Perversely categorified Lagrangian correspondences

Abstract

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 $Symp^n$. 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 $Symp^0$ 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 $Symp^{n-d}$.