Generalized polynomial functors

TL;DR

This paper introduces Schur categories associated with linear categories over a ring, generalizes polynomial functor categories, and establishes Morita equivalences and Schur-Weyl duality in the superalgebra context.

Contribution

It defines Schur categories for linear categories, generalizes polynomial functor categories, and connects these to superalgebras via Morita equivalence and duality.

Findings

Established Morita equivalences between representation categories and supermodules.

Formulated a generalized Schur-Weyl duality for superalgebras.

Extended polynomial functor theory to superalgebra settings.

Abstract

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