In mathematics, an abelian category is a category in which morphisms and objects can be added and in which kernels and cokernels exist and have nice properties. The motivating prototype example of an abelian category is the category of abelian groups, Ab. Abelian category - Definitions. A category is abelian if it has a zero object, it has all pullbacks and pushouts, and all monomorphisms and epimorphisms are normal. By a theorem of Pe ...

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A category is abelian if it has a zero object, it has all pullbacks and pushouts, and all monomorphisms and epimorphisms 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 groups. This means that all hom-sets are abelian groups and the composition of morphisms is bilinear. A preadditive category is ...

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Abelian category, Abelian category - Definitions, Abelian category - Examples, Abelian category - Elementary properties, Abelian category - Related concepts, Abelian category - History

Read more here: » Abelian category: Encyclopedia II - Abelian category - Definitions

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, whil ...

See also:

Abelian category, Abelian category - Definitions, Abelian category - Examples, Abelian category - Elementary properties, Abelian category - Related concepts, Abelian category - History

Read more here: » Abelian category: Encyclopedia II - Abelian category - Elementary properties