Preadditive category

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In mathematics, specifically in category theory, a preadditive category is another name for an Ab-category, i.e., a category that is enriched over the category of abelian groups, Ab. That is, an Ab-category C is a category such that every hom-set Hom(A,B) in C has the structure of an abelian group, and composition of morphisms is bilinear, in the sense that composition of morphisms distributes over the group operation. In formulas: $${\displaystyle f\circ (g+h)=(f\circ g)+(f\circ h)}$$ and $${\displaystyle (f+g)\circ h=(f\circ h)+(g\circ h),}$$ where + is the group operation.

Some authors have used the term additive category for preadditive categories, but this page reserves this term for certain special preadditive categories (see § Special cases below).