DEFINITIONS

A category is abelian if

it has a Zero Object ,

it has all '' Pullbacks '' and '' Pushouts '', and

all Monomorphism s and Epimorphism s are Normal .

By a theorem of Peter Freyd , this definition is equivalent to the following "piecemeal" definition:

A category is '' Preadditive '' if it is Enriched over the Monoidal Category Ab of Abelian Group s. This means that all Hom-set s are abelian groups and the composition of morphisms is Bilinear .

A preadditive category is '' Additive '' if every Finite Set of objects has a Biproduct . This means that we can form finite Direct Sum s and Direct Product s.

An additive category is '' Preabelian '' if every morphism has both a Kernel and a Cokernel .

Finally, a preabelian category is abelian if every Monomorphism and every Epimorphism is Normal . This means that every monomorphism is a kernel of some morphism, and every epimorphism is a cokernel of some morphism.

Note that the enriched structure on Hom-set s is a ''consequence'' of the three Axiom s of the first definition.

EXAMPLES

As mentioned above, the category of all abelian groups is an abelian category. The category of all Finitely Generated Abelian Group s is also an abelian category, as is the category of all finite abelian groups.

If ''R'' is a Ring , then the category of all left (or right) Modules over ''R'' is an abelian category. In fact, it can be shown that any small abelian category is equivalent to a Full Subcategory of such a category of modules ('' Mitchell's Embedding Theorem '').

If ''R'' is a left- Noetherian Ring , then the category of Finitely Generated left modules over ''R'' is abelian. In particular, the category of finitely generated modules over a noetherian Commutative Ring is abelian; in this way, abelian categories show up in Commutative Algebra .

As special cases of the two previous examples: the category of Vector Space s over a fixed Field ''k'' is abelian, as is the category of finite- Dimensional vector spaces over ''k''.

If ''X'' is a Topological Space , then the category of all (real or complex) Vector Bundles on ''X'' is an abelian category. In this way, abelian categories show up in Differential Geometry , Differential Topology , and Algebraic Topology .

If ''X'' is a Topological Space , then the category of all Sheaves of abelian groups on ''X'' is an abelian category. More generally, the category of sheaves of abelian groups on a Grothendieck Site is an abelian category. In this way, abelian categories show up in Algebraic Topology and Algebraic Geometry .

If C is a Small Category and '''A''' is an abelian category, then the category of all Functor s from C to '''A''' forms an abelian category (the morphisms of this category are the Natural Transformation s between functors). If C is small and Preadditive , then the category of all Additive Functor s from C to '''A''' also forms an abelian category. The latter is a generalization of the ''R''-module example, since a ring can be understood as a preadditive category with a single object.

ELEMENTARY PROPERTIES

Given any pair ''A'', ''B'' of objects in an abelian category, there is a special Zero Morphism from ''A'' to ''B''.
This can be defined as the Zero element of the Hom-set Hom(''A'',''B''), since this is an abelian group.
Alternatively, it can be defined as the unique composition ''A'' → 0 → ''B'', where 0 is the Zero Object of the abelian category.

In an abelian category, every morphism ''f'' can be written as the composition of an epimorphism followed by a monomorphism.
This epimorphism is called the '' Coimage '' of ''f'', while the monomorphism is called the '' Image '' of ''f''.

Subobject s and Quotient Object s are Well-behaved in abelian categories.
For example, the Poset of subobjects of any given object ''A'' is a Bounded Lattice .

Every abelian category A is a Module over the monoidal category of finitely generated abelian groups; that is, we can form a Tensor Product of a finitely generated abelian group ''G'' and any object ''A'' of A.
The abelian category is also a Comodule ; Hom(''G'',''A'') can be interpreted as an object of A.
If A is Complete , then we can remove the requirement that ''G'' be finitely generated; most generally, we can form Finitary Enriched Limits in A.

RELATED CONCEPTS

Abelian categories are the most general setting for Homological Algebra .
All of the constructions used in that field are relevant, such as Exact Sequence s, and especially Short Exact Sequence s, and Derived Functor s.
Important theorems that apply in all abelian categories include the Five Lemma (and the Short Five Lemma as a special case), as well as the Snake Lemma (and the Nine Lemma as a special case).

HISTORY

Abelian categories were introduced by s on abelian categories; on the one hand the abelian category of sheaves of abelian groups on a topological space, on the other hand the abelian category of ''G''-modules for a given group ''G''.

REFERENCES

P. Freyd. ''Abelian Categories,'' Harper and Row, New York, 1964. Available online.

Barry Mitchell: ''Theory of Categories'', New York, Academic Press, 1965.

N. Popescu: ''Abelian categories with applications to rings and modules'', Academic Press, London, 1973.

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