Kerodon

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

Remark 5.7.0.8. Let $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}$ be a cocartesian fibration between (small) simplicial sets. We will denote the covariant transport representation of $U$ by $\operatorname{Tr}_{\operatorname{\mathcal{E}}/\operatorname{\mathcal{C}}}$; it can be regarded as a homotopy coherent refinement of the homotopy transport representation $\operatorname{hTr}_{\operatorname{\mathcal{E}}/\operatorname{\mathcal{C}}}$ introduced in Construction 5.2.5.2 (see Remark 5.7.5.8 for a precise statement). We can summarize the situation with the following informal answer to Question 5.7.0.1:

  • For every cocartesian fibration between (small) $\infty $-categories $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}$, the construction $C \mapsto \operatorname{\mathcal{E}}_{C}$ determines a functor of $\infty $-categories $\operatorname{Tr}_{\operatorname{\mathcal{E}}/\operatorname{\mathcal{C}}}: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{QC}}$. Moreover, we can recover $\operatorname{\mathcal{E}}$ (up to equivalence) as the $\infty $-category of elements $\int _{\operatorname{\mathcal{C}}} \operatorname{Tr}_{\operatorname{\mathcal{E}}/\operatorname{\mathcal{C}}}$.