Preadditive category

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 here we follow the current trend of reserving this term for certain special preadditive categories (see § Special cases below).