Kerodon

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

Definition 7.1.3.1. Let $U: \widetilde{\operatorname{\mathcal{C}}} \rightarrow \operatorname{\mathcal{C}}$ be a right fibration of $\infty $-categories. We say that $\widetilde{\operatorname{\mathcal{C}}}$ is represented by an object $Y \in \operatorname{\mathcal{C}}$ if there exists a final object $\widetilde{Y} \in \widetilde{\operatorname{\mathcal{C}}}$ satisfying $U( \widetilde{Y} ) = Y$. In this case, we say that $\widetilde{Y}$ exhibits $\widetilde{\operatorname{\mathcal{C}}}$ as a right fibration represented by $Y$. We say that a right fibration $U: \widetilde{\operatorname{\mathcal{C}}} \rightarrow \operatorname{\mathcal{C}}$ is representable if it it is represented by some object of $\operatorname{\mathcal{C}}$: that is, if the $\infty $-category $\widetilde{\operatorname{\mathcal{C}}}$ has a final object.