The Heisenberg Category of a Category

Starting with a k {\mathbb {k}} -linear or DG category admitting a (homotopy) Serre functor, we construct a k {\mathbb {k}} -linear or DG 2 2 -category categorifying the Heisenberg algebra of the numerical K K -group of the original category. We also define a 2 2 -categorical analogue of the Fock space representation of the Heisenberg algebra. Our construction generalises and unifies various categorical Heisenberg algebra actions appearing in the literature. In particular, we give a full categorical enhancement of the action on derived categories of symmetric quotient stacks introduced by Krug, which itself categorifies a Heisenberg algebra action proposed by Grojnowski.