Module-theoretic approach to dualizable Grothendieck categories

Ryo Kanda

Module-theoretic approach to dualizable Grothendieck categories

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arxiv 2405.16468 v1 pith:2W6LCT77 submitted 2024-05-26 math.CT math.RAmath.RT

Ryo Kanda This is my paper

classification math.CTmath.RAmath.RT

keywords grothendieckcategorydualizableapproachcategoriesclasslinearmodule-theoretic

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We prove that every dualizable Grothendieck category whose dual is again a Grothendieck category satisfies Grothendieck's conditions Ab6 and Ab4*, by taking a module-theoretic approach based on the Gabriel-Popescu embedding. Combining this with a result by Stefanich, we conclude that the class of dualizable linear cocomplete categories is precisely the class of linear Grothendieck category satisfying Ab6 and Ab4*. This provides a complete answer to a modified conjecture on the dualizability, originally posed by Brandenburg, Chirvasitu, and Johnson-Freyd.

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Dualizable Additive Categories
math.AT 2026-08 conditional novelty 8.0 of 10

Dualizable additive categories are characterized intrinsically and via almost modules, and a universal finitary localizing invariant (prestable motives) is constructed whose unit corepresents algebraic K-theory.

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