Remark 5.5.1.9 (01Z4): Comparison with Topological Spaces

Remark 5.5.1.9 (Comparison with Topological Spaces). Let $\operatorname{Top}$ denote the category of topological spaces and continuous functions, endowed with the simplicial enrichment described in Example 2.4.1.5. The geometric realization construction $X \mapsto |X|$ determines a functor of simplicial categories $| \bullet |: \operatorname{Kan}\rightarrow \operatorname{Top}$ (see Construction 3.6.5.1). Moreover, if $X$ and $Y$ are Kan complexes, then Proposition 3.6.5.2 guarantees that the induced map

\[ \operatorname{Fun}(X,Y) = \operatorname{Hom}_{\operatorname{Kan}}(X,Y)_{\bullet } \rightarrow \operatorname{Hom}_{\operatorname{Top}}( |X|, |Y| )_{\bullet } \]

is a homotopy equivalence of Kan complexes. Applying Corollary 4.6.8.8, we deduce that the induced map

\[ \operatorname{\mathcal{S}}= \operatorname{N}_{\bullet }^{\operatorname{hc}}( \operatorname{Kan}) \xrightarrow { | \bullet | } \operatorname{N}_{\bullet }^{\operatorname{hc}}( \operatorname{Top}) \]

is a fully faithful functor of $\infty $-categories. The essential image of this functor is the full subcategory $\operatorname{\mathcal{T}}_0 \subseteq \operatorname{N}_{\bullet }^{\operatorname{hc}}( \operatorname{Top})$ spanned by those topological spaces which have the homotopy type of a CW complex (Proposition 3.6.5.3). We therefore obtain an equivalence of $\infty $-category $\operatorname{\mathcal{S}}\xrightarrow { | \bullet |} \operatorname{\mathcal{T}}_{0}$ (Theorem 4.6.2.21), which has a homotopy inverse induced by the simplicial functor $X \mapsto \operatorname{Sing}_{\bullet }(X)$.