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The aim is to formulate basic concepts and techniques of additive combinatorics using categorical language.

A Categorical Approach to Additive Combinatorics

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The aim is to formulate basic concepts and techniques of additive combinatorics using categorical language.
Defined categories for additive sets and Freiman homomorphisms.
Studied limit and colimit constructions within this category.
Analyzed subcategories and their additive structures using the additive doubling constant.
Related the categorical framework to finite sets and abelian groups.
Demonstrated that additive sets and Freiman homomorphisms construct a category.
Identified various limit and colimit objects and their additive properties.
Related the existence of universal ambient groups to adjunction in the categorical context.

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This categorical framework reveals structures in additive combinatorics, highlighting relationships among groups.

Blanco et al. (2026) studied this question.

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