Pre-Abelian category

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In mathematics, specifically in category theory, a pre-abelian category is an additive category that has all kernels and cokernels.

Spelled out in more detail, this means that a category C is pre-abelian if:

Note that the zero morphism in item 3 can be identified as the identity element of the hom-set Hom(A,B), which is an abelian group by item 1; or as the unique morphism A → O → B, where O is a zero object, guaranteed to exist by item 2.

Contents

Examples

The original example of an additive category is the category Ab of abelian groups. Ab is preadditive because it is a closed monoidal category, the biproduct in Ab is the finite direct sum, the kernel is inclusion of the ordinary kernel from group theory and the cokernel is the quotient map onto the ordinary cokernel from group theory.

Other common examples:

These will give you an idea of what to think of; for more examples, see abelian category (every abelian category is pre-abelian).

Elementary properties

Every pre-abelian category is of course an additive category, and many basic properties of these categories are described under that subject. This article concerns itself with the properties that exist specifically because of the existence of kernels and cokernels.

Although kernels and cokernels are special kinds of equalisers and coequalisers, a pre-abelian category actually has all equalisers and coequalisers. We simply construct the equaliser of two morphisms f and g as the kernel of their difference g − f; similarly, their coequaliser is the cokernel of their difference. (The alternative term "difference kernel" for binary equalisers derives from this fact.) Since pre-abelian categories have all finite products and coproducts (the biproducts) and all binary equalisers and coequalisers (as just described), then by a general theorem of category theory, they have all limits and colimits. That is, pre-abelian categories are finitely complete.

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