Preadditive Category

In mathematics, specifically in category theory, a preadditive category is a category that is enriched over the monoidal category of abelian groups. In other words, the category C is preadditive if every hom-set Hom(A,B) in C has the structure of an abelian group, and composition of morphisms is bilinear over the integers.

A preadditive category is also called an Ab-category, after the notation Ab for the category of abelian groups. Some authors have used the term additive category for preadditive categories, but Wikipedia follows the current trend of reserving this word for certain special preadditive categories (see special cases below).

Read more about Preadditive CategoryExamples, Elementary Properties, Additive Functors, Biproducts, Kernels and Cokernels, Special Cases

Other articles related to "preadditive category, category, preadditive":

Yoneda Lemma - Preadditive Categories, Rings and Modules
... A preadditive category is a category where the morphism sets form abelian groups and the composition of morphisms is bilinear examples are categories of abelian groups or modules ... In a preadditive category, there is both a "multiplication" and an "addition" of morphisms, which is why preadditive categories are viewed as ... Rings are preadditive categories with one object ...
Preadditive Category - Special Cases
... Most of these special cases of preadditive categories have all been mentioned above, but they're gathered here for reference ... A ring is a preadditive category with exactly one object ... An additive category is a preadditive category with all finite biproducts ...

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