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

Abstract

AbstractWe 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.