Kerodon

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

Remark 2.4.2.5. Let $\operatorname{\mathcal{C}}$ be an ordinary category and let $\operatorname{\mathcal{D}}_{\bullet }$ be a simplicial category. Applying Proposition 1.1.4.5 (and Remark 2.4.1.12), we deduce that the restriction map

\[ \{ \text{Simplicial functors $\underline{\operatorname{\mathcal{C}}}_{\bullet } \rightarrow \operatorname{\mathcal{D}}_{\bullet }$} \} \simeq \{ \text{Functors $\operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}_0$} \} , \]

is bijective. In other words, the fully faithful embedding

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

of Remark 2.4.2.4 is left adjoint to the forgetful functor

\[ \operatorname{Cat_{\Delta }}\rightarrow \operatorname{Cat}\quad \quad \operatorname{\mathcal{D}}_{\bullet } \mapsto \operatorname{\mathcal{D}}_0. \]

of Example 2.4.1.4.