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

Develops a theory of module categories over Grothendieck–Verdier categories, showing that suitable module categories can be realised as modules over an internal End algebra. A partially defined relative Serre functor then gives such algebras a Grothendieck–Verdier Frobenius structure.

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.

Recommended citation: J. Fuchs, G. Schaumann, C. Schweigert and S. Wood, Adv. Math. 475, 110325 (2025)
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