# Preadditive category

## Mathematical category whose hom sets form Abelian groups / From Wikipedia, the free encyclopedia

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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:

and

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).