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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. 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 here we follow the current trend of reserving this word for certain special preadditive categories (see special cases below). == 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. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Preadditive category」の詳細全文を読む スポンサード リンク
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