Kerodon

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$\Newextarrow{\xhookrightarrow}{10,10}{0x21AA}$

Proposition 2.4.4.7. Let $G$ be a directed graph, let $\operatorname{Path}[G]$ denote the path category of Construction 1.2.6.1, and let $\underline{\operatorname{Path}[G]}_{\bullet }$ denote the associated constant simplicial category (Example 2.4.2.3). Then the comparison map $u: G_{\bullet } \rightarrow \operatorname{N}_{\bullet }( \operatorname{Path}[G] ) \simeq \operatorname{N}_{\bullet }^{\operatorname{hc}}( \underline{\operatorname{Path}[G]} )$ exhibits $\underline{\operatorname{Path}[G]}_{\bullet }$ as a path category of the simplicial set $G_{\bullet }$.

Proof. Unwinding the definitions, we must show that for every simplicial category $\operatorname{\mathcal{D}}_{\bullet }$, the composite map

\begin{eqnarray*} \operatorname{Hom}_{\operatorname{Cat_{\Delta }}}( \underline{\operatorname{Path}[G]}_{\bullet }, \operatorname{\mathcal{D}}_{\bullet } ) & \rightarrow & \operatorname{Hom}_{\operatorname{Cat}}( \operatorname{Path}[G], \operatorname{\mathcal{D}}) \\ & \rightarrow & \operatorname{Hom}_{\operatorname{Set_{\Delta }}}( G_{\bullet }, \operatorname{N}_{\bullet }(\operatorname{\mathcal{D}}) ) \\ & \rightarrow & \operatorname{Hom}_{\operatorname{Set_{\Delta }}}( G_{\bullet }, \operatorname{N}_{\bullet }^{\operatorname{hc}}(\operatorname{\mathcal{D}}) ) \end{eqnarray*}

is a bijection. Here the first map is bijective because the simplicial category $\underline{\operatorname{Path}[G]}_{\bullet }$ is constant (Remark 2.4.2.5), the second by virtue of Proposition 1.2.6.5, and the third because $G_{\bullet }$ has dimension $\leq 1$ and the comparison map $\operatorname{N}_{\bullet }(\operatorname{\mathcal{D}}) \rightarrow \operatorname{N}_{\bullet }^{\operatorname{hc}}(\operatorname{\mathcal{D}})$ is an isomorphism on simplices of dimension $\leq 1$ (Example 2.4.3.9). $\square$