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In category theory and its applications to mathematics, a biproduct of a finite collection of objects in a category with zero object is both a product and a coproduct. In a preadditive category the notions of product and coproduct coincide for finite collections of objects.〔Borceux, 4-5〕 The biproduct is a generalization of finite direct sums of modules. ==Definition== Let C be a category with zero object. Given objects ''A''1,...,''A''''n'' in C, their ''biproduct'' is an object ''A''1 ⊕ ··· ⊕ ''A''''n'' together with morphisms *''p''''k'': ''A''1 ⊕ ··· ⊕ ''A''''n'' → ''A''''k'' in C (the ''projection morphisms'') *''i''''k'': ''A''''k'' → ''A''1 ⊕ ··· ⊕ ''A''''n'' (the ''injection morphisms'') satisfying *''p''''k'' ∘ ''i''''k'' = 1''A''''k'', the identity morphism of ''A''''k'' *''p''''l'' ∘ ''i''''k'' = 0, the zero morphism ''A''''k'' → ''A''''l'', for ''k'' ≠ ''l''. and such that *(''A''1 ⊕ ··· ⊕ ''A''''n'',''p''''k'') is a product for the ''A''''k'' *(''A''1 ⊕ ··· ⊕ ''A''''n'',''i''''k'') is a coproduct for the ''A''''k''. An empty, or nullary, product is always a terminal object in the category, and the empty coproduct is always an initial object in the category. Since our category C has a zero object, the empty biproduct exists and is isomorphic to the zero object. 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「Biproduct」の詳細全文を読む スポンサード リンク
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