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, ${\displaystyle \mathbf {Ab} }$. That is, an Ab-category ${\displaystyle {\mathcal {C}}}$ is a category such that every hom-set ${\displaystyle \mathrm {Hom} (A,B)}$ in ${\displaystyle {\mathcal {C}}}$ 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 article reserves that term for certain special preadditive categories (see § Special cases below).