Kerodon

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

Remark 6.3.1.18. Let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ and $G: \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{E}}$ be morphisms of simplicial sets, and let $W$ be a collection of edges of $\operatorname{\mathcal{C}}$. If any two of the following three conditions is satisfied, then so is the third:

  • The morphism $F$ exhibits $\operatorname{\mathcal{D}}$ as a localization of $\operatorname{\mathcal{C}}$ with respect to $W$.

  • The morphism $G \circ F$ exhibits $\operatorname{\mathcal{E}}$ as a localization of $\operatorname{\mathcal{C}}$ with respect to $W$.

  • The morphism $G$ is a categorical equivalence of simplicial sets.