Abelian category

In mathematics, an abelian category is a certain kind of category in which morphisms and objects can be added and in which kernels and cokernels[?] exist and have the usual properties. The motivating prototype example of an abelian category is the category of abelian groups, whence the name.

Table of contents 1 Definitions 2 Examples 3 Elementary properties 4 Related concepts 5 To do

Definitions

It's possible to define abelian categories in a very piecemeal fashion. First, recall that a category is preadditive if it is enriched over the monoidal category[?] Ab of abelian groups. Next, recall that a preadditive category is additive if every finite set of objects has a biproduct. Then, recall that 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.

It's also possible to define abelian categories all at once. A category is abelian if it has:

• a zero object;
• all finitary[?] limits;
• all finitary colimits;
• only normal monomorphisms;
• only normal epimorphisms.

Most notably, this latter definition doesn't mention the enrichment over Ab that began the piecemeal definition; that enrichment can be constructed from the assumptions above.

Examples

• As mentioned above, the category of all abelian groups is an abelian category. The category of all finitely generated abelian groups is also an abelian category, as is the category of all finite 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 abelian category is equivalent to a full subcategory[?] of such a category of modules (Mitchell's embedding theorem[?]).
• If C is a small[?] category and A is an abelian category, then the category of all functors from C to A forms an abelian category. If C is small and preadditive, then the category of all additive functors from C to A also forms an abelian category. The latter is a generalization of the previous example, since a ring can be understood as a preadditive category with a single object.
• If K is a commutative noetherian ring, then the category of finitely generated modules[?] over K is abelian. In this way, abelian categories show up in commutative algebra.
• The category of sheaves[?] on a scheme (or, more generally, on a ringed space[?]) is an abelian category. In this way, abelian categories show up in algebraic geometry.

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.

Subobjects[?] and quotient objects[?] 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 concept for homological algebra. All of the constructions used in that field are relevant, such as exact sequences, and especially short exact sequences, and derived functors[?]. Important theorems that apply in all abelian categories include the five lemma, the short five lemma, and the snake lemma.

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To do

There are still several facts listed in Preadditive category, Additive category, and Preabelian category that should be repeated here when this is the most common context in which they're used.