Additive category

In mathematics, specifically in , an additive category is a C admitting all finitary biproducts. There are two equivalent definitions of an additive category: One as a category equipped with additional structure, and another as a category equipped with no extra structure but whose objects and morphisms satisfy certain equations. A category C is preadditive if all its hom-sets are abelian groups and composition of morphisms is bilinear; in other words, C is over the of abelian groups. In a preadditive category, every finitary (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. We give an alternative definition. Define a semiadditive category to be a category (note: not a preadditive category) which admits a zero object and all binary biproducts. It is then a remarkable theorem that the Hom sets naturally admit an abelian monoid structure. A proof of this fact is given below. An additive category may then be defined as a semiadditive category in which every morphism has an additive inverse. This then gives the Hom sets an abelian group structure instead of merely an abelian monoid structure. More generally, one also considers additive for a commutative ring R. These are categories enriched over the monoidal category of and admitting all finitary biproducts. The original example of an additive category is the Ab. The zero object is the trivial group, the addition of morphisms is given pointwise, and biproducts are given by direct sums. More generally, every over a ring R is additive, and so in particular, the over a field K is additive. The algebra of matrices over a ring, thought of as a category as described below, is also additive.

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