit is taken over the category of simplices of X.

Geometric realization

There is a functor |•|: sSet → CGHaus called the geometric realization taking a simplicial set X to its corresponding realization in the category of compactly-generated Hausdorff topological spaces. Intuitively, the realization of X is the topological space obtained if every n-simplex of X is replaced by a topological n-simplex and these topological simplices are glued together in the fashion the simplices of X hang together. In this process the orientation of the simplices of X is lost.
To define the realization functor, we first define it on standard n-simplices Δn as follows: the geometric realization |Δn| is the standard topological n-simplex in general position given by
The definition then naturally extends to any simplicial set X by setting
where the colimit is taken over the n-simplex category of X. The geometric realization is functorial on sSet.
It is significant that we use the category CGHaus of compactly-generated Hausdorff spaces, rather than the category Top of topological spaces, as the target category of geometric realization: like sSet and unlike Top, the category CGHaus is cartesian closed; the categorical product is defined differently in the categories Top and CGHaus, and the one in CGHaus corresponds to the one in sSet via geometric realization.

Singular set for a space

The singular set of a topological space Y is the simplicial set SY defined by
Every order-preserving map φ:→ induces a continuous map |Δn|→|Δm| in a natural way, which by composition yields SY : SY → SY. This definition is analogous to a standard idea in singular homology of "probing" a target topological space with standard topological n-simplices. Furthermore, the singular functor S is right adjoint to the geometric realization functor described above, i.e.:
for any simplicial set X and any topological space Y. Intuitively, this adjunction can be understood as follows: a continuous map from the geometric realization of X to a space Y is uniquely specified if we associate to every simplex of X a continuous map from the corresponding standard topological simplex to Y, in such a fashion that these maps are compatible with the way the simplices in X hang together.

Homotopy theory of simplicial sets

In order to define a model structure on the category of simplicial sets, one has to define fibrations, cofibrations and weak equivalences. One can define fibrations to be Kan fibrations. A map of simplicial sets is defined to be a weak equivalence if its geometric realization is a weak equivalence of spaces. A map of simplicial sets is defined to be a cofibration if it is a monomorphism of simplicial sets. It is a difficult theorem of Daniel Quillen that the category of simplicial sets with these classes of morphisms satisfies the axioms for a proper closed simplicial model category.
A key turning point of the theory is that the geometric realization of a Kan fibration is a Serre fibration of spaces. With the model structure in place, a homotopy theory of simplicial sets can be developed using standard homotopical algebra methods. Furthermore, the geometric realization and singular functors give a Quillen equivalence of closed model categories inducing an equivalence
between the homotopy category for simplicial sets and the usual homotopy category of CW complexes with homotopy classes of continuous maps between them. It is part of the general
definition of a Quillen adjunction that the right adjoint functor carries fibrations to fibrations.

Simplicial objects

A simplicial object X in a category C is a contravariant functor
or equivalently a covariant functor
where Δ still denotes the simplex category. When C is the category of sets, we are just talking about the simplicial sets that were defined above. Letting C be the category of groups or category of abelian groups, we obtain the categories sGrp of simplicial groups and sAb of simplicial abelian groups, respectively.
Simplicial groups and simplicial abelian groups also carry closed model structures induced by that of the underlying simplicial sets.
The homotopy groups of simplicial abelian groups can be computed by making use of the Dold–Kan correspondence which yields an equivalence of categories between simplicial abelian groups and bounded chain complexes and is given by functors
and

History and uses of simplicial sets

Simplicial sets were originally used to give precise and convenient descriptions of classifying spaces of groups. This idea was vastly extended by Grothendieck's idea of
considering classifying spaces of categories, and in particular by Quillen's work of algebraic K-theory. In this work, which earned him a Fields Medal, Quillen
developed surprisingly efficient methods for manipulating
infinite simplicial sets. Later these methods were used in other areas on the border between algebraic geometry and topology. For instance, the André–Quillen homology of a ring is a "non-abelian homology", defined and studied in this way.
Both the algebraic K-theory and the André–Quillen homology are defined using algebraic data to write down a simplicial set, and then taking the homotopy groups of this simplicial set.
Simplicial methods are often useful when one wants to prove that a space is a loop space. The basic idea is that if is a group with classifying space, then is homotopy equivalent to the loop space. If itself is a group, we can iterate the procedure, and is homotopy equivalent to the double loop space . In case is an abelian group, we can actually iterate this infinitely many times, and obtain that is an infinite loop space.
Even if 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 is an infinite loop space. In this way, one can prove that the algebraic -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.