Kerodon

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

Remark 1.4.7.4. Let $G$ be a directed graph and let $\operatorname{\mathcal{C}}$ be an ordinary category. Combining Proposition 1.4.7.3 with Variant 1.4.6.7, we deduce that the canonical map

\[ \operatorname{Hom}_{\operatorname{Set_{\Delta }}}( \operatorname{N}_{\bullet }( \operatorname{Path}[G] ), \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}}) ) \rightarrow \operatorname{Hom}_{\operatorname{Set_{\Delta }}}( G_{\bullet }, \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}}) ) \]

is bijective. Combining this observation with Proposition 1.2.2.1, we obtain a bijection

\[ \operatorname{Hom}_{\operatorname{Cat}}( \operatorname{Path}[G], \operatorname{\mathcal{C}}) \rightarrow \operatorname{Hom}_{\operatorname{Set_{\Delta }}}( G_{\bullet }, \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}}) ). \]

Allowing $\operatorname{\mathcal{C}}$ to vary, we recover the assertion that $u: G_{\bullet } \rightarrow \operatorname{N}_{\bullet }(\operatorname{Path}[G] )$ exhibits $\operatorname{Path}[G]$ as the homotopy category of $G_{\bullet }$ (Proposition 1.2.6.5).