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

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 $\mathcal{A}\hbox{-}\mathrm{Mod}$.