Dualizable locally presentable linear categories are Roos categories

Prove that every dualizable locally presentable linear category is a Grothendieck category satisfying Grothendieck’s conditions Ab6 and Ab4*, that is, a Roos category.

Background

Brandenburg, Chirvasitu, and Johnson-Freyd conjectured that dualizable locally presentable linear categories are strongly generated by compact projective objects, but subsequent work produced a counterexample to that claim. Motivated by these developments, the paper proposes a modified conjecture asserting that dualizability forces the category to be a Roos category, i.e., a Grothendieck category satisfying Ab6 and Ab4*.

The authors develop a module-theoretic approach via the Gabriel-Popescu embedding and, combined with Stefanich’s results, aim to confirm this modified conjecture by identifying dualizable linear cocomplete categories precisely with Roos categories.