Editing Additive category (section)

== Definition ==
There are two equivalent definitions of an additive category: One as a [[category (mathematics)|category]] equipped with additional structure, and another as a category equipped with ''no extra structure'' but whose objects and [[morphism]]s satisfy certain equations.

=== Via preadditive categories ===
A category '''C''' is preadditive if all its [[hom-set]]s are [[abelian group]]s and composition of morphisms is [[bilinear map|bilinear]]; in other words, '''C''' is [[enriched category|enriched]] over the [[monoidal category]] of abelian groups.

In a preadditive category, every finitary [[product (category theory)|product]] is necessarily a [[coproduct]], and hence a [[biproduct]], and [[converse (logic)|converse]]ly every finitary coproduct is necessarily a product (this is a consequence of the definition, not a part of it). The empty product, is  a [[Initial_and_terminal_objects|final object]] and  the empty product in the case of an empty diagram, an [[Initial_and_terminal_objects|initial object]]. Both being limits, they are not finite products nor coproducts. 

Thus an additive category is equivalently described as a preadditive category admitting all finitary products and with the null object or a preadditive category admitting all finitary coproducts and with the null object

=== Via semiadditive categories ===
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 [[biproduct]]s. It is then a remarkable theorem that the Hom sets naturally admit an [[abelian monoid]] structure. A [[mathematical proof|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.

=== Generalization ===

More generally, one also considers additive [[Preadditive category#R-linear categories|{{mvar|R}}-linear categories]] for a [[commutative ring]] {{mvar|R}}. These are categories enriched over the monoidal category of {{mvar|R}}-[[module (mathematics)|module]]s and admitting all finitary biproducts.