Cartesian fibrations of (∞,2)–categories

Gagna, Andrea¹ ; Harpaz, Yonatan² ; Lanari, Edoardo¹

¹ Institute of Mathematics, Czech Academy of Sciences, Prague, Czech Republic
² Institut Galilée, Université Paris 13, Villeta-neuse, France

Algebraic and Geometric Topology, Tome 24 (2024) no. 9, pp. 4731-4778

Résumé

We introduce four variance flavors of (co)cartesian fibrations of ∞–bicategories with ∞–bicategorical fibers, in the framework of scaled simplicial sets. Given a map p: ℰ→ℬ of ∞–bicategories, we define p–(co)cartesian arrows and inner/outer triangles by means of lifting properties against p, leading to a notion of 2–inner/outer (co)cartesian fibrations as those maps with enough (co)cartesian lifts for arrows and enough inner/outer lifts for triangles, together with a compatibility property with respect to whiskerings in the outer case. By doing so, we also recover in particular the case of ∞–bicategories fibered in ∞–categories studied in previous work. We also prove that equivalences of such fibrations can be tested fiberwise. As a motivating example, we show that the domain projection d ⁡ : Fun ⁡ gr(Δ1,𝒞) →𝒞 is a prototypical example of a 2–outer cartesian fibration, where Fun ⁡ gr(X,Y ) denotes the ∞–bicategory of functors, lax natural transformations and modifications. We then define 2–inner and 2–outer flavors of (co)cartesian fibrations of categories enriched in ∞–categories, and we show that a fibration p: ℰ→ℬ of such enriched categories is a (co)cartesian 2–inner/outer fibration if and only if the corresponding map N ⁡ sc ⁡ (p): N ⁡ sc ⁡ ℰ→ N ⁡ sc ⁡ ℬ is a fibration of this type between ∞–bicategories.

Détail

DOI : 10.2140/agt.2024.24.4731

Keywords: fibrations, higher categories, bicategories