# The Heisenberg category of a category

In 90s Nakajima and Grojnowski identified the total cohomology of the Hilbert schemes of points on a smooth projective surface with the Fock space representation of the Heisenberg algebra associated to its cohomology lattice. Later, Krug lifted this to derived categories and generalised it to the symmetric quotient stacks of any smooth projective variety.

On the other hand, Khovanov introduced a categorification of the free boson Heisenberg algebra, i.e. the one associated to the rank 1 lattice. It is a monoidal category whose morphisms are described by a certain planar diagram calculus which categorifies the Heisenberg relations. A similar categorification was constructed by Cautis and Licata for the Heisenberg algebras of ADE type root lattices.

We show how to associate the Heisenberg 2-category to any smoooth and proper DG category and then define its Fock space 2-representation. This construction unifies all the results above and extends them to what can be viewed as the generality of arbitrary noncommutative smooth and proper schemes.