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 $\sigma \subset V(K)$ is a simplex and $\tau \subset \sigma$, $\tau \ne \varnothing$, then $\tau$ is also a simplex.

We say $\tau$ is a face of $\sigma$. If $\sigma \in S(K)$ 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 $k^{\mathrm{th}}$ 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 $\sigma =\{v_0,v_1,\dots ,v_n\}$, with $v_0<v_1<\dots <v_n$ then, for instance, $s_0(\sigma )=\{v_0,v_0,v_1,\dots ,v_n\}$.

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 $\mid K\mid$.

This can be constructed by taking a copy $K(\sigma )$ 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 $e_1,e_2,\dots ,e_{p+1}$ in ${ℝ}^{p+1}$, to represent each abstract $p$-simplex, $\sigma \in S(K)$, and then ‘gluing’ faces together, so whenever $\tau$ is a face of $\sigma$ we identify $K(\tau )$ with the corresponding face of $K(\sigma )$. This space is usually denoted ${\Delta }^p$.

Canonical construction

There is a canonical way of constructing $\mid K\mid$ as follows:

$\mid K\mid$ is the set of all functions from $V(K)$ to the closed interval $[\mathrm{0,1}]$ such that

• if $\alpha \in \mid K\mid$, the set

$$\{v\in V(K)\mid \alpha (v)\ne 0\}$$\{v \in V(K) \mid \alpha(v) \neq 0\}

is a simplex of $K$;

• for each $v\in V(K)$,

  $$\sum _{\alpha \in V(K)}\alpha (v)=1.$$\sum_{\alpha \in V(K)} \alpha (v) = 1.

We can put a metric $d$ on $\mid K\mid$ by

$$d(\alpha ,\beta )=(\sum _{v\in V(K)}(p_v(\alpha )-p_v(\beta ){)}^2{)}^{\frac{1}{2}}.$$d(\alpha,\beta) = \Big(\sum_{v\in V(K)} (p_v(\alpha) - p_v(\beta))^2\Big)^\frac{1}{2}.

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