Additive category

In mathematics, specifically in category theory, an additive category is a preadditive category C admitting all finitary biproducts. In mathematics, specifically in category theory, an additive category is a preadditive category C admitting all finitary biproducts. A category C is preadditive if all its hom-sets are abelian groups and composition of morphisms is bilinear; in other words, C is enriched over the monoidal category of abelian groups. In a preadditive category, every finitary product (including the empty product, i.e., a final object) is necessarily a coproduct (or initial object in the case of an empty diagram), and hence a biproduct, and conversely every finitary coproduct is necessarily a product (this is a consequence of the definition, not a part of it).