Kerodon

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

Remark 2.4.2.5. It follows from Corollary 1.1.4.8 and Remark 2.4.1.12 that the construction

\[ \operatorname{Cat}\rightarrow \operatorname{Cat_{\Delta }}\quad \quad \quad \operatorname{\mathcal{C}}\mapsto \underline{\operatorname{\mathcal{C}}}_{\bullet } \]

of Example 2.4.2.4 is fully faithful. Its essential image consists of those simplicial categories $\operatorname{\mathcal{E}}_{\bullet }$ having the property that, for every pair of objects $X,Y \in \operatorname{Ob}(\operatorname{\mathcal{E}}_{\bullet })$, the simplicial set $\operatorname{Hom}_{\operatorname{\mathcal{E}}}(X,Y)_{\bullet }$ is discrete (Definition 1.1.4.9). We will sometimes abuse notation by not distinguishing between the ordinary category $\operatorname{\mathcal{C}}$ and the constant simplicial category $\underline{\operatorname{\mathcal{C}}}_{\bullet }$.