Notes on duality theories of abelian groups

TL;DR

This paper explores duality theories of abelian groups with a focus on categorical and topological aspects, analyzing the evaluation homomorphism and properties of special groups like LCA, compact, and discrete groups.

Contribution

It provides an initial, detailed study of the evaluation homomorphism in abelian groups, emphasizing duality, quasi-convexity, and properties of specific classes of groups.

Findings

Analysis of the evaluation homomorphism A --> A^^ in abelian groups

Insights into quasi-convexity and properties of special groups

Preliminary results on duality theories for LCA groups

Abstract

In this notebook, I present duality theory (or theories) of abelian groups with some categorical and categorical topological flavour. I consider writing this notebook as a longer-term project, and its current content and presentation is "under development." In other words, all questions, comments, suggestions, and criticism is more than usually welcome. In Chapter 1, which is the only more-or-less ready part, the evaluation homomorphism A --> A^^ (of a group A into its bidual A^^) is studied, with particular attention to quasi-convexity and special groups (compact, discrete, LCA) and subgroups (open, compact). The treatment of LCA groups is incomplete, but the ultimate goal is to make it as self-contained as possible.