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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 ?
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 κ-presentable iff it has a κ-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.
added also pointer to:
for the fact that Grothendieck abelian categories are locally presentable.
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:
added pointer to:
for proof that RMod is Grothendieck abelian internal to any Grothendieck topos (this example)
Two bicategories of k-linear Grothendieck categories as a bicategories of fractions
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