A categorical approach to additive combinatorics
Saúl A. Blanco, Esfandiar Haghverdi
TL;DR
The paper introduces two categories of additive sets and Freiman k-homomorphisms, FR_k and FR_k^0, to bring additive combinatorics into a categorical setting and studies their limits/colimits. It shows that the universal ambient group construction (and related Pontryagin-duality-based viewpoints) arise as adjunctions, connecting to finite-set and abelian-group contexts. The work establishes a complete and well-behaved normalized subcategory FR_k^0, with pullbacks, pushouts, equalizers, and coequalizers, while FR_k lacks some of these. These results lay groundwork for applying categorical tools (and possibly homological methods) to Freiman-type structure theorems and additive combinatorics problems, offering a framework for future deeper theorems and cross-pollination with other disciplines.
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