Kerodon

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

Remark 2.4.6.19. Let $\operatorname{\mathcal{C}}_{\bullet }$ be a simplicial category, let $\mathrm{h}_{2} \mathit{\operatorname{\mathcal{C}}}$ be the homotopy $2$-category of $\operatorname{\mathcal{C}}$, and let $(\mathrm{h}_{2} \mathit{\operatorname{\mathcal{C}}})_{\bullet }$ denote the simplicial category obtained from $\mathrm{h}_{2} \mathit{\operatorname{\mathcal{C}}}$ by applying the construction of Example 2.4.2.7. Then there is a simplicial functor $U: \operatorname{\mathcal{C}}_{\bullet } \rightarrow ( \mathrm{h}_{2} \mathit{\operatorname{\mathcal{C}}} )_{\bullet }$, given on objects by the identity map and on morphism spaces by the tautological maps

\[ \operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y)_{\bullet } \rightarrow \operatorname{N}_{\bullet }( \mathrm{h} \mathit{ \operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y)_{\bullet } }). \]

Passing to the homotopy coherent nerve (and invoking Example 2.4.3.11), we obtain a map of simplicial sets $V: \operatorname{N}_{\bullet }^{\operatorname{hc}}(\operatorname{\mathcal{C}}) \rightarrow \operatorname{N}_{\bullet }^{\operatorname{D}}( \mathrm{h}_{2} \mathit{\operatorname{\mathcal{C}}} )$, which restricts to the identity on the nerve $\operatorname{N}_{\bullet }(\operatorname{\mathcal{C}})$ (which we can regard as a simplicial subset of both $\operatorname{N}_{\bullet }^{\operatorname{hc}}(\operatorname{\mathcal{C}})$ and $\operatorname{N}_{\bullet }^{\operatorname{D}}( \mathrm{h}_{2} \mathit{\operatorname{\mathcal{C}}} )$).