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).