  We associate a graded monoidal supercategory $\mathcal{H}\mathit{eis}_{F,k}$
to every graded Frobenius superalgebra $F$ and integer $k$. These categories,
which categorify a broad range of lattice Heisenberg algebras, recover many
previously defined Heisenberg categories as special cases. In this way, the
categories $\mathcal{H}\mathit{eis}_{F,k}$ serve as a unifying and generalizing
framework for Heisenberg categorification. Even in the case of previously
defined Heisenberg categories, we obtain new, more efficient, presentations of
these categories, based on an approach of Brundan. When $k=0$, our construction
yields new versions of the affine oriented Brauer category depending on a
graded Frobenius superalgebra.
