nLab Gabriel-Popescu theorem

Giraud’s subcategories and Gabriel-Popescu theorem

The Gabriel–Popescu theorem, also quoted as Popesco–Gabriel theorem because of that spelling in the original published paper (cf. the spelling in that order, which is sometimes considered wrong when in the order of the authors!) asserts that every Grothendieck category can be represented as a reflective subcategory of a category ${}_R Mod$ of modules over a unital ring, where the localization functor is exact (one calls such subcategories of module categories Giraud subcategories). The right adjoint (reflection) is in this setup sometimes called the section functor.

This is the Ab-enriched analogue of the fact that sheaf toposes are equivalently the left exact reflective subcategories of presheaf toposes. See a more general statement at enriched sheaf.

Literature

• Nicolae Popesco, Pierre Gabriel, Caractérisation des catégories abéliennes avec générateurs et limites inductives exactes, Les Comptes rendus de l’Académie des sciences Paris 258: 4188–4190 MR 0166241
• wikipedia Gabriel-Popesco theorem
• Nicolae Popescu, Abelian categories with applications to rings and modules, London Math. Soc. Monographs 3, Academic Press 1973. xii+467 pp. MR0340375
• Mitsuhiro Takeuchi, A simple proof of Gabriel and Popesco’s theorem, J. Alg. 18, 112-113 (1971) pdf

There are various generalizations, e.g.

• Wendy Lowen, A generalization of the Gabriel-Popescu theorem, Journal of Pure and Applied Algebra 190 (1) (2004): 197–211, doi MR2043328

N. Kuhn had related results in the study of Steenrod algebra, cf. also

• Nicholas J. Kuhn, Generic representations of the finite general linear groups and the Steenrod algebra. I, Amer. J. Math. 116 (2): 327-360 (1994) doi
• Jacob Lurie, A theorem of Gabriel-Kuhn-Popesco, MIT math 917 notes, lecture 8

An analogue in triangulated setup: every triangulated category which is well generated in the sense of Amnon Neeman and algebraic (over ground ring $k$) in the sense of Bernhard Keller is $k$-linearly triangle equivalent to a localization of the derived category of a small pretriangulated dg-category by a localizing subcategory generated by a set of objects:

• Marco Porta, The Popescu-Gabriel theorem for triangulated categories, Adv. Math. 225 (2010) 1669–1715 doi

In the setup of (pretriangulated) dg-categories and t-structures

• Francesco Genovese, Julia Ramos González, A derived Gabriel–Popescu theorem for t-structures via derived injectives, Intern. Math. Res. Notices 2023:6 (2023) 4695–4760 doi arXiv:2105.02561

• Francesco Genovese, A derived Gabriel-Popescu Theorem for t-structures via derived injectives, talk at Toposes Online 2021 yt