Image (category theory)
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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 f = hg. - For any object Z with a morphism
and a monomorphism 
such that f = lk, there exists a unique morphism 
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.
[edit] Examples
In the category of sets the image of a morphism 
is the inclusion from the ordinary image 
to Y. 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 f can be expressed as follows:
- im f = ker coker f
This holds especially in abelian categories.
