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In category theory, the coproduct, or categorical sum, is a category-theoretic construction which includes as examples the disjoint union of sets and of topological spaces, the free product of groups, and the direct sum of modules and vector spaces. The coproduct of a family of objects is essentially the "least specific" object to which each object in the family admits a morphism. It is the category-theoretic dual notion to the categorical product, which means the definition is the same as the product but with all arrows reversed. Despite this seemingly innocuous change in the name and notation, coproducts can be and typically are dramatically different from products. ==Definition== Let ''C'' be a category and let ''X''1 and ''X''2 be objects in that category. An object is called the coproduct of these two objects, written ''X''1 ∐ ''X''2 or ''X''1 ⊕ ''X''2 or sometimes simply ''X''1 + ''X''2, if there exist morphisms ''i''1 : ''X''1 → ''X''1 ∐ ''X''2 and ''i''2 : ''X''2 → ''X''1 ∐ ''X''2 satisfying a universal property: for any object ''Y'' and morphisms ''f''1 : ''X''1 → Y and ''f''2 : ''X''2 → Y, there exists a unique morphism ''f'' : ''X''1 ∐ ''X''2 → ''Y'' such that ''f''1 = ''f'' ∘ ''i''1 and ''f''2 = ''f'' ∘ ''i''2. That is, the following diagram commutes: The unique arrow ''f'' making this diagram commute may be denoted ''f''1 ∐ ''f''2 or ''f''1 ⊕ ''f''2 or ''f''1 + ''f''2 or (''f''2 ). The morphisms ''i''1 and ''i''2 are called ''canonical injections'', although they need not be injections nor even monic. The definition of a coproduct can be extended to an arbitrary family of objects indexed by a set ''J''. The coproduct of the family is an object ''X'' together with a collection of morphisms ''ij'' : ''Xj'' → ''X'' such that, for any object ''Y'' and any collection of morphisms ''fj'' : ''Xj'' → ''Y'', there exists a unique morphism ''f'' from ''X'' to ''Y'' such that ''fj'' = ''f'' ∘ ''ij''. That is, the following diagrams commute (for each ''j'' ∈ ''J''): The coproduct of the family is often denoted : or : Sometimes the morphism ''f'' may be denoted : to indicate its dependence on the individual ''f''''j''. 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「coproduct」の詳細全文を読む スポンサード リンク
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