A categorical approach to additive combinatorics

TL;DR

This paper develops a categorical framework for additive combinatorics, representing additive sets and Freiman homomorphisms as a category, and explores its structure, limits, colimits, and connections to classical algebraic concepts.

Contribution

It introduces a novel categorical formulation of additive combinatorics, including the construction of a category of additive sets and homomorphisms, and relates it to existing algebraic categories.

Findings

Categorical structure of additive sets and Freiman homomorphisms established

Limit and colimit constructions analyzed within the category

Connection to classical results like the universal ambient group by Konyagin and Lev

Abstract

Motivated by the definition of Freiman homomorphism, we explore the possibilities of formulating some basic notions and techniques of additive combinatorics in a categorical language. We show that additive sets and Freiman homomorphisms form a category and we study several limit and colimit constructions in this, and in an interesting subcategory of this category. Moreover, we study the additive structure of these (co)limit objects using additive doubling constant. We relate this category to that of finite sets and mappings, and that of abelian groups and group homomorphisms. We show that the Konyagin and Lev result on the existence of universal ambient groups is an instance of adjunction