Preadditive in a sentence

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.

Preadditive categories, rings and modules A preadditive category is a category where the morphism sets form abelian groups and the composition of morphisms is bilinear ; examples are categories of abelian groups or modules.

Some authors have used the term additive category for preadditive categories, but here we follow the current trend of reserving this word for certain special preadditive categories (see special cases below).

Additive functors between preadditive categories generalize the concept of ring homomorphism, and ideals in additive categories can be defined as sets of morphisms closed under addition and under composition with arbitrary morphisms.

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.

Additivity If C and D are preadditive categories and F : C ← D is an additive functor with a right adjoint G : C → D, then G is also an additive functor and the hom-set bijections : are, in fact, isomorphisms of abelian groups.

An additive category is a preadditive category in which all finite biproduct exist.

Examples The most obvious example of a preadditive category is the category Ab itself.

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.

Further facts about kernels and cokernels in preadditive categories that are mainly useful in the context of pre-Abelian categories may be found under that subject.

In this way, preadditive categories can be seen as a generalisation of rings.

It is therefore natural to consider arbitrary preadditive categories to be generalizations of rings.

Kernels and cokernels Because the hom-sets in a preadditive category have zero morphisms, the notion of kernel and cokernel make sense.

Rings are preadditive categories with one object.

Special cases Most of these special cases of preadditive categories have all been mentioned above, but they're gathered here for reference.

That is, if f: A → B is a morphism in a preadditive category, then the kernel of f is the equaliser of f and the zero morphism from A to B, while the cokernel of f is the coequaliser of f and this zero morphism.

The latter is a generalization of the R-module example, since a ring can be understood as a preadditive category with a single object.

The preadditive categories most commonly studied are in fact Abelian categories; for example, Ab is an Abelian category.

This is because the biproduct in well known preadditive categories like Ab is the direct sum.

This ring is the endomorphism ring of A. Conversely, every ring (with identity ) is the endomorphism ring of some object in some preadditive category.