Editing Simplicial set (section)

== Intuition ==

Simplicial sets can be viewed as a higher-dimensional generalization of [[Graph (discrete mathematics)#Directed graph|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''&nbsp;→&nbsp;''C'', ''A''&nbsp;→&nbsp;''C'' and ''A''&nbsp;→&nbsp;''B''. In general, an ''n''-simplex is an object made up from a list of ''n''&nbsp;+&nbsp;1 vertices (which are 0-simplices) and ''n''&nbsp;+&nbsp;1 faces (which are (''n''&nbsp;−&nbsp;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 complex]]es, which generalize [[graph (discrete mathematics)|simple undirected graphs]] rather than directed multigraphs.

Formally, a simplicial set ''X'' is a collection of sets ''X''<sub>''n''</sub>, ''n''&nbsp;=&nbsp;0,&nbsp;1,&nbsp;2,&nbsp;..., together with certain maps between these sets: the ''face maps'' ''d''<sub>''n'',''i''</sub>&nbsp;:&nbsp;''X''<sub>''n''</sub>&nbsp;→&nbsp;''X''<sub>''n''−1</sub> (''n''&nbsp;=&nbsp;1,&nbsp;2,&nbsp;3,&nbsp;... and 0&nbsp;≤&nbsp;''i''&nbsp;≤&nbsp;''n'') and ''degeneracy maps'' ''s''<sub>''n'',''i''</sub>&nbsp;:&nbsp;''X''<sub>''n''</sub>→''X''<sub>''n''+1</sub> (''n''&nbsp;=&nbsp;0,&nbsp;1,&nbsp;2,&nbsp;... and 0&nbsp;≤&nbsp;''i''&nbsp;≤&nbsp;''n''). We think of the elements of ''X''<sub>''n''</sub> as the ''n''-simplices of ''X''. The map ''d''<sub>''n'',''i''</sub> assigns to each such ''n''-simplex its ''i''-th face, the face "opposite to" (i.e. not containing) the ''i''-th vertex. The map ''s''<sub>''n'',''i''</sub> 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 ''d''<sub>''n'',''i''</sub> and ''s''<sub>''n'',''i''</sub>. 

Rather than requiring these ''simplicial identities'' explicitly as part of the definition, the short modern definition uses the language of [[category theory]].