Categorically proper homomorphisms of topological groups

TL;DR

This paper generalizes concepts of c-compact and h-complete topological groups to the morphism level, analyzing their stability properties and comparing them with classical topological notions, often without assuming Hausdorffness.

Contribution

It introduces and studies the properties of categorically proper homomorphisms of topological groups, extending existing notions to a broader context.

Findings

Closure under direct products for the new classes

Comparison with classical topological properties

Results valid without assuming Hausdorffness

Abstract

We extend the Dikranjan-Uspenskij notions of c-compact and h-complete topological group to the morphism level, study the stability properties of the newly defined types of maps, such as closure under direct products, and compare them with their counterparts in topology. We assume Hausdorffness only when our proofs require us to do so, which leads to new results and the affirmation of some facts that were known in a Hausdorff context.