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

By R. Haugseng, F. Hebestreit, Sil Linskens, Joost Nuiten. Proceedings of the London Mathematical Society, 2020

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We provide a calculus of mates for functors to the ∞$\infty$ ‐category of ∞$\infty$ ‐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 ∞$\infty$ ‐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 ∞$\infty$ ‐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 (∞,2)$(\infty , 2)$ ‐categories.

Subfields: Algebra and Number Theory, Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/a3dc64188558e0ddd8a43416620c1da8240a0abe