2020/11/17 by Haugseng, Rune, Hebestreit, Fabian, Linskens, Sil +1 · 5 citations
#Algebraic Topology (math.AT) #Category Theory (math.CT) #FOS: Mathematics
paper · doi:10.48550/arxiv.2011.08808
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 (∞, 2)-categories.