Kerodon

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

Corollary 6.3.3.10. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category. Then there is a canonical bijection

\[ \xymatrix@R =50pt@C=50pt{ \{ \textnormal{Localizing collections of morphisms of $\operatorname{\mathcal{C}}$} \} \ar [d]^{\sim } \\ \{ \textnormal{Reflective replete subcategories of $\operatorname{\mathcal{C}}$} \} , } \]

which carries a localizing collection of morphisms $W$ to the full subcategory $\operatorname{\mathcal{C}}' \subseteq \operatorname{\mathcal{C}}$ spanned by the $W$-local objects.

Proof. Combine Proposition 6.3.3.5 with Proposition 6.3.3.9. $\square$