# Kerodon

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

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. $\square$