Perversely categorified Lagrangian correspondences (Q1691328)

In this article, the authors consider a weak \(2\)-category Lag\((S)\), for an \(n\)-shifted derived stack \(S\). In this category, the objects are derived Lagrangians in \(S\), \(1\)-morphisms are relative Lagrangian correspondences and \(2\)-morphisms are relative Lagrangian correspondences between relative Lagrangian correspondences. As a consequence they obtain a \(2\)-category \(\mathrm{Symp}^n\) with objects \(n\)-shifted derived stacks. Due to the high nonlinearity of these categories, the authors define a partially linearized \(2\)-category LSymp. In this category, the objects are \(0\)-shifted symplectic derived stacks with line bundles, \(1\)-morphisms are oriented Lagrangian correspondences and \(2\)-morphisms are given by the hypercohomology of the perverse sheaf associated to the \((-1)\)-shifted symplectic derived Lagrangian intersections.