Category Theory/Additive categories

Definition (biproduct):

Let be a category, and let be a family of objects in . A biproduct of is an object of that is usually denoted as

and for which there exist arrows

and

for all that have the following properties:

  1. and
  2. , together with the morphisms , constitutes a coproduct in the category
  3. , together with the morphisms , constitutes a product in the category

Definition (additive category):

An additive category is a category that satisfies each of the following requirements:

  1. Every morphism in has a kernel and a cokernel
  2. For every two objects of , there exists a biproduct
  3. For every two objects of , the assignment , where is the morphism that arises from postcomposing the morphism (where shall denote the diagonal) with the anti-diagonal , turns into an abelian group