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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 ${}_R Mod$ 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 $Ab$-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.

• 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.

• 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 $\kappa$-presentable iff it has a $\kappa$-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.

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