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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: 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. See also | ||||||||
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