Kerodon

$\Newextarrow{\xRightarrow}{5,5}{0x21D2}$
$\Newextarrow{\xhookrightarrow}{10,10}{0x21AA}$

Variant 3.1.6.8. Let $\operatorname{\mathcal{C}}$ be a category, and let $\operatorname{\mathcal{E}}' \subseteq \operatorname{Fun}( \operatorname{Set_{\Delta }}, \operatorname{\mathcal{C}})$ be the full subcategory spanned by those functors $F: \operatorname{Set_{\Delta }}\rightarrow \operatorname{\mathcal{C}}$. which carry weak homotopy equivalences of simplicial sets to isomorphisms in the category $\operatorname{\mathcal{C}}$. Then:

$(a)$

For every functor $F \in \operatorname{\mathcal{E}}'$, the restriction $F|_{\operatorname{Kan}}$ 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 Variant 3.1.6.8. 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 3.1.6.7, it will suffice to show that the restriction functor $F \mapsto F|_{\operatorname{Kan}}$ induces an equivalence of categories $\operatorname{\mathcal{E}}' \rightarrow \operatorname{\mathcal{E}}$. Using Proposition 3.1.6.1, we can choose a functor $Q: \operatorname{Set_{\Delta }}\rightarrow \operatorname{Kan}$ and a natural transformation $u: \operatorname{id}_{\operatorname{Set_{\Delta }}} \rightarrow Q$ with the property that, for every Kan complex $X$, the induced map $u_{X}: X \rightarrow Q(X)$ is anodyne. For every morphism of simplicial sets $f: X \rightarrow Y$, we have a commutative diagram

\[ \xymatrix { X \ar [r]^-{f} \ar [d]^{u_ X} & Y \ar [d]^{ u_ Y} \\ Q(X) \ar [r]^{ Q(f) } & Q(Y), } \]

where the vertical maps are weak homotopy equivalences (Proposition 3.1.5.13). It follows that if $f$ is a weak homotopy equivalence, then $Q(f)$ is also a weak homotopy equivalence (Remark 3.1.5.16) and therefore a homotopy equivalence (Proposition 3.1.5.12). In other words, the functor $Q$ carries weak homotopy equivalences of simplicial sets to homotopy equivalences of Kan complexes. It follows that precomposition with $Q$ induces a functor $\theta : \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{E}}'$. We claim that $\theta $ is homotopy inverse to the restriction functor $\operatorname{\mathcal{E}}' \rightarrow \operatorname{\mathcal{E}}$. This follows from the following pair of observations:

  • For every functor $F: \operatorname{Set_{\Delta }}\rightarrow \operatorname{\mathcal{C}}$, $u$ induces a natural transformation $F \rightarrow F|_{\operatorname{Kan}} \circ Q$, which depends functorially on $F$ and is an isomorphism for $F \in \operatorname{\mathcal{E}}'$.

  • For every functor $F_0: \operatorname{Kan}\rightarrow \operatorname{\mathcal{C}}$, $u$ induces a natural transformation $F_0 \rightarrow (F_0 \circ Q)|_{\operatorname{Kan}}$, which depends functorially on $F_0$ and is an isomorphism for $F_0 \in \operatorname{\mathcal{E}}$.

$\square$