A local-global principle for parametrized $\infty $ -categories

Abstract We prove a local-global principle for parametrized $\infty $ -categories: we show that any functor $\mathcal {B} \to \mathcal {C}$ is determined by the following data: the collection of fibers $\mathcal {B}_X$ for X running through the set of equivalence classes of objects of $\mathcal {C}$ endowed with the action of the space of automorphisms $\mathrm {Aut}_X(\mathcal {B})$ on the fiber, the local data, together with a locally cartesian fibration ${\mathcal D} \to \mathcal {C}$ and $\mathrm {Aut}_X(\mathcal {B})$ -linear equivalences ${\mathcal D}_X \simeq {\mathcal P}(\mathcal {B}_X)$ to the $\infty $ -category of presheaves on $\mathcal {B}_X$ , the gluing data. As applications we compute the mapping spaces of the conditionally existing internal hom of $\infty \mathrm {Cat}_{/\mathcal {C}}$ and extend the $\infty $ -categorical Grothendieck-construction by proving that $\infty $ -categories over any $\infty $ -category $\mathcal {C}$ are classified by normal lax 2-functors to a double $\infty $ -category of correspondences.