Additive Categories - Additive Functors

Additive Functors

Recall that a functor 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, though, then a functor is additive if and only if it preserves all biproduct diagrams.

That is, if B is a biproduct of A₁, … , A_n in C with projection morphisms p_k and injection morphisms k_j, then F(B) should be a biproduct of F(A₁), … , F(A_n) in D with projection morphisms F(p_k) and injection morphisms F(i_k).

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 here), and most interesting functors studied in all of category theory are adjoints.