Grothendieck-Verdier module categories, Frobenius algebras and relative Serre functors

TL;DR

This paper develops a theory of module categories over Grothendieck-Verdier categories, introducing new structures like relative Serre functors and Frobenius algebras, extending duality concepts beyond rigidity.

Contribution

It introduces a framework for module categories over Grothendieck-Verdier categories, including the concepts of subcategories, internal End algebras, and relative Serre functors, expanding duality theory.

Findings

Internal End of a generator is a Frobenius algebra.

Relative Serre functor provides an equivalence between subcategories.

Internal End objects have structures related to Frobenius algebras.

Abstract

We develop the theory of module categories over a Grothendieck-Verdier category, i.e. a monoidal category with a dualizing object and hence a duality structure more general than rigidity. Such a category C comes with two monoidal structures which are related by non-invertible morphisms and which we treat on an equal footing. Quite generally, non-invertible structure morphisms play a dominant role in this theory. In any Grothendieck-Verdier module category M we find two important subcategories M' and M''. The internal End of an object in M' that is a C-generator is an algebra such that its category of modules is equivalent to M as a module category. We also introduce a partially defined relative Serre functor S which furnishes an equivalence between M' and M''. Any isomorphism between an object m of M' and S(m) in M'' endows the internal End of m with the structure of a Grothendieck-Verdier Frobenius algebra.