Editing Abstract simplicial complex (section)

=== Categorical definition ===
Alternatively, let <math>\mathcal{K}</math> denote the category whose objects are the faces of {{mvar|K}} and whose morphisms are inclusions. Next choose a [[total order]] on the vertex set of {{mvar|K}} and define a [[functor]] ''F'' from <math>\mathcal{K}</math> to the category of topological spaces as follows. For any face ''X'' in ''K'' of dimension ''n'', let {{math|''F''(''X'') {{=}} Δ<sup>''n''</sup>}} be the standard ''n''-simplex. The order on the vertex set then specifies a unique [[bijection]] between the elements of {{mvar|X}} and vertices of {{math|Δ<sup>''n''</sup>}}, ordered in the usual way {{math|''e''<sub>0</sub> < ''e''<sub>1</sub> < ... < ''e<sub>n</sub>''}}. If {{math|''Y'' ⊆ ''X''}} is a face of dimension {{math|''m'' < ''n''}}, then this bijection specifies a unique ''m''-dimensional face of {{math|Δ<sup>''n''</sup>}}. Define {{math|''F''(''Y'') →  ''F''(''X'')}} to be the unique [[affine transformation|affine]] linear [[embedding]] of {{math|Δ<sup>''m''</sup>}} as that distinguished face of {{math|Δ<sup>''n''</sup>}}, such that the map on vertices is order-preserving.

We can then define the geometric realization <math>|K|</math>  as the [[colimit]] of the functor ''F''. More specifically <math>|K|</math>  is the [[Quotient space (topology)|quotient space]] of the [[disjoint union]]

:<math>\coprod_{X \in K}{F(X)}</math>

by the [[equivalence relation]] that identifies a point {{math|''y'' ∈ ''F''(''Y'')}} with its image under the map {{math|''F''(''Y'') → ''F''(''X'')}}, for every inclusion {{math|''Y'' ⊆ ''X''}}.