Lax monoidal adjunctions, two‐variable fibrations and the calculus of mates

Abstract

We provide a calculus of mates for functors to the ‐category of ‐categories and extend Lurie's unstraightening equivalences to show that (op)lax natural transformations correspond to maps of (co)cartesian fibrations that do not necessarily preserve (co)cartesian edges. As a sample application, we obtain an equivalence between lax symmetric monoidal structures on right adjoint functors and oplax symmetric monoidal structures on the left adjoint functors between symmetric monoidal ‐categories that is compatible with both horizontal and vertical composition of such structures. As the technical heart of the paper, we study various new types of fibrations over a product of two ‐categories. In particular, we show how they can be dualised over one of the two factors and how they encode functors out of the Gray tensor product of ‐categories.

On straightening for Segal spaces

“…Proposition 2.12. Let C and D be categories and let p = (p 1 , p 2 ) : X −→ C × D be a functor such that: Functors p : X −→ C × D satisfying conditions (i) and (ii) have been studied in detail in [HHLN23a,HHLN23b], where they are called local orthofibrations. Let us point out the close analogy between the discussion preceding Proposition 2.12 and the discussion in [HHLN23b,Construction 2.3.5].…”

“…On the other hand, the unstraightening of a diagram F : C −→ Cat is convenient to describe 'lax constructions' with categories, stemming from the fact that Un(F ) is the oplax colimit of F (see [GHN17]). For example, a lax natural transformation between two diagrams F and G can be described by a map Un(F ) −→ Un(G) over C that need not preserve cocartesian arrows (see, e.g., [GHL20,HHLN23b] for a discussion in the ∞-categorical setting).…”