Additive category
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).
Thus an additive category is equivalently described as a preadditive category admitting all finitary products, or a preadditive category admitting all finitary coproducts.
Another, yet equivalent, way to define an additive category is a category (not assumed to be preadditive) which has a zero object, finite coproducts and finite products and such that the canonical map from the coproduct to the product
is an isomorphism. This isomorphism can be used to equip with a commutative monoid structure. The last requirement is that this is in fact an abelian group. Unlike the afore-mentioned definitions, this definition does not need the auxiliary additive group structure on the Hom sets as a datum, but rather as a property.
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