# 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:

- and
- , together with the morphisms , constitutes a coproduct in the category
- , 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:

- Every morphism in has a kernel and a cokernel
- For every two objects of , there exists a biproduct
- 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