Grothendieck-Verdier duality in categories of bimodules and weak module functors

TL;DR

This paper explores Grothendieck-Verdier duality in monoidal categories, focusing on bimodules and weak module functors, and investigates conditions under which certain distributors are isomorphisms, with concrete examples.

Contribution

It introduces a framework for understanding duality structures in categories of bimodules using weak module functors and internal Homs, extending existing theory.

Findings

Distributors in Grothendieck-Verdier categories can be isomorphisms under certain conditions.

Concrete examples are provided in categories of bimodules over associative algebras.

The perspective of module categories over monoidal categories clarifies duality structures.

Abstract

Various monoidal categories, including suitable representation categories of vertex operator algebras, admit natural Grothendieck-Verdier duality structures. We recall that such a Grothendieck-Verdier category comes with two tensor products which should be related by distributors obeying pentagon identities. We discuss in which circumstances these distributors are isomorphisms. This is achieved by taking the perspective of module categories over monoidal categories, using in particular the natural weak module functor structure of internal Homs and internal coHoms. As an illustration, we exhibit these concepts concretely in the case of categories of bimodules over associative algebras.