Abstract Simplicial Complex - Definitions

Definitions

A nonempty family Δ of finite subsets of a universal set S is an abstract simplicial complex if, for every set X in Δ, and every subset Y ⊂ X, Y also belongs to Δ. Equivalently, it is an abstract simplicial complex if and only if there do not exist two sets Y ⊂ X such that X belongs to Δ but Y does not.

Note that the empty set belongs to every non-empty abstract simplicial complex, because it is a subset of every other set X in the complex. 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. So for every vertex v of Δ, the set {v} is a face of the complex. 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 a simplicial complex L such that every face of L belongs to Δ; that is, L ⊂ Δ and L is a 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

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 ƒ 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 set ƒ(X) is a face of Γ.