Variant 8.6.5.3 (04DV): Cartesian Duality

Variant 8.6.5.3 (Cartesian Duality). Let $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}$ and $U': \operatorname{\mathcal{E}}' \rightarrow \operatorname{\mathcal{C}}$ be cartesian fibrations of simplicial sets. We say that $U'$ is a cartesian dual of $U$ if the cocartesian fibration $U'^{\operatorname{op}}: \operatorname{\mathcal{E}}'^{\operatorname{op}} \rightarrow \operatorname{\mathcal{C}}^{\operatorname{op}}$ is a cocartesian dual of $U^{\operatorname{op}}: \operatorname{\mathcal{E}}^{\operatorname{op}} \rightarrow \operatorname{\mathcal{C}}^{\operatorname{op}}$. It follows from Theorem 8.6.5.1 that every cartesian fibration $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}$ admits a cartesian dual $U': \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}$ which is uniquely determined up to equivalence. Moreover, Corollary 8.6.4.10 implies that the (contravariant) homotopy transport representation of $U'$ is given by the composition

\[ \mathrm{h} \mathit{\operatorname{\mathcal{C}}}^{\operatorname{op}} \xrightarrow { \operatorname{hTr}_{\operatorname{\mathcal{E}}/\operatorname{\mathcal{C}}} } \mathrm{h} \mathit{\operatorname{QCat}} \xrightarrow { \operatorname{\mathcal{A}}\mapsto \operatorname{\mathcal{A}}^{\operatorname{op}} } \mathrm{h} \mathit{\operatorname{QCat}}. \]

In particular, for every vertex $C \in \operatorname{\mathcal{C}}$, the fiber $\operatorname{\mathcal{E}}'_{C} \{ C\} \times _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{E}}'$ is equivalent to the opposite of the $\infty $-category $\operatorname{\mathcal{E}}_{C} = \{ C\} \times _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{E}}$.