Editing Simplicial set (section)

==Geometric realization==
There is a functor |•|: '''sSet''' ''→'' '''CGHaus''' called the '''geometric realization''' taking a simplicial set ''X'' to its corresponding realization in the category '''CGHaus''' of [[compactly-generated space|compactly-generated]] [[Hausdorff topological space]]s. Intuitively, the realization of ''X'' is the topological space (in fact a [[CW complex]]) obtained if every ''n-''simplex of ''X'' is replaced by a topological ''n-''simplex (a certain ''n-''dimensional subset of (''n''&nbsp;+&nbsp;1)-dimensional Euclidean space defined below) and these topological simplices are glued together in the fashion the simplices of ''X'' hang together. In this process the orientation of the simplices of ''X'' is lost.

To define the realization functor, we first define it on standard n-simplices Δ<sup>''n''</sup> as follows: the geometric realization |Δ<sup>''n''</sup>| is the standard topological ''n''-[[simplex]] in general position given by

:<math>|\Delta^n| = \{(x_0, \dots, x_n) \in \mathbb{R}^{n+1}: 0\leq x_i \leq 1, \sum x_i = 1 \}.</math>

The definition then naturally extends to any simplicial set ''X'' by setting

:|X| = lim<sub>Δ<sup>''n''</sup> → ''X''</sub> | Δ<sup>''n''</sup>|

where the [[Limit (category theory)|colimit]] is taken over the n-simplex category of ''X''. The geometric realization is functorial on '''sSet'''.

It is significant that we use the category '''CGHaus''' of compactly-generated Hausdorff spaces, rather than the category '''Top''' of topological spaces, as the target category of geometric realization: like '''sSet''' and unlike '''Top''', the category '''CGHaus''' is [[Cartesian closed category|cartesian closed]]; the [[Product (category theory)|categorical product]] is defined differently in the categories '''Top''' and '''CGHaus''', and the one in '''CGHaus''' corresponds to the one in '''sSet''' via geometric realization.