Editing Abstract simplicial complex (section)

==Definitions==
A collection {{math|Δ}} of non-empty finite subsets of a [[Set (mathematics)|set]] ''S'' is called a set-family.

A set-family {{math|Δ}} is called an '''abstract simplicial complex''' if, for every set {{mvar|X}} in {{math|Δ}}, and every non-empty subset {{math|''Y'' ⊆ ''X''}}, the set {{mvar|Y}} also belongs to {{math|Δ}}.

The finite sets that belong to {{math|Δ}} are called '''faces''' of the complex, and a face {{mvar|Y}} is said to belong to another face {{mvar|X}} if {{math|''Y'' ⊆ ''X''}}, so the definition of an abstract simplicial complex can be restated as saying that every face of a face of a complex {{math|Δ}} is itself a face of {{math|Δ}}. The '''vertex set''' of {{math|Δ}} is defined as {{math|''V''(Δ) {{=}} ∪Δ}}, the union of all faces of {{math|Δ}}. The elements of the vertex set are called the '''vertices''' of the complex. For every vertex ''v'' of {{math|Δ}}, the set {''v''} is a face of the complex, and every face of the complex is a finite subset of the vertex set.

The maximal faces of {{math|Δ}} (i.e., faces that are not subsets of any other faces) are called '''facets''' of the complex. The '''dimension of a face''' {{mvar|X}} in {{math|Δ}} is defined as {{math|dim(''X'') {{=}} {{!}}''X''{{!}} − 1}}: faces consisting of a single element are zero-dimensional, faces consisting of two elements are one-dimensional, etc. The '''dimension of the complex''' {{math|dim(Δ)}} is defined as the largest dimension of any of its faces, or infinity if there is no finite bound on the dimension of the faces.

The complex {{math|Δ}} is said to be '''finite''' if it has finitely many faces, or equivalently if its vertex set is finite. Also, {{math|Δ}} is said to be '''pure''' if it is finite-dimensional (but not necessarily finite) and every facet has the same dimension. In other words, {{math|Δ}} is pure if {{math|dim(Δ)}} is finite and every face is contained in a facet of dimension {{math|dim(Δ)}}.

One-dimensional abstract simplicial complexes are mathematically equivalent to [[simple graph|simple]] [[undirected graph]]s: the vertex set of the complex can be viewed as the vertex set of a graph, and the two-element facets of the complex correspond to undirected edges of a graph. In this view, one-element facets of a complex correspond to isolated vertices that do not have any incident edges.

A '''subcomplex''' of {{math|Δ}} is an abstract simplicial complex ''L'' such that every face of ''L'' belongs to {{math|Δ}}; that is, {{math|''L'' ⊆ Δ}} and ''L'' is an abstract simplicial complex. A subcomplex that consists of all of the subsets of a single face of {{math|Δ}} is often called a '''simplex''' of {{math|Δ}}. (However, some authors use the term "simplex" for a face or, rather ambiguously, for both a face and the subcomplex associated with a face, by analogy with the non-abstract (geometric) [[simplicial complex]] terminology. To avoid ambiguity, we do not use in this article the term "simplex" for a face in the context of abstract complexes).

The '''[[d-skeleton|''d''-skeleton]]''' of {{math|Δ}} is the subcomplex of {{math|Δ}} consisting of all of the faces of {{math|Δ}} that have dimension at most ''d''. In particular, the [[skeleton (topology)|1-skeleton]] is called the '''underlying graph''' of {{math|Δ}}. The 0-skeleton of {{math|Δ}} can be identified with its vertex set, although formally it is not quite the same thing (the vertex set is a single set of all of the vertices, while the 0-skeleton is a family of single-element sets).

The '''link''' of a face {{mvar|Y}} in {{math|Δ}}, often denoted {{math|Δ/''Y''}} or {{math|lk<sub>Δ</sub>(''Y'')}}, is the subcomplex of {{math|Δ}} defined by

:<math> \Delta/Y := \{ X\in \Delta \mid X\cap Y = \varnothing,\, X\cup Y \in \Delta \}. </math>

Note that the link of the empty set is {{math|Δ}} itself.

=== 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}}.