2-C∗-categories with non-simple units

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 ιA the 1-unit corresponding to an object A, the space Hom(ιA,ιA) 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.