Kerodon

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

Remark 1.4.2.3 (Diagrams of Dimension $\leq 1$). Let $\operatorname{\mathcal{C}}$ be an $\infty $-category and let $K_{\bullet }$ be a simplicial set of dimension $\leq 1$, corresponding to a directed graph $G$ (Proposition 1.1.5.9). In this case, a diagram $K_{\bullet } \rightarrow \operatorname{\mathcal{C}}$ can be identified with a pair $( \{ C_{v} \} _{v \in \operatorname{Vert}(G)}, \{ f_{e} \} _{e \in \operatorname{Edge}(G)} )$, where each $C_{v}$ is an object of the $\infty $-category $\operatorname{\mathcal{C}}$ and each $f_{e}: C_{s(e)} \rightarrow C_{t(e)}$ is a morphism of $\operatorname{\mathcal{C}}$ (here $s(e)$ and $t(e)$ denote the source and target of the edge $e$). It is often convenient to specify diagrams $K_{\bullet } \rightarrow \operatorname{\mathcal{C}}$ by drawing a graphical representation of $G$ (as in Remark 1.1.5.3), where each node is labelled by an object of $\operatorname{\mathcal{C}}$ and each arrow is labelled by a morphism in $\operatorname{\mathcal{C}}$ (having the indicated source and target).