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Motivated by the work of of A. Zelevinsky on positive self-adjoint Hopfalgebras, we define what we call a symmetric self-adjoint Hopf structure for acertain kind of semisimple abelian categories. It is known that every positiveself-adjoint Hopf algebra admits a natural action of the associated Heisenbergdouble. We construct canonical morphisms lifting the relations that define thisaction on the algebra level and define an object that we call a categoricalHeisenberg double that is a natural setting for considering these morphisms. Asexamples, we exhibit the symmetric self-adjoint Hopf structure on thecategories of polynomial functors and equivariant polynomial functors. In thecase of the category of polynomial functors we obtain categorification of theFock space representation of the infinite-dimensional Heisenberg algebra.

Symmetric self-adjoint Hopf categories and a categorical Heisenberg double