Symmetric Self-adjoint Hopf categories and a categorical Heisenberg double

We use the language of higher category theory to define what we call a symmetric self-adjoint Hopf (SSH) structure on a semisimple abelian category. SSH categories are the categorical analog of positive self-adjoint Hopf algebras studied by A.Zelevinsky. It follows from his work that for every positive self-adjoint Hopf algebra the Heisenberg double is equipped with a natural action on the algebra. We obtain categorical analogs of the Heisenberg double and its action from the SSH structure on a category in a canonical way. We exhibit the SSH structure on the category of polynomial functors. The categorical Heisenberg double in this case provides a categorification of the infinite dimensional Heisenberg algebra related to the categorification proposed by M. Khovanov.
The preprint is available on arXiv:1406.3973.