Simplicial set

In mathematics, a simplicial set is a sequence of sets with internal order structure and maps between them. Simplicial sets are higher-dimensional generalizations of directed graphs.
Every simplicial set gives rise to a "nice" topological space, known as its geometric realization. This realization consists of geometric simplices, glued together according to the rules of the simplicial set. Indeed, one may view a simplicial set as a purely combinatorial construction designed to capture the essence of a topological space for the purposes of homotopy theory. Specifically, the category of simplicial sets carries a natural model structure, and the corresponding homotopy category is equivalent to the familiar homotopy category of topological spaces.
Formally, a simplicial set may be defined as a contravariant functor from the simplex category to the category of sets. Simplicial sets were introduced in 1950 by Samuel Eilenberg and Joseph A. Zilber.
Simplicial sets are used to define quasi-categories, a basic notion of higher category theory. 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 X_n, n = 0, 1, 2, ..., together with certain maps between these sets: the face maps ''dn'',i : X_n → X_n−1 and degeneracy maps ''sn'',i : X_n→X_n+1. 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" the i-th vertex. The map s_n,''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 d_n,''i and s''_n,''i.
Rather than requiring these simplicial identities'' explicitly as part of the definition, the short modern definition uses the language of category theory.

Formal definition

Let Δ denote the simplex category. The objects of Δ are nonempty totally ordered finite sets. Each object is uniquely order isomorphic to an object 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. Given a simplicial set X, we often write X_n instead of X.
Simplicial sets form a category, usually denoted sSet, whose objects are simplicial sets and whose morphisms are natural transformations between them. This is the category of presheaves on Δ. As such, it is a topos.

Face and degeneracy maps and simplicial identities

The morphisms 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, 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:

if i < j.
if i < j.
if i = j or i = j + 1.
if i > j + 1.
if i ≤ j.

Conversely, given a sequence of sets X_n 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, called the nerve of S, as follows: for every object of Δ we set NS = hom_poset, the set of 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 NS_n = NS, can be thought of as ordered length- sequences of elements from S: . 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 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 → ... → a_n. The face map d₀ 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 a_i and composes the i-th and -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 Δⁿ, is a simplicial set defined as the functor hom_Δ 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 → Δ^m 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 CGHaus 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.