Geometric realisation of simplicial map

Question

Typically texts give a good definition of the geometric realisation $|\Delta|$ of a simplicial complex $\Delta$. I'm supposing that the geometric realisation forms a functor $|-|:\text{sComp} \to \text{Top}$ from the category of simplicial sets and simplicial maps to topological spaces and continuous functions.

I was wondering where I could find a concrete description of the construction of the geometric realisation $|f|:|\Delta| \to |\Delta'|$ of a simplicial map $f:\Delta \to \Delta'$?

Answer 1

One way to define the realization of a simplicial complex $\Delta$ with vertex set $V$ is:

The space $|\Delta|=X$ is the set of all sums $\sum_V a_v v$ where

• $a_v\ge0$ for all $v\in V$,
• the set $\{v\in V \mid a_v>0\}$ is a simplex in $\Delta$,
• $\sum_V a_v = 1$,

in which a subset $A$ is closed if and only if $A$ intersects each closed simplex $|\sigma| = \{ \sum_v a_v v \mid a_w>0 \implies w\in \sigma \}$ in a closed set (where we take on $|\sigma|$ the topology of the standard-$\text{dim}(\sigma)$-simplex).

Given a simplicial map $f:\Delta\to\Delta'$, we define $|f|:|\Delta|\to|\Delta'|$ as the map $$ \sum a_v v \mapsto \sum a_v f(v) $$