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    • CommentRowNumber1.
    • CommentAuthorzskoda
    • CommentTimeApr 21st 2014

    I strongly disagree with the statement in Grothendieck category that the Grothendieck category is small. The main examples like RMod{}_R Mod are not! What did the writer of that line have in mind ?

    • CommentRowNumber2.
    • CommentAuthorZhen Lin
    • CommentTimeApr 21st 2014

    Probably “locally small”?

    • CommentRowNumber3.
    • CommentAuthorzskoda
    • CommentTimeApr 22nd 2014

    An AbAb-enriched category is by definition locally small – abelian group is not a class. So “abelian locally small” is also a nonsense. I think we should erase this and leave at that place within the definition simply abelian category.

    • CommentRowNumber4.
    • CommentAuthorTodd_Trimble
    • CommentTimeJun 11th 2017

    I added to the Properties section of Grothendieck category the observation that such admits an injective cogenerator (Theorem 9.6.3 of Kashiwara-Schapira).

    • CommentRowNumber5.
    • CommentAuthorDmitri Pavlov
    • CommentTimeJun 30th 2017
    • (edited Jan 12th 2021)

    Does the class of locally presentable Grothendieck categories coincide with the class of categories of sheaves of abelian groups on a site, i.e., categories of abelian groups in a Grothendieck topos?

    It seems to me that it is so, using the following argument:

    A locally presentable Grothendieck category can be defined as a locally presentable abelian category such that filtered colimits preserve finite limits.

    Any locally presentable abelian category is a reflective localization of the category of presheaves of abelian groups on a small category.

    The commutation of filtered colimits and finite limits implies that the reflection functor preserves finite limits.

    A left exact localization of the category of presheaves of abelian groups on a small category can be identified with the category of sheaves of abelian groups on the associated site (the covering sieves are those sieves that the reflector sends to an isomorphism).

    • CommentRowNumber6.
    • CommentAuthorHurkyl
    • CommentTimeJan 12th 2021

    Added that a Grothendieck category is locally presentable, by the Gabriel-Popescu theorem.

    diff, v22, current

    • CommentRowNumber7.
    • CommentAuthorRichard Williamson
    • CommentTimeJan 12th 2021
    • (edited Jan 12th 2021)

    [Edit: ignore this!]

    • CommentRowNumber8.
    • CommentAuthorDmitri Pavlov
    • CommentTimeJan 12th 2021

    Corrected the claim about local presentability, mentioned Vopěnka’s principle.

    diff, v23, current

    • CommentRowNumber9.
    • CommentAuthorHurkyl
    • CommentTimeJan 13th 2021

    Oh! I had seen it stated unconditionally that Grothendieck category implies locally presentable. See

    • The citation from wikipedia is appendix A of

    I hadn’t yet had a chance to fill out the sketched argument; is the issue about the smallness of the generator? Should the correct statement be something like a Grothendieck category is κ\kappa-presentable iff it has a κ\kappa-compact generator?

    • CommentRowNumber10.
    • CommentAuthorDmitri Pavlov
    • CommentTimeJan 13th 2021
    • (edited Jan 13th 2021)

    Re #9: If a Grothendieck abelian category has a small generator, then it is locally presentable.

    Concerning the cited paper by Jan Šťovíček, I do not see how to show in Lemma A.1 that G’ in his notation (the image of the embedding functor in the category of modules) is closed under κ-directed colimits for some κ.

    He refers to pages 198–199 in Stenström’s book, but these pages are from the section “Construction of Modules of Quotients”, and I cannot find any relevant statements there.

    The most straightforward way to establish this claim is to assume that the generator G is κ-small.

    But perhaps I am missing something. You may want to email the author directly.

    • CommentRowNumber11.
    • CommentAuthorMarc Hoyois
    • CommentTimeJan 13th 2021

    @Dmitri The version of Gabriel-Popescu in that book (Theorem 4.1 on page 220) states that any Grothendieck category is equivalent to a category of F-closed A-modules, where F is some collection of ideals in A and a module M is F-closed if Hom(A,M) → Hom(I,M) is an isomorphism for any ideal I ∈ F. Since A-modules are locally presentable, F-closed A-modules are clearly an accessible localization. So if this version of Gabriel-Popescu is correct, it implies that every Grothendieck category is locally presentable.

    • CommentRowNumber12.
    • CommentAuthorDmitri Pavlov
    • CommentTimeJan 13th 2021

    Re #11: I see, page 198 gives the definition of F-closed modules used in his version of the Gabriel–Popescu theorem.

    Indeed, it appears that this argument proves this claim unconditionally.

    I wonder if there is a more direct way to see this, without passing through the rather technical proof of Theorem 4.1 on page 220.

    • CommentRowNumber13.
    • CommentAuthorzskoda
    • CommentTimeJan 14th 2021

    I added a comprehensive and reliable survey of Grothendieck categories by Garkusha which also mentions some connections to model theory.

    • CommentRowNumber14.
    • CommentAuthorDmitri Pavlov
    • CommentTimeJan 27th 2021

    Added a reference to Krause’s proof that all Grothendieck categories are locally presentable (without Vopěnka’s principle).

    Added a reference to a generalization for κ-filtered colimits.

    diff, v25, current

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