Image (category theory)
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Given a category C and a morphism <math>f:X\rightarrow Y</math> in C, the image of f is a monomorphism <math>h:I\rightarrow Y</math> satisfying the following universal property:
- There exists a morphism <math>g:X\rightarrow I</math> such that f = hg.
- For any object Z with a morphism <math>k:X\rightarrow Z</math> and a monomorphism <math>l:Z\rightarrow Y</math> such that f = lk, there exists a unique morphism <math>m:I\rightarrow Z</math> such that k = mg and h = lm.
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 <math>f : X \to Y</math> is the inclusion from the ordinary image <math>\{f(x) ~|~ x \in X\}</math> to <math>Y</math>. 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 <math>f</math> can be expressed as follows:
- im f = ker coker f
This holds especially in abelian categories.