The notion of left (resp. right) regular object of a tensor C*-category
equipped with a faithful tensor functor into the category of Hilbert spaces is
introduced. If such a category has a left (resp. right) regular object, it can
be interpreted as a category of corepresentations (resp. representations) of
some multiplicative unitary. A regular object is an object of the category
which is at the same time left and right regular in a coherent way. A category
with a regular object is endowed with an associated standard braided symmetry.
Conjugation is discussed in the context of multiplicative unitaries and their
associated Hopf C*-algebras. It is shown that the conjugate of a left regular
object is a right regular object in the same category. Furthermore the
representation category of a locally compact quantum group has a conjugation.
The associated multiplicative unitary is a regular object in that category.Comment: 48 pages, Late
