Kerodon

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

Example 6.2.2.6. Let $\operatorname{\mathcal{S}}$ denote the $\infty $-category of spaces (Construction 5.6.1.1) and let $\operatorname{\mathcal{QC}}$ denote the $\infty $-category of (small) $\infty $-categories (Construction 5.6.4.1). Then $\operatorname{\mathcal{S}}$ is a reflective and coreflective subcategory of $\operatorname{\mathcal{QC}}$. If $\operatorname{\mathcal{C}}$ is a small $\infty $-category, then the inclusion map $\operatorname{\mathcal{C}}^{\simeq } \hookrightarrow \operatorname{\mathcal{C}}$ exhibits the core $\operatorname{\mathcal{C}}^{\simeq }$ as a $\operatorname{\mathcal{S}}$-coreflection of $\operatorname{\mathcal{C}}$ (this follows by combining Proposition 4.4.3.16 with Remark 5.6.4.6), and the comparison map $\operatorname{\mathcal{C}}\rightarrow \operatorname{Ex}^{\infty }(\operatorname{\mathcal{C}})$ exhibits the Kan complex $\operatorname{Ex}^{\infty }(\operatorname{\mathcal{C}})$ as a $\operatorname{\mathcal{S}}$-reflection of $\operatorname{\mathcal{C}}$ (this follows by combining Proposition 3.3.6.7 with Remark 5.6.4.6).