2-C*-categories with non-simple units

TL;DR

This paper explores the structure of 2-C*-categories with non-simple units, showing they can be viewed as sections of Hilbert bundles and illustrating their relation to bundles of finite Hopf-algebras.

Contribution

It introduces a framework for understanding 2-C*-categories with non-simple units, including their fiber structure and examples involving Hopf-algebras.

Findings

2-arrows as continuous sections in Hilbert bundles

Description of fiber behavior in categorical structure

Example of 2-C*-category related to Hopf-algebra bundles

Abstract

We study the general structure of 2-C*-categories closed under conjugation, projections and direct sums. We do not assume units to be simple, i.e. for i_A the 1-unit corresponding to an object A, the space Hom(i_A, i_A) is a commutative unital C*-algebra. We show that 2-arrows can be viewed as continuous sections in Hilbert bundles and describe the behaviour of the fibres with respect to the categorical structure. We give an example of a 2-C*-Category giving rise to bundles of finite Hopf-algebras in duality. We make some remarks concerning Frobenius algebras and Q-systems in the general context of tensor C*-categories with non-simple units.