A Categorical Approach to Additive Combinatorics

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 category and in one of its interesting subcategories. Moreover, we study the additive structure of these (co)limit objects using the additive doubling constant. We relate this category to that of finite sets and mappings, and to 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 an adjunction.