Editing Abstract simplicial complex (section)

=== Simplicial maps ===
{{Main|Simplicial map}}
Given two abstract simplicial complexes, {{math|Δ}} and {{math|Γ}}, a '''[[simplicial map]]''' is a [[Function (mathematics)|function]] {{math|&nbsp;''f''&thinsp;}} that maps the vertices of {{math|Δ}} to the vertices of {{math|Γ}} and that has the property that for any face {{mvar|X}} of {{math|Δ}}, the [[Image (mathematics)|image]] {{math|&nbsp;''f''&thinsp;(''X'')}} is a face of {{math|Γ}}.  There is a [[Category (mathematics)|category]] '''SCpx''' with abstract simplicial complexes as objects and simplicial maps as [[morphism]]s.  This is equivalent to a suitable category defined using non-abstract [[simplicial complexes]].

Moreover, the categorical point of view allows us to tighten the relation between the underlying set ''S'' of an abstract simplicial complex {{math|Δ}} and the vertex set {{math|''V''(Δ) ⊆ ''S''}} of {{math|Δ}}: for the purposes of defining a category of abstract simplicial complexes, the elements of ''S'' not lying in {{math|''V''(Δ)}} are irrelevant.  More precisely, '''SCpx''' is equivalent to the category where:
* an object is a set ''S'' equipped with a collection of non-empty finite subsets {{math|Δ}} that contains all singletons and such that if {{mvar|X}} is in {{math|Δ}} and {{math|''Y'' ⊆ ''X''}} is non-empty, then {{mvar|Y}} also belongs to {{math|Δ}}.
* a morphism from {{math|(''S'', Δ)}} to {{math|(''T'', Γ)}} is a function {{math|''f'' : ''S'' → ''T''}} such that the image of any element of {{math|Δ}} is an element of {{math|Γ}}.