Example 2.4.2.16 (00KJ): Topologically Enriched Categories

Example 2.4.2.16 (Topologically Enriched Categories). Let $\operatorname{Top}$ denote the category of topological spaces. The formation of singular simplicial sets (Construction 1.2.2.2) determines a functor

\[ \operatorname{Sing}_{\bullet }: \operatorname{Top}\rightarrow \operatorname{Set_{\Delta }}\quad \quad X \mapsto \operatorname{Sing}_{\bullet }(X) \]

which preserves finite products (in fact, it preserves all small limits), and can therefore be regarded as a monoidal functor from $\operatorname{Top}$ to $\operatorname{Set_{\Delta }}$ (where we endow both $\operatorname{Top}$ and $\operatorname{Set_{\Delta }}$ with the cartesian monoidal structure). Applying Remark 2.1.7.4, we see that every topologically enriched category $\operatorname{\mathcal{C}}$ can be regarded as a simplicial category $\operatorname{\mathcal{C}}_{\bullet }$ having the same objects, with morphism simplicial sets given by

\[ \operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y)_{\bullet } = \operatorname{Sing}_{\bullet }( \underline{\operatorname{Hom}}_{\operatorname{\mathcal{C}}}(X,Y) ); \]

here $\underline{\operatorname{Hom}}_{\operatorname{\mathcal{C}}}(X,Y)$ denotes the set of morphisms from $X$ to $Y$, endowed with the topology determined by the topological enrichment of $\operatorname{\mathcal{C}}$ (see Example 2.1.7.8).