Bundles of C∗-categories

We introduce the notions of multiplier C∗-category and continuous bundle of C∗-categories, as the categorical analogues of the corresponding C∗-algebraic notions. Every symmetric tensor C∗-category with conjugates is a continuous bundle of C∗-categories, with base space the spectrum of the C∗-algebra associated with the identity object. We classify tensor C∗-categories with fibre the dual of a compact Lie group in terms of suitable principal bundles. This also provides a classification for certain C∗-algebra bundles, with fibres fixed-point algebras of Od.