Towers of graded superalgebras categorify the twisted Heisenberg double

We show that the Grothendieck groups of the categories of finitely-generated graded supermodules and finitely-generated projective graded supermodules over a tower of graded superalgebras satisfying certain natural conditions give rise to twisted Hopf algebras that are twisted dual. Then, using induction and restriction functors coming from such towers, we obtain a categorification of the twisted Heisenberg double, introduced in [9], and its Fock space representation. We show that towers of wreath product algebras (in particular, the tower of Sergeev superalgebras) and the tower of nilCoxeter graded superalgebras satisfy our axioms. In the latter case, one obtains a categorification of the quantum Weyl algebra.