Editing Preadditive category (section)

== Biproducts ==

{{Main|Biproduct}}

Any [[finite set|finite]] [[product (category theory)|product]] in a preadditive category must also be a [[coproduct]], and conversely. In fact, finite products and coproducts in preadditive categories can be characterised by the following ''biproduct condition'':
:The object ''B'' is a '''biproduct''' of the objects ''A''<sub>1</sub>, ..., ''A<sub>n</sub>'' [[if and only if]] there are ''projection morphisms'' ''p''<sub>''j''</sub>:&nbsp;''B''&nbsp;→&nbsp;''A''<sub>''j''</sub> and ''injection morphisms'' ''i''<sub>''j''</sub>:&nbsp;''A''<sub>''j''</sub>&nbsp;→&nbsp;''B'', such that (''i''<sub>1</sub>∘''p''<sub>1</sub>)&nbsp;+ ···&nbsp;+ (''i<sub>n</sub>''∘''p<sub>n</sub>'') is the identity morphism of ''B'', ''p<sub>j</sub>''∘''i<sub>j</sub>'' is the [[identity morphism]] of <var>A</var><sub><var>j</var></sub>, and ''p''<sub>''j''</sub>∘''i<sub>k</sub>'' is the zero morphism from ''A''<sub>''k''</sub> to ''A<sub>j</sub>'' whenever ''j'' and ''k'' are [[Distinct (mathematics)|distinct]].

This biproduct is often written ''A''<sub>1</sub>&thinsp;⊕&thinsp;···&thinsp;⊕&thinsp;''A<sub>n</sub>'', borrowing the notation for the [[direct sum]]. This is because the biproduct in well known preadditive categories like '''Ab''' ''is'' the direct sum. However, although [[Infinity|infinite]] direct sums make sense in some categories, like '''Ab''', infinite biproducts do ''not'' make sense (see {{section link|Category of abelian groups#Properties}}).

The biproduct condition in the case ''n''&thinsp;=&thinsp;0 simplifies drastically; ''B'' is a ''nullary biproduct'' if and only if the identity morphism of ''B'' is the zero morphism from ''B'' to itself, or equivalently if the hom-set Hom(''B'',''B'') is the [[trivial ring]]. Note that because a nullary biproduct will be both [[Terminal object|terminal]] (a nullary product) and [[Initial object|initial]] (a nullary coproduct), it will in fact be a '''[[zero object]]'''.
Indeed, the term "zero object" originated in the study of preadditive categories like '''Ab''', where the zero object is the [[trivial group|zero group]].

A preadditive category in which every biproduct exists (including a zero object) is called ''[[additive category|additive]]''. Further facts about biproducts that are mainly useful in the context of additive categories may be found under that subject.