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

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 ζ with products and a monoidal structure satisfying the appropriate assumptions, we provide bijections between the fp-hom-closed definable subcategories of ζ, the Serre tensor-ideals of ζfp - mod and the closed subsets of a Ziegler-type topology.

For a skeletally small preadditive category A with an additive, symmetric, rigid monoidal structure we show that elementary duality induces a bijection between the fp-hom-closed definable subcategories of Mod - A and the definable tensor-ideals of Mod - A.