Kerodon

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

Convention 1.4.2.12. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category and let $G$ be a directed graph satisfying conditions $(a)$ and $(b)$ of Definition 1.4.2.5, so that the vertex set $\operatorname{Vert}(G)$ inherits a partial ordering (Proposition 1.4.2.6). We will sometimes refer to the notion of a commutative diagram $\sigma $ in $\operatorname{\mathcal{C}}$, which we indicate graphically by a collection of objects $\{ C_ v \} _{v \in \operatorname{Vert}(G)}$ of $\operatorname{\mathcal{C}}$, connected by arrows which are labelled by morphisms $\{ f_{e} \} _{e \in \operatorname{Edge}(G)}$. In this case, it should be understood that $\sigma $ is a diagram $\operatorname{N}_{\bullet }( \operatorname{Vert}(G) ) \rightarrow \operatorname{\mathcal{C}}$, which carries each vertex $v$ of $\operatorname{N}_{\bullet }( \operatorname{Vert}(G) )$ to the object $C_{v} \in \operatorname{\mathcal{C}}$ and each edge $e = (v,w)$ of $G$ to the morphism $f_{e}$ in $\operatorname{\mathcal{C}}$. Beware that in this case, the map $\sigma $ need not be completely determined by the pair $( \{ C_ v \} _{v \in \operatorname{Vert}(G)}, \{ f_ e \} _{e \in \operatorname{Edge}(G)} )$ (this pair can instead be identified with the restriction $\sigma |_{ K_{\bullet } }$, where $K_{\bullet }$ is the $1$-dimensional simplicial subset of $\operatorname{N}_{\bullet }( \operatorname{Vert}(G) )$ corresponding to $G$).