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Given a category ''C'' and a morphism in ''C'', the image of ''f'' is a monomorphism satisfying the following universal property: #There exists a morphism such that . #For any object Z with a morphism and a monomorphism such that , there exists a unique morphism such that . Remarks: # such a factorization does not necessarily exist # ''g'' is unique by definition of monic (= left invertible, abstraction of injectivity) # ''m'' is monic. # ''h''=''lm'' already implies that ''m'' is unique. # ''k''=''mg'' The image of ''f'' is often denoted by im ''f'' or Im(''f''). One can show that a morphism ''f'' is monic if and only if ''f'' = im ''f''. ==Examples== In the category of sets the image of a morphism is the inclusion from the ordinary image to . In many concrete categories such as groups, abelian groups and (left- or right) modules, the image of a morphism is the image of the correspondent morphism in the category of sets. In any normal category with a zero object and kernels and cokernels for every morphism, the image of a morphism can be expressed as follows: :im ''f'' = ker coker ''f'' This holds especially in abelian categories. 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「Image (category theory)」の詳細全文を読む スポンサード リンク
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