Kerodon

Example 5.6.5.5 (Fibrations over a Point). Let $\operatorname{\mathcal{E}}$ be a small $\infty $-category, which we identify with a morphism $\mathscr {F}: \Delta ^0 \rightarrow \operatorname{\mathcal{QC}}$. Then $\mathscr {F}$ is a covariant transport representation of the projection map $U: \operatorname{\mathcal{E}}\rightarrow \Delta ^0$. More precisely, Example 5.6.2.17 supplies an equivalence of $\infty $-categories $\operatorname{\mathcal{E}}\rightarrow \int _{\Delta ^0} \mathscr {F}$ which witnesses $\mathscr {F}$ as a covariant transport representation of $U$. More generally, a functor $\Delta ^0 \rightarrow \operatorname{\mathcal{QC}}$ is a covariant transport representation of $U$ if and only if it corresponds to an $\infty $-category which is equivalent to $\operatorname{\mathcal{E}}$.