Editing Abstract simplicial complex

{{Short description|Mathematical object}}
[[Image:Simplicial complex example.svg|thumb|200px|Geometric realization of a 3-dimensional abstract simplicial complex]]

In [[combinatorics]], an '''abstract simplicial complex''' (ASC), often called an '''abstract complex''' or just a '''complex''', is a [[family of sets]] that is closed under taking [[subset]]s, i.e., every subset of a set in the family is also in the family. It is a purely combinatorial description of the geometric notion of a [[simplicial complex]].<ref name=Lee>[[John M. Lee|Lee, John M.]], Introduction to Topological Manifolds, Springer 2011, {{ISBN|1-4419-7939-5}}, p153</ref> For example, in a 2-dimensional simplicial complex, the sets in the family are the triangles (sets of size 3), their edges (sets of size 2), and their vertices (sets of size 1).

In the context of [[matroid]]s and [[greedoid]]s, abstract simplicial complexes are also called '''[[independence system]]s'''.<ref>{{cite book|author = Korte, Bernhard|author-link = Bernhard Korte|author2=Lovász, László|author2-link=László Lovász|author3=Schrader, Rainer| year = 1991| title = Greedoids | publisher = Springer-Verlag | isbn = 3-540-18190-3 |page = 9}}</ref>

An abstract simplex can be studied algebraically by forming its [[Stanley–Reisner ring]]; this sets up a powerful relation between [[combinatorics]] and [[commutative algebra]].

==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}}.

==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''}}.

==Examples==
1. Let ''V'' be a finite set of [[cardinality]] {{math|''n'' + 1}}. The '''combinatorial ''n''-simplex''' with vertex-set ''V'' is an ASC whose faces are all nonempty subsets of ''V'' (i.e., it is the [[power set]] of ''V''). If {{math|''V'' {{=}} ''S'' {{=}} {0, 1, ..., ''n''},}}  then this ASC is called the '''standard combinatorial ''n''-simplex'''.

2. Let ''G'' be an undirected graph. The '''[[clique complex]]''' '''of ''G''''' is an ASC whose faces are all [[Clique (graph theory)|cliques]] (complete subgraphs) of ''G''. The '''independence complex of ''G''''' is an ASC whose faces are all [[Independent set (graph theory)|independent sets]] of ''G'' (it is the clique complex of the [[complement graph]] of G).  Clique complexes are the prototypical example of [[flag complex]]es. A '''flag complex''' is a complex ''K''  with the property that every set, all of whose 2-element subsets are faces of ''K'', is itself a face of ''K''.

3. Let ''H'' be a [[hypergraph]]. A [[Matching in hypergraphs|matching]] in ''H'' is a set of edges of ''H'', in which every two edges are [[Disjoint sets|disjoint]]. The '''matching complex of ''H''''' is an ASC whose faces are all [[Matching in hypergraphs|matchings]] in ''H''. It is the [[independence complex]] of the [[Line graph of a hypergraph|line graph]] of ''H''.

4. Let ''P'' be a [[partially ordered set]] (poset). The '''order complex''' of ''P'' is an ASC whose faces are all finite [[Total order#Chains|chains]] in ''P''. Its [[Homology (mathematics)|homology]] groups and other [[Topological property|topological invariants]] contain important information about the poset ''P''.

5. Let ''M'' be a [[metric space]] and ''δ'' a real number. The '''[[Vietoris–Rips complex]]''' is an ASC whose faces are the finite subsets of ''M'' with diameter at most ''δ''. It has applications in [[homology theory]], [[hyperbolic group]]s, [[image processing]], and [[mobile ad hoc network]]ing. It is another example of a flag complex.

6. Let <math>I</math> be a square-free [[monomial ideal]] in a [[polynomial ring]] <math>S = K[x_1, \dots, x_n]</math> (that is, an ideal generated by products of subsets of variables). Then the exponent vectors of those square-free monomials of <math>S</math> that are not in <math>I</math> determine an abstract simplicial complex via the map <math>\mathbf{a}\in \{0,1\}^n \mapsto \{i \in [n] : a_i = 1\}</math>. In fact, there is a bijection between (non-empty) abstract simplicial complexes on {{math| ''n''}} vertices and square-free monomial ideals in {{math| ''S''}}. If <math>I_{\Delta}</math> is the square-free ideal corresponding to the simplicial complex <math>\Delta</math> then the [[Quotient ring|quotient]] <math>S/I_{\Delta}</math> is known as the [[Stanley–Reisner ring]] of <math>{\Delta}</math>.

7. For any [[open covering]] ''C'' of a topological space, the '''[[nerve complex]]''' of ''C'' is an abstract simplicial complex containing the sub-families of ''C'' with a non-empty [[intersection]].

==Enumeration==
The number of abstract simplicial complexes on up to ''n'' labeled elements (that is on a set ''S'' of size ''n'') is one less than the ''n''th [[Dedekind number]]. These numbers grow very rapidly, and are known only for {{math|''n'' ≤ 9}}; the Dedekind numbers are (starting with ''n'' = 0):
:1, 2, 5, 19, 167, 7580, 7828353, 2414682040997, 56130437228687557907787, 286386577668298411128469151667598498812365 {{OEIS|id=A014466}}. This corresponds to the number of non-empty [[antichain]]s of subsets of an {{math| ''n''}} set.

The number of abstract simplicial complexes whose vertices are exactly ''n'' labeled elements is given by the sequence "1, 2, 9, 114, 6894, 7785062, 2414627396434, 56130437209370320359966, 286386577668298410623295216696338374471993"  {{OEIS|id=A006126}}, starting at ''n'' = 1. This corresponds to the number of antichain covers of a labeled ''n''-set; there is a clear bijection between antichain covers of an ''n''-set and simplicial complexes on ''n'' elements described in terms of their maximal faces.

The number of abstract simplicial complexes on exactly ''n'' unlabeled elements is given by the sequence "1, 2, 5, 20, 180, 16143, 489996795, 1392195548399980210" {{OEIS|id=A006602}}, starting at ''n'' = 1.

== Computational problems ==
{{Main|Simplicial complex recognition problem}}
The [[simplicial complex recognition problem]] is: given a finite ASC, decide whether its geometric realization is homeomorphic to a given geometric object. This problem is [[Undecidable problem|undecidable]] for any ''d''-dimensional manifolds for ''d'' ≥ 5.<ref>{{citation |last=Stillwell |first=John |title=Classical Topology and Combinatorial Group Theory |url=https://books.google.com/books?id=265lbM42REMC&pg=PA247 |volume=72 |page=247 |year=1993 |series=Graduate Texts in Mathematics |publisher=Springer |isbn=9780387979700 |authorlink=John Stillwell}}.</ref>

== Relation to other concepts ==
An abstract simplicial complex with an additional property called the '''augmentation property''' or the '''exchange property''' yields a '''[[matroid]]'''. The following expression shows the relations between the terms:

HYPERGRAPHS = SET-FAMILIES    ⊃    INDEPENDENCE-SYSTEMS = ABSTRACT-SIMPLICIAL-COMPLEXES    ⊃    MATROIDS.

==See also==
* [[Kruskal–Katona theorem]]
* [[Simplicial set]]

==References==
{{Reflist}}

{{DEFAULTSORT:Abstract Simplicial Complex}}
[[Category:Algebraic topology]]
[[Category:Families of sets]]
[[Category:Simplicial sets]]