Categorical Aspects of Compact Quantum Groups

We show that either of the two reasonable choices for the category of compact quantum groups is nice enough to allow for a plethora of universal constructions, all obtained “by abstract nonsense” via the adjoint functor theorem. This approach both produces new objects (such as the coproduct of a family of compact quantum groups or the compact quantum group freely generated by a locally compact quantum space) and recovers in a uniform setting constructions which have appeared in the literature, such as the quantum Bohr compactification of a locally compact semigroup. We also provide Tannakian descriptions of these universal constructions, and characterize epimorphisms and monomorphisms in the category of compact quantum groups.