The original example of an additive category is the category Ab of abelian groups. Ab is preadditive because it is a closed monoidal category, the biproduct in Ab is the finite direct sum, the kernel is inclusion of the ordinary kernel from group theory and the cokernel is the quotient map onto the ordinary cokernel from group theory.
Other common examples:
- The category of (left) modules over a ring R, in particular:
- the category of vector spaces over a field K.
- The category of (Hausdorff) abelian topological groups.
These will give you an idea of what to think of; for more examples, see abelian category (every abelian category is pre-abelian).
Read more about this topic: Pre-abelian Category
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