Kerodon

Example 8.6.9.8 (Geometric Realization). Let $\operatorname{Top}$ denote the category of topological spaces. The construction $[n] \mapsto | \Delta ^ n |$ determines a functor from the simplex category $\operatorname{{\bf \Delta }}$ to $\operatorname{Top}$, which we denote by $| \Delta ^{\bullet } |$. Let $S_{\bullet }$ be a simplicial set, which we regard as a functor $\operatorname{{\bf \Delta }}^{\operatorname{op}} \rightarrow \operatorname{Set}\subset \operatorname{Top}$. Then the geometric realization $| S_{\bullet } |$ of Notation 1.2.3.3 can be identified with the coend $\operatorname{Coend}(| \Delta ^{\bullet } |, S_{\bullet } )$.