Geometric realization

Remarks

A simplicial map $f \colon K \to L$ induces a continuous map $\abs{f} \colon \abs{K} \to \abs{L}$, so realization is a functor. It is left adjoint to the singular simplicial set functor: continuous maps $\abs{K} \to X$ correspond to simplicial maps $K \to \operatorname{Sing} X$. Each point of $\abs{K}$ has a unique representative $(x, t)$ with $x$ nondegenerate and $t$ in the interior of $\abs{\Delta^n}$. It follows that $\abs{K}$ is a CW complex with one $n$-cell for each nondegenerate $n$-simplex.

Example: for the standard simplex $\Delta^n$, the realization $\abs{\Delta^n}$ is the topological $n$-simplex, since every simplex of $\Delta^n$ is $\theta^*$ of the top simplex $\id_{[n]}$.

Example: the quotient $\Delta^1 / \partial \Delta^1$ has one nondegenerate $0$-simplex and one nondegenerate $1$-simplex whose two faces are equal, so its realization is a circle.

Non-example: the disjoint union $\bigsqcup_n K_n \times \abs{\Delta^n}$ without the relation $\sim$ is not the realization; for $K = \Delta^1$ it has infinitely many components, while $\abs{\Delta^1}$ is an interval.

Some authors use the category of compactly generated spaces, where the realization functor preserves finite products.