Eilenberg-Watts calculus for finite categories and a bimodule Radford $S^4$ theorem

TL;DR

This paper develops Morita invariant versions of Eilenberg-Watts theorems for finite categories, enabling functor category switching, and generalizes Radford's $S^4$-theorem to bimodule categories, with applications to module categories and Serre functors.

Contribution

It introduces Morita invariant Eilenberg-Watts theorems for finite categories and extends Radford's $S^4$-theorem to bimodule categories, connecting functor categories and module structures.

Findings

Morita invariant Eilenberg-Watts theorems for finite categories

Equivalence between left and right exact functor categories

Generalization of Radford's $S^4$-theorem to bimodule categories

Abstract

We obtain Morita invariant versions of Eilenberg-Watts type theorems, relating Deligne products of finite linear categories to categories of left exact as well as of right exact functors. This makes it possible to switch between different functor categories as well as Deligne products, which is often very convenient. For instance, we can show that applying the equivalence from left exact to right exact functors to the identity functor, regarded as a left exact functor, gives a Nakayama functor. The equivalences of categories we exhibit are compatible with the structure of module categories over finite tensor categories. This leads to a generalization of Radford's $S^4$-theorem to bimodule categories. We also explain the relation of our construction to relative Serre functors on module categories that are constructed via inner Hom functors.