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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 ?
Probably “locally small”?
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.
I added to the Properties section of Grothendieck category the observation that such admits an injective cogenerator (Theorem 9.6.3 of Kashiwara-Schapira).
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).
[Edit: ignore this!]
Oh! I had seen it stated unconditionally that Grothendieck category implies locally presentable. See
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?
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.
@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.
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.
I added a comprehensive and reliable survey of Grothendieck categories by Garkusha which also mentions some connections to model theory.
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