---
tag: XHHTS6TDYZ
kind: definition
name: Geometric realization
msc: [55U10, 18N50]
defines: TG6WHHR7CF
uses: [RWBMXQW6S5, A486VR787Z, 0V8046R0P9, 5W95QFHXPR]
sources:
  - url: https://en.wikipedia.org/wiki/Simplicial_set
---

## Definition

Let $K$ be a simplicial set, with $n$-simplices $K_n$ and structure maps $\theta^* \colon K_n \to K_m$ for order-preserving $\theta \colon [m] \to [n]$.
Let $\abs{\Delta^n}$ be the standard topological $n$-simplex, and let $\theta_* \colon \abs{\Delta^m} \to \abs{\Delta^n}$ be the affine map sending the vertex $e_i$ to $e_{\theta(i)}$.
Give each set $K_n$ the discrete topology.
The *geometric realization* of $K$ is the [[quotient space|0V8046R0P9]]
$$\abs{K} = \Bigl( \bigsqcup_{n \ge 0} K_n \times \abs{\Delta^n} \Bigr) \Big/ \sim$$
of the [[disjoint union|A486VR787Z]], where $\sim$ is the equivalence relation generated by
$$(\theta^* x, t) \sim (x, \theta_* t) \quad \text{for all } \theta \colon [m] \to [n], \ x \in K_n, \ t \in \abs{\Delta^m}.$$

## 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|RWBMXQW6S5]] 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|5W95QFHXPR]] 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.