(AB1) pre-abelian category
(AB2) abelian category
(AB5) Grothendieck category
It is easy to see that not every abelian category is equivalent to Mod for some ring . The reason is that has all small limits and colimits. But for instance the category of finitely generated -modules is an abelian category but lacks these properties.
However, we have
This result can be found as Theorem 7.34 on page 150 of (Freyd). (The terminology there is a bit outdated, in that it calls an abelian category “fully abelian” if it admits a full and faithful exact functor to a category of -modules.) A pedagogical discussion is in section 1.6 of (Weibel). See also (Wikipedia) for the idea of the proof.
We can also characterize which abelian categories are equivalent to a category of -modules:
In fact, in this situation we can take where is any compact projective generator. Conversely, if , then has all small coproducts and is a compact projective generator.
This theorem, minus the explicit description of , can be found as Exercise F on page 103 of (Freyd). The first part of this theorem can also be found as Prop. 2.1.7 in (Ginzburg). Conversely, it is easy to see that is a compact projective generator of .
Going further, we can try to characterize functors between categories of -modules that come from tensoring with bimodules. Here we have
If is an an --bimodule, the tensor product functor
This theorem was more or less simultaneously proved by Watts and Eilenberg; a generalization is proved in (Nyman-Smith), and references to the original papers can be found there.
into the strict 2-category of
For more discussion see the -Cafe.
A standard textbook is
Details on the proof and its variants are also in
section 1.6 of
An introductory survey is for instance also in section 3 of