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In the context of abstract algebra or universal algebra, a monomorphism is an injective homomorphism. A monomorphism from ''X'' to ''Y'' is often denoted with the notation . In the more general setting of category theory, a monomorphism (also called a monic morphism or a mono) is a left-cancellative morphism, that is, an arrow such that, for all morphisms , : Monomorphisms are a categorical generalization of injective functions (also called "one-to-one functions"); in some categories the notions coincide, but monomorphisms are more general, as in the examples below. The categorical dual of a monomorphism is an epimorphism, i.e. a monomorphism in a category ''C'' is an epimorphism in the dual category ''C''op. Every section is a monomorphism, and every retraction is an epimorphism. ==Relation to invertibility== Left invertible morphisms are necessarily monic: if ''l'' is a left inverse for ''f'' (meaning ''l'' is a morphism and ), then ''f'' is monic, as : A left invertible morphism is called a split mono. However, a monomorphism need not be left-invertible. For example, in the category Group of all groups and group morphisms among them, if ''H'' is a subgroup of ''G'' then the inclusion is always a monomorphism; but ''f'' has a left inverse in the category if and only if ''H'' has a normal complement in ''G''. A morphism is monic if and only if the induced map , defined by for all morphisms , is injective for all ''Z''. 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「monomorphism」の詳細全文を読む スポンサード リンク
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