Kerodon

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

Corollary 9.3.2.16. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category and suppose we are given a pushout diagram

9.9
\begin{equation} \begin{gathered}\label{equation:localizing-morphisms-closed-under-pushout} \xymatrix@C =50pt@R=50pt{ X \ar [r]^-{w} \ar [d] & Y \ar [d] \\ X' \ar [r]^-{w'} & Y' } \end{gathered} \end{equation}

in $\operatorname{\mathcal{C}}$. If $W$ is a localizing collection of morphisms of $\operatorname{\mathcal{C}}$ which contains $w$, then it also contains $w'$.

Proof. Let $\operatorname{\mathcal{C}}' \subseteq \operatorname{\mathcal{C}}$ be the full subcategory spanned by the $W$-local objects. Since $W$ is localizing, it coincides with the collection of $\operatorname{\mathcal{C}}'$-local equivalences (Proposition 6.2.3.12), and is therefore closed under pushouts by virtue of Variant 9.3.2.15. $\square$