Kerodon

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

Definition 5.6.2.1. Let $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}$ be a cocartesian fibration of simplicial sets. We will say that a commutative diagram of simplicial sets

\[ \xymatrix@R =50pt@C=50pt{ \operatorname{\mathcal{E}}\ar [d]^{U} \ar [r]^-{ \widetilde{\mathscr {F}} } & \operatorname{\mathcal{QC}}_{\operatorname{Obj}} \ar [d]^{V} \\ \operatorname{\mathcal{C}}\ar [r]^-{ \mathscr {F} } & \operatorname{\mathcal{QC}}} \]

witnesses $\mathscr {F}$ as a covariant transport representation of $U$ if the induced map

\[ \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}\times _{\operatorname{\mathcal{QC}}} \operatorname{\mathcal{QC}}_{\operatorname{Obj}} = \int _{\operatorname{\mathcal{C}}} \mathscr {F} \]

is an equivalence of cocartesian fibrations over $\operatorname{\mathcal{C}}$, in the sense of Definition 5.1.6.1. We say that $\mathscr {F}: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{QC}}$ is a covariant transport representation of $U$ if there exists a diagram which witnesses $\mathscr {F}$ as a covariant transport representation of $U$.