Kerodon

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

Corollary 6.2.3.13. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category. Then there is a 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 assigns to each localizing collection $W$ the full subcategory $\operatorname{\mathcal{C}}' \subseteq \operatorname{\mathcal{C}}$ spanned by the $W$-local objects.

Proof. Combine Propositions 6.2.3.12 and 6.2.3.8 (the inverse bijection carries a replete full subcategory $\operatorname{\mathcal{C}}' \subseteq \operatorname{\mathcal{C}}$ to the collection of $\operatorname{\mathcal{C}}'$-local equivalences). $\square$