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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 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 Ab-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 13th 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

    • https://en.wikipedia.org/wiki/Grothendieck_category
    • The citation from wikipedia is appendix A of https://arxiv.org/abs/1005.3251

    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-presentable iff it has a κ-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

    • CommentRowNumber15.
    • CommentAuthorvarkor
    • CommentTimeSep 14th 2022

    Mention that Grothendieck categories may be seen as enriched topoi.

    diff, v27, current

    • CommentRowNumber16.
    • CommentAuthorUrs
    • CommentTimeApr 18th 2023

    added also pointer to:

    for the fact that Grothendieck abelian categories are locally presentable.

    diff, v31, current

    • CommentRowNumber17.
    • CommentAuthorUrs
    • CommentTimeApr 18th 2023
    • (edited Apr 18th 2023)

    made more explicit that categories of vector spaces are an example (here)

    added a reference for the claim that categories of unbounded chain complexes in a Grothendieck abelian category are themselves Grothendieck abelian:

    diff, v31, current

    • CommentRowNumber18.
    • CommentAuthorUrs
    • CommentTimeApr 18th 2023

    added pointer to:

    for proof that RMod is Grothendieck abelian internal to any Grothendieck topos (this example)

    diff, v31, current

    • CommentRowNumber19.
    • CommentAuthorzskoda
    • CommentTimeSep 15th 2024

    Two bicategories of k-linear Grothendieck categories as a bicategories of fractions

    • J. Ramos González, Grothendieck categories as a bilocalization of linear sites, Appl Categor Struct 26, 717–745 (2018) doi

    diff, v34, current