Definition
A category C is additive if
- it has a zero object
- every hom-set Hom(A, B) has an addition, endowing it with the structure of an Abelian group, and such that composition of morphisms is bilinear
- all finitary biproducts exist.
Note that a category is called preadditive if just the second holds, whereas it is called semiadditive if both the first and the third hold.
Also, since the empty biproduct is a zero object in the category, we may omit the first condition. If we do this, however, we need to presuppose that the category C has zero morphisms, or equivalently that C is enriched over the category of pointed sets.
Read more about this topic: Additive Categories
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