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In category theory, the product of two (or more) objects in a category is a notion designed to capture the essence behind constructions in other areas of mathematics such as the cartesian product of sets, the direct product of groups, the direct product of rings and the product of topological spaces. Essentially, the product of a family of objects is the "most general" object which admits a morphism to each of the given objects. ==Definition== Let be a category with some objects and . An object is a product of and , denoted , iff it satisfies this universal property: : there exist morphisms , such that for every object and pair of morphisms , there exists a unique morphism such that the following diagram commutes: The unique morphism is called the product of morphisms and and is denoted . The morphisms and are called the canonical projections or projection morphisms. Above we defined the binary product. Instead of two objects we can take an arbitrary family of objects indexed by some set . Then we obtain the definition of a product. An object is the product of a family of objects iff there exist morphisms , such that for every object and a -indexed family of morphisms there exists a unique morphism such that the following diagrams commute for all : The product is denoted ; if = , then denoted and the product of morphisms is denoted . 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「Product (category theory)」の詳細全文を読む スポンサード リンク
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