Simplicial set

In mathematics, a simplicial set is a construction in categorical homotopy theory that is a pure algebraic model of the notion of a "well-behaved" topological space. Historically, this model arose from earlier work in combinatorial topology and in particular from the notion of simplicial complexes. Simplicial sets are used to define quasi-categories, a basic notion of higher category theory.

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 (this will become clear in the formal definition).

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 have analogous versions for simplicial sets which generalize these results. 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.

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 an ordered list of three vertices A, B, C and three arrows f:A→B, g:B→C and h:A→C. In general, an n-simplex is an object made up from an ordered 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).

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