# Epimorphic quantum subgroups and coalgebra codominions

## TL;DR

This paper explores categorical properties of coalgebras, bialgebras, and Hopf algebras, providing new characterizations and invariance results that generalize classical algebraic group theorems.

## Contribution

It offers novel categorical characterizations of monomorphisms, epimorphisms, and dominions in coalgebra-related categories, extending classical algebraic results to quantum and categorical contexts.

## Key findings

- Characterization of monomorphisms in coalgebra categories
- Invariance of codominions under field extension
- Surjections are regular epimorphisms when codomain is cosemisimple

## Abstract

We prove a number of results concerning monomorphisms, epimorphisms, dominions and codominions in categories of coalgebras. Examples include: (a) representation-theoretic characterizations of monomorphisms in all of these categories that when the Hopf algebras in question are commutative specialize back to the familiar necessary and sufficient conditions (due to Bien-Borel) that a linear algebraic subgroup be epimorphically embedded; (b) the fact that a morphism in the category of (cocommutative) coalgebras, (cocommutative) bialgebras, and a host of categories of Hopf algebras has the same codominion in any of these categories which contain it; (c) the invariance of the Hopf algebra or bialgebra (co)dominion construction under field extension, again mimicking the well-known corresponding algebraic-group result; (d) the fact that surjections of coalgebras, bialgebras or Hopf algebras are regular epimorphisms (i.e. coequalizers) provided the codomain is cosemisimple; (e) in particular, the fact that embeddings of compact quantum groups are equalizers in the category thereof, generalizing analogous results on (plain) compact groups; (f) coalgebra-limit preservation results for scalar-extension functors (e.g. extending scalars along a field extension $\Bbbk\le \Bbbk'$ is a right adjoint on the category of $\Bbbk$-coalgebras).