Please note that the content of this book primarily consists of articles available from Wikipedia or other free sources online. In mathematics, specifically in category theory, a pre-abelian category is an additive category that has all kernels and cokernels. 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. 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. 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.

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