nLab
simplicial complex

Definition:

A simplicial complex K is a set of objects, V(K), called vertices and a set, S(K), of finite non-empty subsets of V(K), called simplices. The simplices satisfy the condition that if σV(K) is a simplex and τσ, τ, then τ is also a simplex.

We say τ is a face of σ. If σS(K) has p+1 elements it is said to be a p-simplex. The set of p-simplices of K is denoted by K p. The dimension of K is the largest p such that K p is non-empty.

Simplicial complexes v. simplicial sets

  • Simplicial complexes are, in some sense, special cases of simplicial sets, but only ‘in some sense’. To get from a simplicial complex to a simplicial set, you must pick a total order on the set of vertices. Without an order on the vertices you cannot speak of the k th face of a simplex, which is an essential feature of a simplicial set! The degeneracies are obtained by repeating an element when listing the vertices of a simplex. If σ={v 0,v 1,,v n}, with v 0<v 1<<v n then, for instance, s 0(σ)={v 0,v 0,v 1,,v n}.

Geometric realisations and Polyhedra

An abstract simplicial complex is a combinatorial gadget that models certain aspects of a spatial configuration. Sometimes it is useful, perhaps even necessary, to produce a topological space from that data in a simplicial complex.

Idea

To each simplicial complex K, one can associate a topological space called the polyhedron of K often also called or geometric realisation of K and denoted K.

This can be constructed by taking a copy K(σ) of a standard topological p-simplex for each p-simplex of K and then ‘gluing’ them together according to the face relations encoded in K.

Definition

The standard (topological) p-simplex is usually taken to be the convex hull of the basis vectors e 1,e 2,,e p+1 in p+1, to represent each abstract p-simplex, σS(K), and then ‘gluing’ faces together, so whenever τ is a face of σ we identify K(τ) with the corresponding face of K(σ). This space is usually denoted Δ p.

Canonical construction

There is a canonical way of constructing K as follows:

K is the set of all functions from V(K) to the closed interval [0,1] such that

  • if αK, the set
{vV(K)α(v)0}\{v \in V(K) \mid \alpha(v) \neq 0\}

is a simplex of K;

  • for each vV(K),

    αV(K)α(v)=1.\sum_{\alpha \in V(K)} \alpha (v) = 1.

We can put a metric d on K by

d(α,β)=( vV(K)(p v(α)p v(β)) 2) 12.d(\alpha,\beta) = \Big(\sum_{v\in V(K)} (p_v(\alpha) - p_v(\beta))^2\Big)^\frac{1}{2}.

This however gives K as a subspace of #(V(K)), and so is usually of much higher dimension then might seem geometrically significant in a given context.