Spherical Morita contexts and relative Serre functors

TL;DR

This paper explores the structure of Morita contexts in finite tensor categories, relating duals, Serre functors, and Radford isomorphisms, with implications for topological field theory.

Contribution

It introduces a bicategorical Radford S^4 theorem, pivotal Morita equivalence, and the concept of spherical module categories, connecting these to topological quantum field theories.

Findings

Double duals expressed via relative Serre functors

Bicategorical Radford S^4 theorem established

Pivotal Morita equivalence characterized

Abstract

The Morita context provided by an exact module category over a finite tensor category gives a two-object bicategory with duals. Right and left duals of objects in the module category are given by internal Homs and coHoms, respectively. We express the double duals in terms of relative Serre functors, which leads to a Radford isomorphism for module categories. There is a bicategorical version of the Radford $S^4$ theorem: on the bicategory of a Morita context, the relative Serre functors assemble into a pseudo-functor, and the Radford isomorphisms furnish a trivialization of the square of this pseudo-functor, i.e. of the fourth power of the duals. We also show that the Morita bicategories coming from pivotal exact module categories are pivotal as bicategories, leading to the notion of pivotal Morita equivalence. This equivalence of tensor categories amounts to the equivalence of their bicategories of pivotal module categories. Furthermore, we introduce the notion of a spherical module category; it ensures that all categories in the Morita context of a spherical module category are spherical. Our results are motivated by and have applications to topological field theory.