Pivotal tricategories and a categorification of inner-product modules

TL;DR

This paper develops a tricategory framework for bimodule categories over finite tensor categories, introducing inner-product structures and dualities relevant to topological field theories.

Contribution

It introduces a tricategory of inner-product bimodule categories over pivotal tensor categories, incorporating dualities and pivotal structures, advancing the categorification of inner-product modules.

Findings

Finite bimodule categories form a tricategory.

Inner-product bimodule categories have dualities and pivotal structures.

Inner-product bimodule categories relate to Frobenius algebras and $*$-Morita equivalence.

Abstract

This article investigates duals for bimodule categories over finite tensor categories. We show that finite bimodule categories form a tricategory and discuss the dualities in this tricategory using inner homs. We consider inner-product bimodule categories over pivotal tensor categories with additional structure on the inner homs. Inner-product module categories are related to Frobenius algebras and lead to the notion of $*$-Morita equivalence for pivotal tensor categories. We show that inner-product bimodule categories form a tricategory with two duality operations and an additional pivotal structure. This is work is motivated by defects in topological field theories.