A monoidal analogue of the 2-category anti-equivalence between ABEX and DEF

TL;DR

This paper establishes a deep duality between certain monoidal structures on small abelian categories and definable additive categories, revealing new correspondences and topological characterizations in category theory.

Contribution

It introduces a monoidal analogue of a known 2-category anti-equivalence, linking abelian categories with definable additive categories under monoidal structures.

Findings

Bijections between definable subcategories and tensor-ideals.

Characterization of these subcategories via a Ziegler-type topology.

Elementary duality induces correspondences between subcategories and tensor-ideals.

Abstract

We prove that the 2-category of skeletally small abelian categories with exact monoidal structures is anti-equivalent to the 2-category of fp-hom-closed definable additive categories satisfying an exactness criterion. For a fixed finitely accessible category $\mathcal{C}$ with products and a monoidal structure satisfying the appropriate assumptions, we provide bijections between the fp-hom-closed definable subcategories of $\mathcal{C}$, the Serre tensor-ideals of $\mathcal{C}^{\mathrm{fp}}\hbox{-}\mathrm{mod}$ and the closed subsets of a Ziegler-type topology. For a skeletally small preadditive category $\mathcal{A}$ with an additive, symmetric, rigid monoidal structure we show that elementary duality induces a bijection between the fp-hom-closed definable subcategories of $\mathrm{Mod}\hbox{-}\mathcal{A}$ and the definable tensor-ideals of $\mathcal{A}\hbox{-}\mathrm{Mod}$.