Simplicial set

In <a href="/facts/Mathematics/pxTouaz4">mathematics</a>, a simplicial set is a sequence of sets with internal order structure (<a href="/facts/Abstract_simplicial_complex/ytwxClsf">abstract simplices</a>) and maps between them. Simplicial sets are higher-dimensional generalizations of <a href="/facts/Directed_graph/YBdfQ0um">directed graphs</a>. 
Every simplicial set gives rise to a "nice" <a href="/facts/Topological_space/v2ut7CbK">topological space</a>, known as its geometric realization. This realization consists of <a href="/facts/Simplex/0FCfNRN8">geometric simplices</a>, 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 <a href="/facts/Homotopy_theory/MJwYyRny">homotopy theory</a>. Specifically, the category of simplicial sets carries a natural <a href="/facts/Model_category/KcxFUgY8">model structure</a>, and the corresponding <a href="/facts/Homotopy_category/JaEnPakj">homotopy category</a> is equivalent to the familiar homotopy category of topological spaces.
Formally, a simplicial set may be defined as a <a href="/facts/Contravariant_functor/l2u3rIBE">contravariant functor</a> from the <a href="/facts/Simplex_category/fDeKuXpX">simplex category</a> to the <a href="/facts/Category_of_sets/LF5ykYl0">category of sets</a>. Simplicial sets were introduced in 1950 by <a href="/facts/Samuel_Eilenberg/vs6aksKX">Samuel Eilenberg</a> and Joseph A. Zilber.
Simplicial sets are used to define <a href="/facts/Quasi-category/YgCfQmMD">quasi-categories</a>, a basic notion of <a href="/facts/Higher_category_theory/7Pp6a0VN">higher category theory</a>. 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.

Simplicial set open-in-new

Simplicial set