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Let $A$ be an abelian category that has a generator and satisfies the AB4 axiom. I would like to understand (better) the relations between various additional "restrictions" on $A$.

So here is my list of additional "axioms":

(1) $A$ is an AB5 category (and so, Grothendieck abelian).

(2) There is an exact conservative functor from $A$ into abelian groups that respects coproducts (and so, colimits).

(3) $A$ has an injective cogenerator, satisfies the AB3* axiom, and sending an object of $A$ into the functor it corepresents on the category $Inj A$ (of injective objects of $A$) gives an equivalence of $A$ with the category of those functors from $Inj A$ into abelian groups that respect products.

In particular, I wonder whether condition (2) implies (3) and whether (3) implies (1). If these implications do not hold, are there any "natural" additional conditions that could be added to make them valid?

Also, does condition (3) (possibly, combined with (2)) have any "nice" reformulations or consequences? Does it possess any "natural strengthenings"?

Any hints would be very welcome! Is there any text where I can read about these matters ("classical" books on abelian categories do not help much)?

My motivation comes from the study of the heart of compactly generated $t$-structures (and more generaly, $t$-structures of finite type); $Inj A$ is related to the right adjacent weight structure, and condition (2) comes from the existence of a "left orthogonal" weight structure satisfying "nice conditions".

P.S. Professor Rickard has given a very nice example demonstrating that (3) does not imply (1). Certainly, any other enlightening examples would also be very welcome!

P.P.S. Certainly, a functor (of abelian categories) is conservative and exact if and only if it is faithful exact.:)

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  • $\begingroup$ Can you give a piece of motivation / context for this question? I guess that it would invite more people to think about it. (Although personally I'm also happy to think about these questions without further context.) $\endgroup$ – HeinrichD Nov 26 '16 at 21:56
  • $\begingroup$ Well, I am studying hearts of $t$-structrures in triangulated categories closed with respect to coproducts. The AB4 condition and the existence of a generator can be proved under very "natural" restrictions; the other conditions are more difficult (still I can prove them under certain restrictions). So, I wonder whether my results are "independent" and also what other conditions for $A$ would be reasonable to look at. $\endgroup$ – Mikhail Bondarko Nov 26 '16 at 22:01
  • $\begingroup$ Do you know an example of where 2 fails? $\endgroup$ – arsmath Nov 27 '16 at 11:28
  • $\begingroup$ What about sheaves in Grothendieck topologies not having enough points? Actually, my "triangulated" method of producing abelian categories does not give much information on them ("easily"); so I wonder which statement of this sort are already "known". $\endgroup$ – Mikhail Bondarko Nov 27 '16 at 11:34
  • $\begingroup$ Concerning abelian category properties of t-structure hearts, and specifically the question when they are Grothendieck categories, you may wish to look into the paper arxiv.org/abs/1708.07540 $\endgroup$ – Leonid Positselski Nov 7 '17 at 20:13
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I don't think (3) implies (1).

For example, the opposite category of the category of abelian groups satisfies (3), but is not AB5.

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Obviously, (2) implies (1). Indeed, if directed colimits are preserved by a conservative exact functor taking values in a category where they are exact, then they are exact in the source category.

Also, (1) implies (3). Indeed, by Grothendieck's theorem, any Grothendieck abelian category has enough injective objects. Taking the coproduct of all quotients of a generator and embedding it into an injective object produces an injective cogenerator.

Further, as it follows from Freyd's Special Adjoint Functor (Existence) Theorem, any cocomplete co-wellpowered category with a generator is complete. In particular, any abelian category satisfying Ab3 and having a generator satisfies Ab3*.

Finally, for any abelian categories $A$ and $B$ such that $A$ has enough injectives, any additive functor $F\colon Inj A\to B$ can be uniquely extended to a left exact functor $G\colon A\to B$. You just present an arbitrary object $X\in A$ as the kernel of a morphism between two injectives $f\colon I\to J$, so $X=\ker f$, and put $G(X)=\ker F(f)$. The functor $G$ preserves limits if and only if the functor $F$ preserves products (notice that kernels always commute with products).

Now, any complete well-powered additive category with a cogenerator, and in particular any abelian category satisfying Ab3* and having a cogenerator, is anti-equivalent to the category of limit-preserving functors from it to the category of abelian groups. This is, once again, a reformulation of (the additive case of) Freyd's adjoint functor existence theorem.

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