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In category theory, the concept of an injective cogenerator is drawn from examples such as Pontryagin duality. Generators are objects which cover other objects as an approximation, and (dually) cogenerators are objects which envelope other objects as an approximation. When working with unfamiliar algebraic objects, one can use these to approximate with the more familiar. More precisely: * A generator of a category with a zero object is an object ''G'' such that for every nonzero object H there exists a nonzero morphism f:''G'' → ''H''. * A cogenerator is an object ''C'' such that for every nonzero object ''H'' there exists a nonzero morphism f:''H'' → ''C''. (Note the reversed order). ==The abelian group case== Assuming one has a category like that of abelian groups, one can in fact form direct sums of copies of ''G'' until the morphism :''f'': Sum(''G'') →''H'' is surjective; and one can form direct products of ''C'' until the morphism :''f'':''H''→ Prod(''C'') is injective. For example, the integers are a generator of the category of abelian groups (since every abelian group is a quotient of a free abelian group). This is the origin of the term ''generator''. The approximation here is normally described as ''generators and relations.'' As an example of a ''cogenerator'' in the same category, we have Q/Z, the rationals modulo the integers, which is a divisible abelian group. Given any abelian group ''A'', there is an isomorphic copy of ''A'' contained inside the product of |A| copies of Q/Z. This approximation is close to what is called the ''divisible envelope'' - the true envelope is subject to a minimality condition. 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「Injective cogenerator」の詳細全文を読む スポンサード リンク
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