Kerodon

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

Exercise 6.2.2.12. In the situation of Corollary 6.2.2.11, let $\operatorname{\mathcal{C}}' \subseteq \operatorname{\mathcal{C}}$ be a full subcategory of $\operatorname{\mathcal{C}}$. Show that $\eta $ exhibits $L$ as a $\operatorname{\mathcal{C}}'$-localization functor if and only if the following conditions are satisfied:

  • For each object $X \in \operatorname{\mathcal{C}}$, the object $L(X)$ is contained in $\operatorname{\mathcal{C}}'$.

  • For each object $Y \in \operatorname{\mathcal{C}}'$, there exists an isomorphism $Y \rightarrow L(X)$ for some object $X \in \operatorname{\mathcal{C}}$.

If the subcategory $\operatorname{\mathcal{C}}' \subseteq \operatorname{\mathcal{C}}$ is replete (Example 4.4.1.11), then it is uniquely determined by these conditions.