Kerodon

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

Example 7.1.7.9. Let $U: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be an inner fibration of $\infty $-categories. Then:

  • A morphism $e$ of $\operatorname{\mathcal{C}}$ is $U$-cartesian (in the sense of Definition 5.1.1.1) if and only if it is a $U$-limit diagram when viewed as a morphism of simplicial sets $(\Delta ^0)^{\triangleleft } \rightarrow \operatorname{\mathcal{C}}$.

  • A morphism $f$ of $\operatorname{\mathcal{C}}$ is $U$-cocartesian (in the sense of Definition 5.1.1.1) if and only if it is a $U$-colimit diagram when viewed as a morphism of simplicial sets $(\Delta ^0)^{\triangleright } \rightarrow \operatorname{\mathcal{C}}$.

This follows by combining Remark 7.1.7.8 with Proposition 5.1.1.13.