# Kerodon

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

Corollary 6.3.5.3. Let $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}$ be a cocartesian fibration of simplicial sets. Suppose that, for every vertex $C \in \operatorname{\mathcal{C}}$, the $\infty$-category $\operatorname{\mathcal{E}}_{C} = \{ C\} \times _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{E}}$ is weakly contractible. Let $W$ be the collection of all edges $e$ of $\operatorname{\mathcal{E}}$ having the property that $U(e)$ is a degenerate edge of $\operatorname{\mathcal{C}}$. Then $U$ exhibits $\operatorname{\mathcal{C}}$ as a localization of $\operatorname{\mathcal{E}}$ with respect to $W$.

Proof. For each vertex $C \in \operatorname{\mathcal{C}}$, let $W_{C}$ be the collection of all morphisms in the $\infty$-category $\operatorname{\mathcal{E}}_{C}$. Since $\operatorname{\mathcal{E}}_{C}$ is weakly contractible, the projection map $\operatorname{\mathcal{E}}_{C} \rightarrow \{ C\}$ exhibits $\{ C\}$ as a localization of $\operatorname{\mathcal{E}}_{C}$ with respect to $W_{C}$ (Proposition 6.3.1.20). The desired result now follows by applying Proposition 6.3.5.2 to the commutative diagram

$\xymatrix@R =50pt@C=50pt{ \operatorname{\mathcal{E}}\ar [dr]^{U} \ar [rr]^{U} & & \operatorname{\mathcal{C}}\ar [dl]_{\operatorname{id}_{\operatorname{\mathcal{C}}} } \\ & \operatorname{\mathcal{C}}. & }$
$\square$