Could not include enriched category theory - contents
additive and abelian categories
(AB1) pre-abelian category
(AB2) abelian category
(AB5) Grothendieck category
left/right exact functor
and
nonabelian homological algebra
The notion of abelian category is an abstraction of basic properties of the category Ab of abelian groups, more generally of the category $R$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.
While additive categories differ significantly from toposes, there is an intimate relation between abelian categories and toposes. See AT category for more on that.
Recall the following fact about pre-abelian categories from this proposition, discussed there:
Every morphism $f \colon A\to B$ in a pre-abelian category has a canonical decomposition
where $p$ is a cokernel, hence an epi, and $i$ is a kernel, and hence monic.
An abelian category is a pre-abelian category satisfying the following equivalent conditions.
For every morphism $f$, the canonical morphism $\bar{f} \colon coker(ker(f)) \to ker(coker(f))$ of prop. 1 is an isomorphism.
Every monomorphism is a kernel and every epimorphism is a cokernel.
These two conditions are indeed equivalent.
The first condition implies that if $f$ is a monomorphism then $f\cong \ker(\coker(f))$ so $f$ is a kernel. Dually if $f$ is an epimorphism it follows that $f \cong coker(ker(f))$. So (1) implies (2).
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
every monomorphism is a regular monomorphism;
every epimorphism is a regular epimorphism.
It follows that every abelian category is a balanced category.
In an abelian category every morphism decomposes uniquely up to a unique isomorphism into the composition of an epimorphism and a monomorphism, via prop 1 combined with def. 1.
Since by remark 2 every monic is regular, hence strong, it follows that $(epi, mono)$ is an orthogonal factorization system in an abelian category; see at (epi, mono) factorization system.
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 $Ab$-enrichment of an abelian category need not be specified a priori. If an arbitrary (not necessarily pre-additive) locally small category $C$ 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 $C$ is abelian with respect to this structure. However, in most examples, the $Ab$-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 last point is of relevance in particular for higher categorical generalizations of additive categories. See for instance remark 2.14, p. 5 of Jacob Lurie’s Stable Infinity-Categories.
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 $R$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 $R Mod$, for some ring $R$. 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 $Ab$, see the discussion at Embedding into Ab below.
First of all, it’s 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 category limits and colimits. For a Noetherian ring $R$ the category of finitely generated $R$-modules is an abelian category that lacks these properties.
(…)
(…)
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 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 $R$-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 $R$-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 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 $R$ is a compact projective generator of $R Mod$.
One can characterize functors between categories of $R$-modules that are either (isomorphic) to functors of the form $B \otimes_R -$ where $B$ 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 $n$-Cafe.
Of course, Ab is abelian, as is the category of modules over any ring.
Therefore in particular the category Vect of vector spaces is an abelian category.
The category of sheaves of abelian groups on any site is abelian.
The category of torsion-free abelian groups is pre-abelian, but not abelian: the monomorphism $2:\mathbb{Z}\to\mathbb{Z}$ 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 accounts include
Saunders MacLane, Categories for the Working Mathematician .
N. Popescu, Abelian categories with applications to rings and modules, London Math. Soc. Monographs 3, Academic Press 1973. xii+467 pp. MR0340375
Pavel Etingof, Shlomo Gelaki, Dmitri Nikshych, Victor Ostrik, chapter 1 of Tensor categories, Mathematical Surveys and Monographs, Volume 205, American Mathematical Society, 2015 (pdf)
Reviews include
Embedding of abelian categories into Ab is discussed in
For more discussion of the Freyd-Mitchell embedding theorem see there.
The proof that $R Mod$ is an abelian category is spelled out for instance in
A discussion about to which extent abelian categories are a general context for homological algebra is archived at nForum here.
See also the catlist 1999 discussion on comparison between abelian categories and topoi (AT categories).