Editing Additive category (section)

== Additive functors ==

A [[functor]] {{math|''F'': '''C''' → '''D'''}} between preadditive categories is ''additive'' if it is an abelian group [[homomorphism]] on each [[hom-set]] in '''C'''. If the categories are additive, then a functor is additive if and only if it preserves all [[biproduct]] diagrams.

That is, if {{mvar|B}} is a biproduct of&nbsp;{{math|''A''<sub>1</sub>, ... , ''A<sub>n</sub>''}} in&nbsp;'''C''' with projection morphisms {{math|''p<sub>k</sub>''}} and injection morphisms {{math|''i<sub>j</sub>''}}, then {{math|''F''(''B'')}} should be a biproduct of&nbsp;{{math|''F''(''A''<sub>1</sub>), ... , ''F''(''A<sub>n</sub>'')}} in&nbsp;'''D''' with projection morphisms {{math|''F''(''p''<sub>''j''</sub>)}} and injection morphisms {{math|''F''(''i<sub>j</sub>'')}}.

Almost all functors studied between additive categories are additive. In fact, it is a theorem that all [[adjoint functors]] between additive categories must be additive functors (see [[Adjoint functors#Additivity|here]]). Most of the interesting functors studied in category theory are adjoints.

=== Generalization ===

When considering functors between {{mvar|R}}-linear additive categories, one usually restricts to {{mvar|R}}-[[preadditive category#R-linear categories|linear functors]], so those functors giving an {{mvar|R}}-[[module homomorphism]] on each hom-set.