additive and abelian categories
(AB1) pre-abelian category
(AB2) abelian category
(AB5) Grothendieck category
left/right exact functor
(also nonabelian homological algebra)
The Freyd–Mitchell embedding theorem says that every abelian category is a full subcategory of a category of modules over some ring $R$ and that the embedding is an exact functor.
It is easy to see that not every abelian category is equivalent to $R$Mod for some ring $R$. The reason is that $R Mod$ has all small limits and colimits. But for instance, for $R$ Noetherian, the category of finitely generated $R$-modules is an abelian category but lacks these properties.
However, we have
Every small abelian category admits a full, faithful and exact functor to the category $R$Mod for some ring $R$.
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 (The terminology there is a bit outdated, in that it calls an abelian category “fully abelian” if it admits a full and faithful exact functor to a category of $R$-modules.)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 $R$-modules:
Let $C$ be an abelian category. If $C$ has all small coproducts and has a compact projective generator, then $C \simeq R Mod$ for some ring $R$.
In fact, in this situation we can take $R = C(x,x)^{op}$ where $x$ is any compact projective generator. Conversely, if $C \simeq R Mod$, then $C$ has all small coproducts and $x = R$ is a compact projective generator.
This theorem, minus the explicit description of $R$, 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 $R$ is a compact projective generator of $R Mod$.
Going further, we can try to characterize functors between categories of $R$-modules that come from tensoring with bimodules. Here we have
If $B$ is an an $S$-$R$-bimodule, the tensor product functor
is right exact and preserves small coproducts. Conversely, if $F\colon Mod_R \to Mod_S$ is right exact and that preserves small coproducts, it is naturally isomorphic to $B \otimes_R -$ where $B$ is the $S$-$R$-bimodule $F R$.
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.
Going still further we should be able to obtain a nice theorem describing the image of the embedding of the 2-category of
into the strict 2-category of
For more discussion see the $n$-Cafe.
A standard textbook is
Details on the proof and its variants are also in
section 1.6 of
and
An introductory survey is for instance also in section 3 of
See also