  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 O_d.
