Kerodon

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

Warning 6.3.3.8. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category and let $W$ be a collection of morphisms of $\operatorname{\mathcal{C}}$ which is both localizing and colocalizing. In this case, Proposition 6.3.3.5 and Variant 6.3.3.7 provide two different concrete realizations of the localization $\operatorname{\mathcal{C}}[W^{-1}]$, given by the full subcategories $\operatorname{\mathcal{C}}' \subseteq \operatorname{\mathcal{C}}\supseteq \operatorname{\mathcal{C}}''$ spanned by the $W$-local and $W$-colocal objects of $\operatorname{\mathcal{C}}$, respectively. Note that $\operatorname{\mathcal{C}}'$ and $\operatorname{\mathcal{C}}''$ are necessarily equivalent as abstract $\infty $-categories. More precisely, if $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{C}}[W^{-1}]$ is a functor which exhibits $\operatorname{\mathcal{C}}[W^{-1}]$ as the localization of $\operatorname{\mathcal{C}}$ with respect to $W$, then the restrictions

\[ \operatorname{\mathcal{C}}' \xrightarrow { F|_{\operatorname{\mathcal{C}}'} } \operatorname{\mathcal{C}}[W^{-1}] \xleftarrow { F|_{\operatorname{\mathcal{C}}''} } \operatorname{\mathcal{C}}'' \]

are equivalences of $\infty $-categories. Beware that $\operatorname{\mathcal{C}}'$ and $\operatorname{\mathcal{C}}''$ usually do not coincide when regarded as subcategories of $\operatorname{\mathcal{C}}$.