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