Abstract simplicial complex

A collection Δ of non-empty finite subsets of a set S is called a set-family.

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

The finite sets that belong to Δ are called faces of the complex, and a face Y is said to belong to another face X if Y ⊆ X, so the definition of an abstract simplicial complex can be restated as saying that every face of a face of a complex Δ is itself a face of Δ. The vertex set of Δ is defined as V(Δ) = ∪Δ, the union of all faces of Δ. The elements of the vertex set are called the vertices of the complex. For every vertex v of 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 Δ (i.e., faces that are not subsets of any other faces) are called facets of the complex. The dimension of a face X in Δ is defined as 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 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 Δ is said to be finite if it has finitely many faces, or equivalently if its vertex set is finite. Also, 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, Δ is pure if dim(Δ) is finite and every face is contained in a facet of dimension dim(Δ).

One-dimensional abstract simplicial complexes are mathematically equivalent to simple undirected graphs: 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 Δ is an abstract simplicial complex L such that every face of L belongs to Δ; that is, L ⊆ Δ and L is an abstract simplicial complex. A subcomplex that consists of all of the subsets of a single face of Δ is often called a simplex of 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 of Δ is the subcomplex of Δ consisting of all of the faces of Δ that have dimension at most d. In particular, the 1-skeleton is called the underlying graph of Δ. The 0-skeleton of 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 Y in Δ, often denoted Δ/Y or lk_Δ(Y), is the subcomplex of Δ defined by

${\displaystyle \Delta /Y:=\{X\in \Delta \mid X\cap Y=\varnothing ,\,X\cup Y\in \Delta \}.}$

Note that the link of the empty set is Δ itself.

Given two abstract simplicial complexes, Δ and Γ, a simplicial map is a function  f  that maps the vertices of Δ to the vertices of Γ and that has the property that for any face X of Δ, the image  f (X) is a face of Γ. There is a category SCpx with abstract simplicial complexes as objects and simplicial maps as morphisms. 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 Δ and the vertex set V(Δ) ⊆ S of Δ: for the purposes of defining a category of abstract simplicial complexes, the elements of S not lying in 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 Δ that contains all singletons and such that if X is in Δ and Y ⊆ X is non-empty, then Y also belongs to Δ.
• a morphism from (S, Δ) to (T, Γ) is a function f : S → T such that the image of any element of Δ is an element of Γ.

We can associate to an abstract simplicial complex K a topological space ${\displaystyle |K|}$, called its geometric realization, which is the carrier of a simplicial complex. The construction goes as follows.

First, define ${\displaystyle |K|}$ as a subset of ${\displaystyle [0,1]^{S}}$ consisting of functions ${\displaystyle t\colon S\to [0,1]}$ satisfying the two conditions:

${\displaystyle \{s\in S:t_{s}>0\}\in K}$

${\displaystyle \sum _{s\in S}t_{s}=1}$

Now think of the set of elements of ${\displaystyle [0,1]^{S}}$ with finite support as the direct limit of ${\displaystyle [0,1]^{A}}$ where A ranges over finite subsets of S, and give that direct limit the induced topology. Now give ${\displaystyle |K|}$ the subspace topology.

Alternatively, let ${\displaystyle {\mathcal {K}}}$ denote the category whose objects are the faces of K and whose morphisms are inclusions. Next choose a total order on the vertex set of K and define a functor F from ${\displaystyle {\mathcal {K}}}$ to the category of topological spaces as follows. For any face X in K of dimension n, let F(X) = Δⁿ be the standard n-simplex. The order on the vertex set then specifies a unique bijection between the elements of X and vertices of Δⁿ, ordered in the usual way e₀ < e₁ < ... < e_n. If Y ⊆ X is a face of dimension m < n, then this bijection specifies a unique m-dimensional face of Δⁿ. Define F(Y) → F(X) to be the unique affine linear embedding of Δ^m as that distinguished face of Δⁿ, such that the map on vertices is order-preserving.

We can then define the geometric realization ${\displaystyle |K|}$ as the colimit of the functor F. More specifically ${\displaystyle |K|}$ is the quotient space of the disjoint union

${\displaystyle \coprod _{X\in K}{F(X)}}$

by the equivalence relation that identifies a point y ∈ F(Y) with its image under the map F(Y) → F(X), for every inclusion Y ⊆ X.

If K is finite, then we can describe ${\displaystyle |K|}$ more simply. Choose an embedding of the vertex set of K as an affinely independent subset of some Euclidean space ${\displaystyle \mathbb {R} ^{N}}$ of sufficiently high dimension N. Then any face X in K can be identified with the geometric simplex in ${\displaystyle \mathbb {R} ^{N}}$ spanned by the corresponding embedded vertices. Take ${\displaystyle |K|}$ to be the union of all such simplices.

If K is the standard combinatorial n-simplex, then ${\displaystyle |K|}$ can be naturally identified with Δⁿ.

1. Let V be a finite set of cardinality n + 1. The combinatorial n-simplex with vertex-set V is an ASC whose faces are all subsets of V (i.e., it is the power set of V). If 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 cliques (complete subgraphs) of G. The independence complex of G is an ASC whose faces are all independent sets of G (it is the clique complex of the complement graph of G). Clique complexes are the prototypical example of flag complexes. A flag complex is a complex K with the property that every set of elements that pairwise belong to faces of K is itself a face of K.

3. Let H be a hypergraph. A matching in H is a set of edges of H, in which every two edges are disjoint. The matching complex of H is an ASC whose faces are all matchings in H. It is the independence complex of the 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 chains in P. Its homology groups and other 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 groups, image processing, and mobile ad hoc networking. It is another example of a flag complex.

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

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.

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 nth Dedekind number. These numbers grow very rapidly, and are known only for n ≤ 8; the Dedekind numbers are (starting with n = 0):

1, 2, 5, 19, 167, 7580, 7828353, 2414682040997, 56130437228687557907787 (sequence A014466 in the OEIS). This corresponds to the number of non-empty antichains of subsets of an 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" (sequence A006126 in the OEIS), 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" (sequence A006602 in the OEIS), starting at n = 1.

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.

• Kruskal–Katona theorem
• Simplicial set