Abstract: In recent years, it has been observed that symplectic and algebraic invariants of algebraic symplectic stacks are related by some remarkable dualities, particularly symplectic duality and relative Langlands duality. The 3-dimensional topological field theory perspective suggests that these relations ought to have their origins in an equivalence of 2-categories, and I will explain how this works in the abelian case. The new ingredient is a 2-category of perverse sheaves of categories, which are expected to play the same role in categorical representation theory that perverse sheaves have played in geometric representation theory. This is based on joint work with Justin Hilburn and Aaron Mazel-Gee, and continuing work with Justin Hilburn.