Kerodon

Variant 8.6.4.13. Let $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}$ and $U^{\vee }: \operatorname{\mathcal{E}}^{\vee } \rightarrow \operatorname{\mathcal{C}}$ be cocartesian fibrations of simplicial sets. In the formulation of Definition 8.6.4.12, we have implicitly assumed that for each vertex $C \in \operatorname{\mathcal{C}}$, the $\infty $-categories $\operatorname{\mathcal{E}}_{C}$ and $\operatorname{\mathcal{C}}^{\vee }_{C}$ are locally small (if this condition is not satisfied, then a balanced profunctor $\mathscr {K}_{C}: \operatorname{\mathcal{E}}^{\vee }_{C} \times \operatorname{\mathcal{E}}_{C} \rightarrow \operatorname{\mathcal{S}}$ cannot exist). However, we will sometimes apply the theory of cocartesian duality in situations where this condition is not satisfied. If $\kappa $ is an uncountable cardinal (not necessarily small), we will say that a morphism $\mathscr {K}: \operatorname{\mathcal{E}}^{\vee } \times _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{S}}^{<\kappa }$ exhibits $U^{\vee }$ as a cocartesian dual of $U$ if it satisfies conditions $(a)$ and $(b)$ of Definition 8.6.4.12. In this case, we can take $\kappa $ to be any uncountable cardinal having the property that, for each vertex $C \in \operatorname{\mathcal{C}}$, the $\infty $-categories $\operatorname{\mathcal{E}}_{C}$ and $\operatorname{\mathcal{E}}^{\vee }_{C}$ are locally $\kappa $-small.