Examples
The most obvious example of a preadditive category is the category Ab itself. More precisely, Ab is a closed monoidal category. Note that commutativity is crucial here; it ensures that the sum of two group homomorphisms is again a homomorphism. In contrast, the category of all groups is not closed. See medial category.
Other common examples:
- The category of (left) modules over a ring R, in particular:
- the category of vector spaces over a field K.
- The algebra of matrices over a ring, thought of as a category as described in the article Additive category.
- Any ring, thought of as a category with only one object, is a preadditive category. Here composition of morphisms is just ring multiplication and the unique hom-set is the underlying abelian group.
These will give you an idea of what to think of; for more examples, follow the links to special cases below.
Read more about this topic: Preadditive Category
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