Kerodon

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

Remark 1.1.5.10. The proof of Proposition 1.1.5.9 gives an explicit description of the inverse equivalence $\operatorname{Graph}\simeq \operatorname{\mathcal{C}}\hookrightarrow \operatorname{Set_{\Delta }}$: it carries a directed graph $G$ to the $1$-dimensional simplicial set $G_{\bullet }$ given by the pushout

\[ ( \underset { v \in \operatorname{Vert}( G )}{\coprod } \Delta ^0 ) \amalg _{ \underset { e \in \operatorname{Edge}( G )}{\coprod } \operatorname{\partial \Delta }^{1} } ( \underset { e \in \operatorname{Edge}( G)}{\coprod } \Delta ^{1} ). \]