$\Newextarrow{\xhookrightarrow}{10,10}{0x21AA}$
Corollary 3.6.5.6. Let $\operatorname{\mathcal{C}}$ be a category, and let $\operatorname{\mathcal{E}}' \subseteq \operatorname{Fun}( \operatorname{Top}, \operatorname{\mathcal{C}})$ be the full subcategory spanned by those functors $F: \operatorname{Top}\rightarrow \operatorname{\mathcal{C}}$. which carry weak homotopy equivalences of topological spaces to isomorphisms in the category $\operatorname{\mathcal{C}}$. Then:
- $(a)$
For every functor $F \in \operatorname{\mathcal{E}}'$, the composite functor
\[ \operatorname{Kan}\xrightarrow { | \bullet | } \operatorname{Top}\xrightarrow {F} \operatorname{\mathcal{C}} \]
factors uniquely as a composition $\operatorname{Kan}\twoheadrightarrow \mathrm{h} \mathit{\operatorname{Kan}} \xrightarrow { \overline{F} } \operatorname{\mathcal{C}}$.
- $(b)$
The construction $F \mapsto \overline{F}$ induces an equivalence of categories $\operatorname{\mathcal{E}}' \rightarrow \operatorname{Fun}( \mathrm{h} \mathit{\operatorname{Kan}}, \operatorname{\mathcal{C}})$.
Proof of Corollary 3.6.5.6.
Let $\operatorname{\mathcal{E}}\subseteq \operatorname{Fun}( \operatorname{Kan}, \operatorname{\mathcal{C}})$ be the full subcategory spanned by those functors $F: \operatorname{Kan}\rightarrow \operatorname{\mathcal{C}}$ which carry homotopy equivalences of Kan complexes to isomorphisms in $\operatorname{\mathcal{C}}$. By virtue of Corollary 3.1.7.7, it will suffice to show that precomposition with the geometric realization functor $| \bullet |: \operatorname{Kan}\rightarrow \operatorname{Top}$ induces an equivalence of categories $\operatorname{\mathcal{E}}' \rightarrow \operatorname{\mathcal{E}}$. We claim that this functor has a homotopy inverse $\operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{E}}'$, given by precomposition with the functor $\operatorname{Sing}_{\bullet }: \operatorname{Top}\rightarrow \operatorname{Kan}$. This follows from the following pair of observations:
For every functor $F: \operatorname{Top}\rightarrow \operatorname{\mathcal{C}}$, the counit map $\overline{F} \circ \operatorname{Sing}_{\bullet } \rightarrow F$ is an isomorphism when $F$ belongs to $\operatorname{\mathcal{E}}'$ (since, for every topological space $X$, the counit map $| \operatorname{Sing}_{\bullet }(X) | \rightarrow X$ is a weak homotopy equivalence; see Corollary 3.6.4.2).
For every functor $F_0: \operatorname{Kan}\rightarrow \operatorname{\mathcal{C}}$, the unit map $F_0 \rightarrow \overline{ F_0 \circ \operatorname{Sing}_{\bullet } }$ is an isomorphism (since, for every simplicial set $Y$, the unit map $Y \rightarrow \operatorname{Sing}_{\bullet }( |Y| )$ is a weak homotopy equivalence of simplicial sets, and therefore induces a homotopy equivalence of topological spaces $|Y| \rightarrow | \operatorname{Sing}_{\bullet }( |Y| ) |$).
$\square$