Examples
The original example of an additive category is the category of abelian groups Ab. The zero object is the trivial group, the addition of morphisms is given point-wise, and biproducts are given by direct sums.
More generally, every module category over a ring R is additive, and so in particular, the category of vector spaces over a field K is additive.
The algebra of matrices over a ring, thought of as a category as described below, is also additive.
Read more about this topic: Additive Categories
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