Kerodon

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

Remark 5.4.4.8 (Comparison with Kan Complexes). Every Kan complex is an $\infty $-category (Example 1.3.0.3). Moreover, if $X$ and $Y$ are Kan complexes, then the simplicial set $\operatorname{Fun}(X,Y)$ is also a Kan complex (Corollary 3.1.3.4), and therefore coincides with its core $\operatorname{Fun}(X,Y)^{\simeq }$. It follows that we can regard the simplicial category $\operatorname{Kan}$ of Construction 5.4.1.1 as a full simplicial subcategory of $\operatorname{QCat}$. Passing to homotopy coherent nerves, we deduce that the $\infty $-category $\operatorname{\mathcal{S}}= \operatorname{N}_{\bullet }^{\operatorname{hc}}( \operatorname{Kan})$ is the full subcategory of $\operatorname{\mathcal{QC}}= \operatorname{N}_{\bullet }^{\operatorname{hc}}( \operatorname{QCat})$ spanned by the Kan complexes.