Simplicial Structures in Topology

46 II Simplicial Complexes
II.2 Abstract Simplicial Complexes
In this section, we shall give the definition of the category Csim of simplicial
complexes and simplicial maps; furthermore, we shall define two important functors
with domain Csim, namely, the geometric realization functor and the homology
functor.
An (abstract) simplicial complex is a pair K =(X,Φ) given by a finite set X and
a set of nonempty subsets of X such that:
K1 (∀x ∈ X) , {x}∈Φ,
K2 (∀σ ∈ Φ)(∀σ ′ ⊂ σ , σ ′ �= /0) , σ ′ ∈ Φ.
The elements of X are the vertices of K. The elements of Φ are the simplexes of K.
If σ is a simplex of K, every non-empty σ ′ ⊂ σ is a face of σ. According to
condition K2, we can say that all faces of a simplex are simplexes. A simplex σ
with n + 1elements(n ≥ 0) is an n-simplex (we also say that σ is a simplex of
dimension n); we adopt the notation dimσ = n. It follows that the 0-simplexes are
vertices. The dimension of K is the maximal dimension of its simplexes; if the
dimensions of all simplexes of K have a maximum n, we say that K has dimension
n or that K is n-dimensional.
(II.2.1) Remark. We explicitly observe that in this book all simplicial complexes
have a finite number of vertices.
Before we present some examples and constructions with simplicial complexes,
we give a definition: a simplicial complex L =(Y,Ψ) is a subcomplex of K =(X,Φ)
if Y ⊂ X and Ψ ⊂ Φ.
(II.2.2) Remark. Let K0 =(X0,Φ0) and K1 =(X1,Φ1) be subcomplexes of a simplicial
complex K; we observe that the union K0 ∪ K1 =(X0 ∪ X1,Φ0 ∪ Φ1) and the
intersection K0 ∩K1 =(X0 ∩X1,Φ0 ∩Φ1) (with X0 ∩X1 �= /0) are subcomplexes of K.
In particular, the union of two disjoint simplicial complexes K0 and K1 (that is to
say, such that X0 ∩ X1 = /0) is a simplicial complex.
Let us now give some examples.
1. Let X be a finite set and let ℘(X)=2 X be the set of all subsets of X; clearly,
the pair K =(X,℘(X) � /0) is a simplicial complex.
2. The set of all simplexes of an Euclidean simplicial complex is an abstract simplicial
complex if we forget the fact that its vertices are points of R n . The
set X is the set of all vertices, while Φ is the set of simplexes. Thus, the examples
of Euclidean polyhedra on p. 45 are examples of abstract simplicial
complexes.
3. Let Γ be a graph (that is to say, a set of vertices X and a symmetric subset Φ
of X × X, called set of edges). It is not hard to prove that (X,Φ) is a simplicial
complex if we assume that σ ∈ Φ ⊂ 2 X whenever σ is a set with just one
element or is the set of the two vertices at the ends of an edge.