Editing Simplicial set (section)

{{Short description|Mathematical construction used in homotopy theory}}
In [[mathematics]], a '''simplicial set''' is a sequence of sets with internal order structure ([[Abstract simplicial complex|abstract simplices]]) and maps between them.  Simplicial sets are higher-dimensional generalizations of [[directed graph]]s. 

Every simplicial set gives rise to a "nice" [[topological space]], known as its geometric realization. This realization consists of [[Simplex|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 category|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.<ref>{{Cite journal|last=Eilenberg|first=Samuel|last2=Zilber|first2=J. A.|date=1950|title=Semi-Simplicial Complexes and Singular Homology|journal=Annals of Mathematics|volume=51|issue=3|pages=499–513|doi=10.2307/1969364|jstor=1969364}}</ref>

Simplicial sets are used to define [[quasi-category|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'''.