In mathematics, a simplicial set is a sequence of sets with internal order structure (abstract simplices) 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 (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 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 (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 → C, A → C and A → 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 − 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 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 (n = 1, 2, 3, ... and 0 ≤ i ≤ n) and degeneracy maps sn,i : Xn→Xn+1 (n = 0, 1, 2, ... and 0 ≤ i ≤ n). 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" (i.e. not containing) the i-th vertex. The map sn,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 dn,i and sn,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, and the morphisms (non-strictly) order-preserving functions. Each object is uniquely isomorphic to an object of the form
[n] = {0, 1, ..., n} with n ≥ 0. A simplicial set X is a contravariant functor
X : Δ → Set where Set is the category of sets. (Alternatively and equivalently, one may define simplicial sets as covariant functors from the opposite category Δop → 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 the category of presheaves on Δ. As such, it is a Grothendieck topos.
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