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 RMod{}_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.


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

An analogue in triangulated setup: every triangulated category which is well generated in the sense of Amnon Neeman and algebraic (over ground ring kk) in the sense of Bernhard Keller is kk-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:

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

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