Editing Simplicial set (section)

==Formal definition==

Let Δ denote the [[simplex category]]. The objects of Δ are nonempty [[total order|totally ordered]] sets. Each object is uniquely order isomorphic to an object of the form
:[''n''] = {0, 1, ..., ''n''}
with ''n'' ≥ 0. The morphisms in Δ are (non-strictly) [[order-preserving function]]s between these sets.

A '''simplicial set''' ''X'' is a [[functor#Covariance and contravariance|contravariant functor]]

:''X'' : Δ → '''Set'''

where '''Set''' is the [[category of sets]]. (Alternatively and equivalently, one may define simplicial sets as [[functor#Covariance and contravariance|covariant functors]] from the [[Dual (category theory)|opposite category]] Δ<sup>op</sup> ''→''f '''Set'''.) Given a simplicial set ''X,'' we often write ''X<sub>n</sub>'' instead of ''X''([''n'']).

Simplicial sets form a category, usually denoted '''sSet''', whose objects are simplicial sets and whose morphisms are [[natural transformations]] between them. This is the category of [[presheaf (category theory)|presheaves]] on Δ. As such, it is a [[topos]].

===Face and degeneracy maps and simplicial identities===
The morphisms (maps) of the simplex category Δ are generated by two particularly important families of morphisms, whose images under a given simplicial set functor are called the '''face maps''' and '''degeneracy maps''' of that simplicial set.

The ''face maps'' of a simplicial set ''X'' are the images in that simplicial set of the morphisms <math>\delta^{n,0},\dotsc,\delta^{n,n}\colon[n-1]\to[n]</math>, where <math>\delta^{n,i}</math> is the only (order-preserving) injection <math>[n-1]\to[n]</math> that "misses" <math>i</math>.
Let us denote these face maps by <math>d_{n,0},\dotsc,d_{n,n}</math> respectively, so that <math>d_{n,i}</math> is a map <math>X_n \to X_{n-1}</math>. If the first index is clear, we write <math>d_i</math> instead of <math>d_{n,i}</math>.

The ''degeneracy maps'' of the simplicial set ''X'' are the images in that simplicial set of the morphisms <math>\sigma^{n,0},\dotsc,\sigma^{n,n}\colon[n+1]\to[n]</math>, where <math>\sigma^{n,i}</math> is the only (order-preserving) surjection <math>[n+1]\to[n]</math> that "hits" <math>i</math> twice.
Let us denote these degeneracy maps by <math>s_{n,0},\dotsc,s_{n,n}</math> respectively, so that <math>s_{n,i}</math> is a map <math>X_n \to X_{n+1}</math>. If the first index is clear, we write <math>s_i</math> instead of <math>s_{n,i}</math>.

The defined maps satisfy the following '''simplicial identities''':

#<math>d_i d_j = d_{j-1} d_i</math>  if ''i'' < ''j''. (This is short for <math>d_{n-1,i} d_{n,j} = d_{n-1,j-1} d_{n,i}</math> if 0 ≤ ''i'' < ''j'' ≤ ''n''.)
#<math>d_i s_j = s_{j-1}d_i</math>  if ''i'' < ''j''.
#<math>d_i s_j = \text{id}</math> if ''i'' = ''j'' or ''i'' = ''j''&nbsp;+&nbsp;1.
#<math>d_i s_j = s_j d_{i-1}</math>  if ''i'' > ''j''&nbsp;+&nbsp;1.
#<math>s_i s_j = s_{j+1} s_i</math>  if ''i'' ≤ ''j''.

Conversely, given a sequence of sets ''X<sub>n</sub>'' together with maps <math>d_{n,i} : X_n \to X_{n-1}</math> and <math>s_{n,i} : X_n \to X_{n+1}</math> that satisfy the simplicial identities, there is a unique simplicial set ''X'' that has these face and degeneracy maps. So the identities provide an alternative way to define simplicial sets.