Kerodon

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

Example 4.4.1.11 (Replete Subcategories). Let $\operatorname{\mathcal{C}}$ be an $\infty $-category and let $\operatorname{\mathcal{C}}' \subseteq \operatorname{\mathcal{C}}$ be a subcategory (Definition 4.1.2.2). The following conditions are equivalent:

$(1)$

The inclusion functor $\operatorname{\mathcal{C}}' \hookrightarrow \operatorname{\mathcal{C}}$ is an isofibration.

$(2)$

If $u: X \rightarrow Y$ is an isomorphism in $\operatorname{\mathcal{C}}$ and the object $Y$ belongs to the subcategory $\operatorname{\mathcal{C}}'$, then the isomorphism $u$ also belongs to the subcategory $\operatorname{\mathcal{C}}'$ (and, in particular, the object $X$ also belongs to $\operatorname{\mathcal{C}}'$).

$(3)$

If $u: X \rightarrow Y$ is an isomorphism in $\operatorname{\mathcal{C}}$ and the object $X$ belongs to the subcategory $\operatorname{\mathcal{C}}'$, then the isomorphism $u$ also belongs to the subcategory $\operatorname{\mathcal{C}}'$ (and, in particular, the object $Y$ also belongs to $\operatorname{\mathcal{C}}'$).

If these conditions are satisfied, then we say that the subcategory $\operatorname{\mathcal{C}}' \subseteq \operatorname{\mathcal{C}}$ is replete.