Preadditive category

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In mathematics, specifically in category theory, a preadditive category is a category that is enriched over the monoidal category of abelian groups. In other words, the category C is preadditive if every hom-set Hom(A,B) in C has the structure of an abelian group, and composition of morphisms is bilinear over the integers.

A preadditive category is also called an Ab-category, after the notation Ab for the category of abelian groups. Some authors have used the term additive category for preadditive categories, but Wikipedia follows the current trend of reserving this word for certain special preadditive categories (see special cases below).

Contents

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 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.

Elementary properties

Because every hom-set Hom(A,B) is an abelian group, it has a zero element 0. This is the zero morphism from A to B. Because composition of morphisms is bilinear, the composition of a zero morphism and any other morphism (on either side) must be another zero morphism. If you think of composition as analogous to multiplication, then this says that multiplication by zero always results in a product of zero, which is a familiar intuition. Extending this analogy, the fact that composition is bilinear in general becomes the distributivity of multiplication over addition.

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