nLab Gabriel-Popescu theorem

Idea

For Grothendieck categories

The Gabriel–Popescu theorem [Popescu & Gabriel 1964] asserts that every Grothendieck category is equivalent to a reflective subcategory of a category of modules over a unital ring, where the left adjoint localization functor is exact (one calls such subcategories of module categories Giraud subcategories). The fully faithful right adjoint is then also called the section functor.

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

For triangulated categories

An analogous statement holds for triangulated categories [Rota 2010]:

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.

References

The original article:

• Nicolae Popescu, Pierre Gabriel: Caractérisation des catégories abéliennes avec générateurs et limites inductives exactes, Les Comptes rendus de l’Académie des sciences Paris 258 (1964) 4188-4190 [MR 0166241 BnF]

Other accounts:

• Mitsuhiro Takeuchi: A simple proof of Gabriel and Popesco’s theorem, J. Alg. 18 (1971) 112-113 (1971) [pdf]

• Nicolae Popescu: Abelian categories with applications to rings and modules, London Math. Soc. Monographs 3, Academic Press (1973) [MR0340375]

• Barry Mitchell: A quick proof of the Gabriel-Popesco theorem, Journal of Pure and Applied Algebra

  20 3 (1981) 313-315 [doi:10.1016/0022-4049(81)90065-7]

See also:

• Wikipedia: Gabriel-Popesco theorem

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 analogous statement for triangulated categories:

• 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