Categorical aspects of compact quantum groups

TL;DR

This paper explores the categorical structure of compact quantum groups, demonstrating the existence of various universal constructions and characterizing morphisms within the category.

Contribution

It introduces new categorical constructions for compact quantum groups and characterizes morphisms, unifying and extending existing quantum group theory.

Findings

Existence of universal constructions via adjoint functor theorem

Recovery of known quantum group constructions like quantum Bohr compactification

Introduction of new constructions such as coproducts and free quantum groups

Abstract

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 recovers constructions which have appeared in the literature, such as the quantum Bohr compactification of a locally compact semigroup, and provides new ones, such as the coproduct of a family of compact quantum groups, and the compact quantum group freely generated by a locally compact quantum space. In addition, we characterize epimorphisms and monomorphisms in the category of compact quantum groups.