Proposition 4.4.3.6. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category. Then the core $\operatorname{\mathcal{C}}^{\simeq }$ is a replete subcategory of $\operatorname{\mathcal{C}}$ (Example 4.4.1.12): that is, the inclusion $\operatorname{\mathcal{C}}^{\simeq } \hookrightarrow \operatorname{\mathcal{C}}$ is an isofibration of $\infty $-categories
$\Newextarrow{\xhookrightarrow}{10,10}{0x21AA}$
Proof. Combining Example 4.1.2.4, Remark 4.1.2.6, and Remark 4.4.3.2, we deduce that the inclusion map $\operatorname{\mathcal{C}}^{\simeq } \hookrightarrow \operatorname{\mathcal{C}}$ is an inner fibration; in particular, $\operatorname{\mathcal{C}}^{\simeq }$ is an $\infty $-category. The repleteness is immediate from the definition (since $\operatorname{\mathcal{C}}^{\simeq }$ contains every isomorphism in $\operatorname{\mathcal{C}}$). $\square$