Editing Simplicial set (section)

==Examples==
Given a [[partially ordered set]] (''S'', ≤), we can define a simplicial set ''NS'', called the [[nerve (category theory)|nerve]] of ''S'', as follows: for every object [''n''] of Δ we set ''NS''([''n'']) = hom<sub>'''poset'''</sub>( [''n''] , ''S''), the set of order-preserving maps from [''n''] to ''S''. Every morphism &phi;: [''n''] → [''m''] in Δ is an order preserving map, and via composition induces a map ''NS''(&phi;) : ''NS''([''m'']) → ''NS''([''n'']). It is straightforward to check that ''NS'' is a contravariant functor from Δ to '''Set''': a simplicial set.

Concretely, the ''n''-simplices of the nerve ''NS'', i.e. the elements of ''NS''<sub>''n''</sub> = ''NS''([''n'']), can be thought of as ordered length-(''n''+1) sequences of elements from ''S'':  (''a''<sub>0</sub>&nbsp;≤&nbsp;''a''<sub>1</sub>&nbsp;≤ ... ≤&nbsp;''a''<sub>''n''</sub>). The face map ''d''<sub>''i''</sub> drops the ''i''-th element from such a list, and the degeneracy maps ''s''<sub>''i''</sub> duplicates the ''i''-th element.

A similar construction can be performed for every category ''C'', to obtain the nerve ''NC'' of ''C''. Here, ''NC''([''n'']) is the set of all functors from [''n''] to ''C'', where we consider [''n''] as a category with objects 0,1,...,''n'' and a single morphism from ''i'' to ''j'' whenever ''i''&nbsp;≤&nbsp;''j''.

Concretely, the ''n''-simplices of the nerve ''NC'' can be thought of as sequences of ''n'' composable morphisms in ''C'': ''a''<sub>0</sub>&nbsp;→&nbsp;''a''<sub>1</sub>&nbsp;→&nbsp;...&nbsp;→&nbsp;''a''<sub>''n''</sub>. (In particular, the 0-simplices are the objects of ''C'' and the 1-simplices are the morphisms of ''C''.) The face map ''d''<sub>0</sub> drops the first morphism from such a list, the face map ''d''<sub>''n''</sub> drops the last, and the face map ''d''<sub>''i''</sub> for 0&nbsp;<&nbsp;''i''&nbsp;<&nbsp;''n'' drops ''a<sub>i</sub>'' and composes the ''i''-th and (''i''&nbsp;+&nbsp;1)-th morphisms. The degeneracy maps ''s''<sub>''i''</sub> lengthen the sequence by inserting an identity morphism at position&nbsp;''i''.

We can recover the poset ''S'' from the nerve ''NS'' and the category ''C'' from the nerve ''NC''; in this sense simplicial sets generalize posets and categories.

Another important class of examples of simplicial sets is given by the singular set ''SY'' of a topological space ''Y''. Here ''SY''<sub>''n''</sub> consists of all the continuous maps from the standard topological ''n''-simplex to ''Y''. The singular set is further explained below.