Kerodon

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

Remark 2.4.3.8 (Comparison with the Nerve). Let $\operatorname{\mathcal{C}}_{\bullet }$ be a simplicial category and let $\operatorname{\mathcal{C}}= \operatorname{\mathcal{C}}_0$ denote the underlying ordinary category. For every partially ordered set $Q$, composition with the simplicial functor $\operatorname{Path}[Q]_{\bullet } \rightarrow Q$ of Remark 2.4.3.2 induces a monomorphism

\[ \{ \text{Ordinary functors $Q \rightarrow \operatorname{\mathcal{C}}$} \} \hookrightarrow \{ \text{Simplicial functors $\operatorname{Path}[Q]_{\bullet } \rightarrow \operatorname{\mathcal{C}}_{\bullet }$} \} . \]

Restricting this construction to partially ordered sets of the form $[n] = \{ 0 < 1 < \cdots < n \} $, we obtain a monomorphism of simplicial sets $\operatorname{N}_{\bullet }(\operatorname{\mathcal{C}}) \hookrightarrow \operatorname{N}_{\bullet }^{\operatorname{hc}}(\operatorname{\mathcal{C}})$, where $\operatorname{N}_{\bullet }(\operatorname{\mathcal{C}})$ is the nerve of Construction 1.2.1.1 and $\operatorname{N}_{\bullet }^{\operatorname{hc}}(\operatorname{\mathcal{C}})$ is the homotopy coherent nerve of Definition 2.4.3.5.