2-CARTESIAN FIBRATIONS II: A GROTHENDIECK CONSTRUCTION FOR $\infty $-BICATEGORIES

Abstract In this work, we conclude our study of fibred $\infty $ -bicategories by providing a Grothendieck construction in this setting. Given a scaled simplicial set S (which need not be fibrant) we construct a 2-categorical version of Lurie’s straightening-unstraightening adjunction, thereby furnishing an equivalence between the $\infty $ -bicategory of 2-Cartesian fibrations over S and the $\infty $ -bicategory of contravariant functors with values in the $\infty $ -bicategory of $\infty $ -bicategories. We provide a relative nerve construction in the case where the base is a 2-category, and use this to prove a comparison to existing bicategorical Grothendieck constructions.