Editing Preadditive category (section)

== Additive functors ==

If <math>C</math> and <math>D</math> are preadditive categories, then a [[functor]] <math>F : C \rightarrow D</math> is '''additive''' if it too is [[enriched functor|enriched]] over the category <math>Ab</math>. That is, <math>F</math> is additive [[if and only if]], given any objects <math>A</math> and <math>B</math> of <math>C</math>, the [[function (mathematics)|function]] <math>F:\text{Hom}(A,B)\rightarrow \text{Hom}(F(A),F(B))</math> is a [[group homomorphism]]. Most functors studied between preadditive categories are additive.

For a simple example, if the rings <math>R</math> and <math>S</math> are represented by the one-object preadditive categories <math>C_R</math> and <math>C_S</math>, then a [[ring homomorphism]] from <math>R</math> to <math>S</math> is represented by an additive functor from <math>C_R</math> to <math>C_S</math>, and conversely.

If <math>C</math> and <math>D</math> are categories and <math>D</math> is preadditive, then the [[functor category]] <math>D^C</math> is also preadditive, because [[natural transformation]]s can be added in a natural way.
If <math>C</math> is preadditive too, then the category <math>\text{Add}(C,D)</math> of additive functors and all natural transformations between them is also preadditive.

The latter example leads to a generalization of [[module (mathematics)|module]]s over rings: If <math>C</math> is a preadditive category, then <math>\text{Mod}(C)\mathbin{:=} \text{Add}(C,Ab)</math> is called the '''module category''' over <math>C</math>.{{fact|date=June 2015}} When <math>C</math> is the one-object preadditive category corresponding to the ring <math>R</math>, this reduces to the ordinary category of [[category of modules|(left) <math>R</math>-modules]]. Again, virtually all concepts from the theory of modules can be generalised to this setting.