Generalized polynomial functors

We define Schur categories, ${\mathrm{\Gamma }}^d\mathcal{C}$, associated to a $\mathbb{k}$-linear category $\mathcal{C}$, over a commutative ring $\mathbb{k}$. The corresponding representation categories, $\mathbf{r}\mathbf{e}\mathbf{p}{\mathrm{\Gamma }}^d\mathcal{C}$, generalize categories of strict polynomial functors. Given a $\mathbb{k}$-superalgebra $A$, we show that for certain categories $\mathcal{V}={\mathcal{V}}_A$, ${\mathcal{E}}_A$ of $A$-supermodules, there is a Morita equivalence between $\mathbf{r}\mathbf{e}\mathbf{p}{\mathrm{\Gamma }}^d\mathcal{V}$ and the category of supermodules over a generalized Schur superalgebra of the form $S^A(m\mathrm{\mid }n,d)$ and $S^A(n,d)$, respectively. We also describe a formulation of generalized Schur-Weyl duality from the viewpoint of the category $\mathbf{r}\mathbf{e}\mathbf{p}{\mathrm{\Gamma }}^d{\mathcal{E}}_A$.