Perversely categorified Lagrangian correspondences

In this article, we construct a 2-category of Lagrangians in a fixed shifted symplectic derived stack S. The objects and morphisms are all given by Lagrangians living on various fiber products. A special case of this gives a 2-category of n-shifted symplectic derived stacks Sympⁿ. This is a 2-category version of Weinstein's symplectic category in the setting of derived symplectic geometry. We introduce another 2-category Symp^or of 0-shifted symplectic derived stacks where the objects and morphisms in Symp⁰ are enhanced with orientation data. Using this, we define a partially linearized 2-category LSymp. Joyce and his collaborators defined a certain perverse sheaf on any oriented (-1)-shifted symplectic derived stack. In LSymp, the 2-morphisms in Symp^or are replaced by the hypercohomology of the perverse sheaf assigned to the (-1)-shifted symplectic derived Lagrangian intersections. To define the compositions in LSymp we use a conjecture by Joyce, that Lagrangians in (-1)-shifted symplectic stacks define canonical elements in the hypercohomology of the perverse sheaf over the Lagrangian. We refine and expand his conjecture and use it to construct LSymp and a 2-functor from Symp^or to LSymp. We prove Joyce's conjecture in the most general local model. Finally, we define a 2-category of d-oriented derived stacks and fillings. Taking mapping stacks into a n-shifted symplectic stack defines a 2-functor from this category to Symp^n-d.