Editing Abstract simplicial complex (section)

==Geometric realization==
We can associate to any abstract simplicial complex (ASC) ''K'' a [[topological space]] <math>|K|</math>, called its '''geometric realization'''. There are several ways to define <math>|K|</math>.

=== Geometric definition ===
Every [[geometric simplicial complex]] (GSC) determines an ASC:''<ref name=":0">{{Cite Matousek 2007}}, Section 4.3</ref>''{{Rp|page=14|location=}} the vertices of the ASC are the vertices of the GSC, and the faces of the ASC are the vertex-sets of the faces of the GSC. For example, consider a GSC with 4 vertices {1,2,3,4}, where the maximal faces are the triangle between {1,2,3} and the lines between {2,4} and {3,4}. Then, the corresponding ASC contains the sets {1,2,3}, {2,4}, {3,4}, and all their subsets. We say that the GSC is the '''geometric realization''' of the ASC.  

Every ASC has a geometric realization. This is easy to see for a finite ASC.''<ref name=":0" />''{{Rp|page=14|location=}} Let <math>N := |V(K)|</math>. Identify the vertices in <math>V(K)</math> with the vertices of an (''N-1'')-dimensional simplex in <math>\R^N</math>. Construct the GSC  {[[convex hull|conv]](F): F is a face in K}. Clearly, the ASC associated with this GSC is identical to ''K'', so we have indeed constructed a geometric realization of ''K.'' In fact, an ASC can be realized using much fewer dimensions. If an ASC is ''d''-dimensional (that is, the maximum cardinality of a simplex in it is ''d''+1), then it has a geometric realization in <math>\R^{2d+1}</math>, but might not have a geometric realization in <math>\R^{2d}</math> ''<ref name=":0" />{{Rp|page=16|location=}}''  The special case ''d''=1 corresponds to the well-known fact, that any [[Graph (discrete mathematics)|graph]] can be plotted in <math>\R^{3}</math> where the edges are straight lines that do not intersect each other except in common vertices, but not any [[Graph (discrete mathematics)|graph]] can be plotted in <math>\R^{2}</math> in this way. 

If ''K'' is the standard combinatorial ''n''-simplex, then <math>|K|</math>  can be naturally identified with {{math|Δ<sup>''n''</sup>}}.

Every two geometric realizations of the same ASC, even in Euclidean spaces of different dimensions, are [[Homeomorphism|homeomorphic]].''<ref name=":0" />''{{Rp|page=14|location=}} Therefore, given an ASC ''K,'' one can speak of ''the'' geometric realization of ''K''.

=== Topological definition ===
The construction goes as follows. First, define <math>|K|</math> as a subset of <math>[0, 1]^S</math> consisting of functions <math>t\colon S\to [0, 1]</math> satisfying the two conditions:
:<math>\{s\in S:t_s>0\}\in K</math>
:<math>\sum_{s\in S}t_s=1</math>
Now think of the set of elements of <math>[0, 1]^S</math>  with finite support as the [[direct limit]] of <math>[0, 1]^A</math> where ''A'' ranges over finite subsets of ''S'', and give that direct limit the [[final topology|induced topology]].  Now give <math>|K|</math> the [[subspace topology]].

=== 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''}}.