nLab abelian category

Idea

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.

Definition

Recall the following fact about pre-abelian categories from this proposition, discussed there:

Proposition

Every morphism $f \colon A\to B$ in a pre-abelian category has a canonical de-composition

(1)$$A \overset{\; p \;}{\twoheadrightarrow} \coker\big(\ker f\big) \xrightarrow{\; \overline{f} \;} \ker\big(\coker f\big) \xhookrightarrow{\; i \;} B$$

where:

• $p$ is a cokernel, hence an epimorphism,

• $i$ is a kernel, hence a monomorphism.

Definition

An abelian category is a pre-abelian category satisfying the following equivalent conditions.

1. For every morphism $f$, the canonical morphism $\bar{f} \colon coker(ker(f)) \to ker(coker(f))$ (1) from prop. is an isomorphism (hence providing an image factorization $A \twoheadrightarrow im(f) \hookrightarrow B$).

2. Every monomorphism is a kernel and every epimorphism is a cokernel.

Proposition

These two conditions are indeed equivalent.

Proof

The first condition implies that if $f$ is a monomorphism then $f \cong \ker(\coker(f))$ (in the category of objects over $B$) 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).

Properties

General

Remark

The notion of abelian category is self-dual: opposite of any abelian category is abelian.

Remark

By the second formulation of the definition , 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.

Proposition

In an abelian category, pullback preserves epimorphisms and pushout preserves monomorphisms.

Because every abelian category is a regular category. For an explicit proof see, e.g., Selick, Prop. 1.3.13.

Factorization of morphisms

Proposition

In an abelian category every morphism decomposes uniquely up to a unique isomorphism into the composition of an epimorphism and a monomorphism, via prop combined with def. .

Since by remark 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.

Remark

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.

Canonical $Ab$-enrichment

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.

Relation to exactness properties of toposes

The exactness properties of abelian categories have many features in common with exactness properties of toposes or of pretoposes.

Freyd (1999) 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.

Embedding theorems

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.

Counterexamples

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.

Embedding into $Ab$

(…)

(Bergman 1974)

(…)

Freyd-Mitchell embedding into $R Mod$

Mitchell’s Embedding Theorem

Every small abelian category admits a full, faithful and exact functor to the category $R Mod$ for some ring $R$.

Proof

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:

Theorem

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.

Proof

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

• rings
• bimodules
• bimodule homomorphisms

into the strict 2-category of

• abelian categories
• right exact functors
• natural transformations.

For more discussion see the $n$-Cafe.

Examples

• Of course, Ab is abelian,

• the category $R$Mod of (left) modules over any ring $R$ is abelian

• Therefore in particular the category Vect of vector spaces over any field is an abelian category

• The full subcategory of $R$Mod whose objects are the Noetherian left $R$-modules is abelian, since it contains any submodule or quotient module of any of its objects (see Theorem 2.3.8 p.103 of Berrick and Keating in the textbook references below).

• Similarly, the full subcategory of $R$Mod whose objects are the Artinian left $R$-modules is abelian, since it contains any submodule or quotient module of any of its objects (loc. cit.).

• Also similarly, the full subcategory of $R$Mod whose objects are the Artinian semisimple modules is abelian, since it contains any submodule or quotient module of any of its objects (loc. cit.) .

• as is the category of representations of a group (e.g. here)

• The category of sheaves of abelian groups on any site is abelian.

Counter-examples:

• 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.

Related concepts

• additive and abelian categories

• abelian subcategory

• Deligne tensor product of abelian categories

• pseudo-abelian category

• quasi-abelian category

• semi-abelian category

• length of an object

References

Maybe the first reference on abelian categories, then still called exact categories is

• D. A. Buchsbaum, Exact categories and duality, Transactions of the American Mathematical Society Vol. 80, No. 1 (1955), pp. 1-34 (JSTOR)

Further foundations of the theory were then laid in

• Alexander Grothendieck, Sur quelques points d'algèbre homologique, Tôhoku Math. J. vol 9, n.2, 3, 1957.

Other classic references:

• Pierre Gabriel, Des Catégories Abéliennes, Bulletin de la Société Mathématique de France 90 (1962) 323-448 [numdam:BSMF_1962__90__323_0]

• 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 3 (2003) [tac:tr3, pdf]

Textbook accounts:

• Nicolae Popescu, Abelian categories with applications to rings and modules, London Math. Soc. Monographs 3, Academic Press (1973) [MR0340375]

• Saunders MacLane, Chapter VIII of: Categories for the Working Mathematician (1978)

• A. J. Berrick and M. E. Keating, Categories and Modules, with K-theory in View, Cambridge Studies in Advanced Mathematics 67, Cambridge University Press (2000)

• Masaki Kashiwara, Pierre Schapira, Section 8 of: Categories and Sheaves, Grundlehren der Mathematischen Wissenschaften 332, Springer (2006) [doi:10.1007/3-540-27950-4, pdf]

• Pavel Etingof, Shlomo Gelaki, Dmitri Nikshych, Victor Ostrik, Chapter 1 of: Tensor Categories, AMS Mathematical Surveys and Monographs 205 (2015) [ISBN:978-1-4704-3441-0, pdf]

Further review:

• Rankey Datta, An introduction to abelian categories (2010) (pdf)

• Paul Selick, Homological Algebra Notes (pdf,pdf)

On diagram chasing and elements in an abelian category:

• George Bergman, A note on abelian categories – translating element-chasing proofs, and exact embedding in abelian groups (1974) [pdf, pdf]

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

• Rankeya Datta, The category of modules over a commutative ring and abelian categories (pdf)

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).

Formalization of abelian categories as univalent categories in univalent foundations of mathematics (homotopy type theory):

• unimath, Abelian Categories [UniMath.CategoryTheory.Abelian]