Could not include enriched category theory - contents
(AB1) pre-abelian category
(AB2) abelian category
(AB5) Grothendieck category
The notion of abelian category is an abstraction of basic properties of the category Ab of abelian groups, more generally of the category Mod of modules over some ring, and still more generally of categories of sheaves of abelian groups and of modules. It is such that much of the homological algebra of chain complexes can be developed inside every abelian category.
The concept of abelian categories is one in a sequence of notions of additive and abelian categories.
An abelian category is a pre-abelian category satisfying the following equivalent conditions.
These two conditions are indeed equivalent.
The converse can be found in, among other places, Chapter VIII of (MacLane).
The notion of abelian category is self-dual: opposite of any abelian category is abelian.
By the second formulation of the definition 1, in an abelian category
It follows that every abelian category is a balanced category.
Some references claim that this property characterizes abelian categories among pre-abelian ones, but it is not clear to the authors of this page why this should be so, although we do not currently have a counterexample; see this discussion.
The -enrichment of an abelian category need not be specified a priori. If an arbitrary (not necessarily pre-additive) locally small category has a zero object, binary products and coproducts, kernels, cokernels and the property that every monic is a kernel arrow and every epi is a cokernel arrow (so that all monos and epis are normal), then it can be equipped with a unique addition on the morphism sets such that composition is bilinear and is abelian with respect to this structure. However, in most examples, the -enrichment is evident from the start and does not need to be constructed in this way. (A similar statement is true for additive categories, although the most natural result in that case gives only enrichment over abelian monoids; see semiadditive category.)
The exactness properties of abelian categories have many features in common with exactness properties of toposes or of pretoposes. In a fascinating post to the categories mailing list, Peter Freyd gave a sharp description of the properties shared by these categories, introducing a new concept called AT categories (for “abelian-topos”), and showing convincingly that the difference between the A and the T can be concentrated precisely in the difference of the behavior of the initial object.
Not every abelian category is a concrete category such as Ab or Mod. But for many proofs in homological algebra it is very convenient to have a concrete abelian category, for that allows one to check the behaviour of morphisms on actual elements of the sets underlying the objects.
The following embedding theorems, however, show that under good conditions an abelian category can be embedded into Ab as a full subcategory by an exact functor, and generally can be embedded this way into , for some ring . This is the celebrated Freyd-Mitchell embedding theorem discussed below.
This implies for instance that proofs about exactness of sequences in an abelian category can always be obtained by a naive argument on elements – called a “diagram chase” – because that does hold true after such an embedding, and the exactness of the embedding means that the notion of exact sequences is preserved by it.
Alternatively, one can reason with generalized elements in an abelian category, without explicitly embedding it into a larger concrete category, see at element in an abelian category. But under suitable conditions this comes down to working subject to an embedding into , see the discussion at Embedding into Ab below.
First of all, it’s easy to see that not every abelian category is equivalent to Mod for some ring . The reason is that has all small category limits and colimits. For a Noetherian ring the category of finitely generated -modules is an abelian category that lacks these properties.
This result can be found as Theorem 7.34 on page 150 of Peter Freyd’s book Abelian Categories. His terminology is a bit outdated, in that he calls an abelian category “fully abelian” if admits a full and faithful exact functor to a category of -modules. See also the Wikipedia article for the idea of the proof.
For more see at Freyd-Mitchell embedding theorem.
We can also characterize which abelian categories are equivalent to a category of -modules:
Let be an abelian category. If has all small coproducts and has a compact projective generator, then for some ring . 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 Peter Freyd’s book Abelian Categories. The first part of this theorem can also be found as Prop. 2.1.7. of Victor Ginzburg’s Lectures on noncommutative geometry. Conversely, it is easy to see that is a compact projective generator of .
One can characterize functors between categories of -modules that are either (isomorphic) to functors of the form where is a bimodule or those which look as Hom-modules. For the characterization of the tensoring functors see Eilenberg-Watts theorem.
Going still further one should be able to obtain a nice theorem describing the image of the embedding of the weak 2-category of
into the strict 2-category of
For more discussion see the -Cafe.
Therefore in particular the category Vect of vector spaces is an abelian category.
The category of torsion-free abelian groups is pre-abelian, but not abelian: the monomorphism is not a kernel.
Maybe the first reference on abelian categories, then still called exact categories is
Further foundations of the theory were then laid in
Other classic references, now available online, include:
Peter Freyd, Abelian Categories – An Introduction to the theory of functors, originally published by Harper and Row, New York(1964), Reprints in Theory and Applications of Categories, No. 3, 2003 (TAC, pdf)
Textbook discussion is also in
N. Popescu, Abelian categories with applications to rings and modules, London Math. Soc. Monographs 3, Academic Press 1973. xii+467 pp. MR0340375
Embedding of abelian categories into Ab is discussed in
For more discussion of the Freyd-Mitchell embedding theorem see there.
The proof that is an abelian category is spelled out for instance in