The dual of a compact group can be characterized at abstract levels, as a category of endomorphism of a C* algebra or as an abstract tensor category, independently of any assumption about the existence of a representation functor in the category of Hilbert spaces. But no satisfactory proper analog of this result yet exists for compact quantum groups.

However if we consider the tensor category generated by the regular representation, it can be characterized intrinsecally at the same abstract levels for locally compact groups, quantum groups, actually more in general for multiplicative unitaries; the C*-algebra involved is automatically simple and is moreover separable if and only if the intrinsic dimension of the regular representation is countable.

(Joint work with Claudia Pinzari and John E.Roberts, IJM to appear).