This work applies a Grothendieck construction to demonstrate equivalence in 2-Cartesian fibrations and contravariant functors in infinity-bicategories.
In this work, we conclude our study of fibred ∞ -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 ∞ -bicategory of 2-Cartesian fibrations over S and the ∞ -bicategory of contravariant functors with values in the ∞ -bicategory of ∞ -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.
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Abellan et al. (2025) studied this question.
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