Kerodon

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

Remark 3.1.4.11. Let $\operatorname{Kan}$ denote the full subcategory of $\operatorname{Set_{\Delta }}$ spanned by the Kan complexes, and let $\operatorname{\mathcal{C}}$ be any category. Then precomposition with the quotient map $\operatorname{Kan}\rightarrow \mathrm{h} \mathit{\operatorname{Kan}}$ induces an isomorphism from the functor category $\operatorname{Fun}( \mathrm{h} \mathit{\operatorname{Kan}}, \operatorname{\mathcal{C}})$ to the full subcategory of $\operatorname{Fun}( \operatorname{Kan}, \operatorname{\mathcal{C}})$ spanned by those functors $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{Kan}$ which satisfy the following condition:

  • If $X$ and $Y$ are Kan complexes and $u_0, u_1: X \rightarrow Y$ are homotopic morphisms, then $F(u_0) = F(u_1)$ in $\operatorname{Hom}_{\operatorname{\mathcal{C}}}( F(X), F(Y) )$.