Kerodon

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

Example 7.1.4.9. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category. Then an object $Y \in \operatorname{\mathcal{C}}$ is final (in the sense of Definition 7.1.2.1) if and only if the map

\[ (\emptyset )^{\triangleleft } \simeq \Delta ^0 \xrightarrow {Y} \operatorname{\mathcal{C}} \]

is a limit diagram in $\operatorname{\mathcal{C}}$. Similarly, $Y$ is initial if and only if the map

\[ (\emptyset )^{\triangleright } \simeq \Delta ^0 \xrightarrow {Y} \operatorname{\mathcal{C}} \]

is a colimit diagram in $\operatorname{\mathcal{C}}$.