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We show that the Grothendieck groups of the categories of finitely-generatedgraded supermodules and finitely-generated projective graded supermodules overa tower of graded superalgebras satisfying certain natural conditions give riseto twisted Hopf algebras that are twisted dual. Then, using induction andrestriction functors coming from such towers, we obtain a categorification ofthe twisted Heisenberg double and its Fock space representation. We show thattowers of wreath product algebras (in particular, the tower of Sergeevsuperalgebras) and the tower of nilcoxeter graded superalgebras satisfy ouraxioms. In the latter case, one obtains a categorification of the quantum Weylalgebra.

Towers of graded superalgebras categorify the twisted Heisenberg double