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Note that the link of the empty set is {{math|Δ}} itself.
=== Simplicial maps ===
Given two abstract simplicial complexes, {{math|Δ}} and {{math|Γ}}, a '''simplicial map''' is a [[Function (mathematics)|function]] {{math| ''f'' }} 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| ''f'' (''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]].▼
{{Main|Simplicial map}}
▲Given two abstract simplicial complexes, {{math|Δ}} and {{math|Γ}}, a '''[[simplicial map]]''' is a [[Function (mathematics)|function]] {{math| ''f'' }} 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| ''f'' (''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:
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