Preadditive Category - Additive Functors

Additive Functors

If C and D are preadditive categories, then a functor F: C → D is additive if it too is enriched over the category Ab. That is, F is additive if and only if, given any objects A and B of C, the function f: Hom(A,B) → Hom(F(A),F(B)) is a group homomorphism. Most functors studied between preadditive categories are additive.

For a simple example, if the rings R and S are represented by the one-object preadditive categories R and S, then a ring homomorphism from R to S is represented by an additive functor from R to S, and conversely.

If C and D are categories and D is preadditive, then the functor category Fun(C,D) is also preadditive, because natural transformations can be added in a natural way. If C is preadditive too, then the category Add(C,D) of additive functors and all natural transformations between them is also preadditive.

The latter example leads to a generalization of modules over rings: If C is a preadditive category, then Mod(C) := Add(C,Ab) is called the module category over C. When C is the one-object preadditive category corresponding to the ring R, this reduces to the ordinary category of (left) R-modules. Again, virtually all concepts from the theory of modules can be generalised to this setting.