# Objects of which Grothendieck abelian categories have elements?

The Freyd-Mitchell embedding theorem is a very useful tool for dealing with small abelian categories. However, it does not allow to use "elements" of objects of an abelian category $A$ in those statements that involve "infinite constructions".

So I wonder: for which Grothendieck abelian $A$ (this certainly implies that $A$ is not small if it is non-zero) there exists an exact conservative functor $F$ into abelian groups (this is a certain weak substitute of "having elements")? Does the Gabriel-Popescu theorem help here (so, what can one say if $A$ is described as a "nice" localization of certain category of modules)?

Also, how would you call a functor $F$ possessing this property; does "a stalk functor" sound fine? What is the relation of the existence of $F$ condition to the existence of compact generators for $D(A)$? If $A$ is a category of sheaves for certain Grothendieck topology then can one relate $F$ to the points of this topology?

Any hints, references or examples are very welcome!

• What I know is that there is an analog of Freyd-Mitchell for elementary toposes which does not require any smallness. It is called the Barr cover I think, you embed your topos into double negation sheaves on the (closed) complement of the element "true" in the subobject classifier. – მამუკა ჯიბლაძე Dec 6 '16 at 11:23
• If the idea is to reason about "infinite constructions" as though in a category of modules, then why don't you require your embedding to preserve certain (co)limits? – Tim Campion Feb 26 '18 at 9:18
• Yes, this is what I want to be true.:) So, which results in this direction are known? – Mikhail Bondarko Feb 27 '18 at 15:09

I am not an expert of the abelian world, but I think I can answer.

Since my background is not precisely abelian, I will start with an example in category theory.

Thm. Let $$\mathcal{K}$$ be a locally $$\kappa$$-presentable category, then there is a faithful and conservative functor to Set that preserves $$\kappa$$-directed colimits.

Proof. Since $$\mathcal{K}$$ is a locally $$\kappa$$-presentable category, it has a strong generator $$\mathcal{G}$$ made by $$\kappa$$-presentable objects. Then, the functor $$\coprod_{G \in \mathcal{G}} \text{hom}_{\mathcal{K}}(G, \_)$$ is faithful and conservative. Moreover it preserves $$\kappa$$-directed colimits and connected limits.

It looks to me that with the same argument one can prove the following statement.

Thm. Let $$\mathcal{A}$$ be an abelian category with a strong projective generator, then it has an exact, faithful and conservative functor into $$\mathbb{Ab}$$. Moreover, if the generator is made by $$\kappa$$-presentable objects, the functor preserves $$\kappa$$-directed colimits.

Observe that the functor is left exact because $$\mathbb{Ab}$$ is AB4.

Moreover, the following holds.

Thm. Let $$\mathcal{A}$$ be a Grothendieck category with a faithful and conservative functor in $$\mathbb{Ab}$$ which is accessible, exact and preserve connected limits. Then $$\mathcal{A}$$ has a projective strong generator.

Proof. Call $$F$$ such a functor, then it must have a multiadjoint. This is equivalent to the fact that $$F \cong \coprod_{G \in \mathcal{G}} \text{hom}_{\mathcal{A}}(G, \_)$$ for a small family $$\mathcal{G}$$. Since $$F$$ is faithful and conservative, $$\mathcal{G}$$ must be a strong generator.

• The main shortcoming, I think, is that this functor will typically not be right exact. – Tim Campion Sep 27 '18 at 22:04
• I edited, does it help? – Ivan Di Liberti Sep 27 '18 at 22:10
• Sure. I suppose that's the best one can hope for. – Tim Campion Sep 27 '18 at 22:12
• $F \cong \text{hom}_{\mathbb{Ab}}(\mathbb{Z}, F) \cong \coprod_{G \in \mathcal{G}(\mathbb{Z})} \text{hom}_{\mathcal{A}}(G, \_)$ – Ivan Di Liberti Sep 28 '18 at 7:10
• If there are enough points (and this is certainly the case for any sort of "ordinary" sheaves) then the direct sum of stalks is an exact conservative functor into abelian groups; yet this functor is only ind-corepresentable. – Mikhail Bondarko Sep 28 '18 at 17:45