(AB1) pre-abelian category
(AB2) abelian category
(AB5) Grothendieck category
Grothendieck categories are those abelian categories
In terms of the AB hierarchy discussed at additive and abelian categories we have
This means that a Grothendieck category is an abelian category
that admits a generator;
that admits small colimits;
Dually a co-Grothendieck category is an AB5 category with a cogenerator. The category of abelian groups is not a co-Grothendieck category. Any abelian category which is simultaneously Grothendieck and co-Grothendieck has just a single object (see Freyd’s book, p.116).
A Grothendieck category satisfies the following properties.
it admits small limits;
it satisfies Pierre Gabriel’s sup property: every small family of subobjects of a given object has a supremum which is a subobject of
Much of the localization theory of rings generalizes to general Grothendieck categories.
for a Grothendieck category, the category of complexes in is again a Grothendieck category.
Grothendieck categories are mentioned at the end of section 8.3 in
The relation to complexes is in section 14.1.
The proof that filtered colimits in are exact is spelled out for instance in
The duality of Grothendieck categories with categories of modules over linearly compact rings is discussed in
Discussion of model structures on chain complexes in Grothendieck abelian categories is in