Image (category theory)

Given a category C and a morphism $f:X\rightarrow Y$ in C, the image of f is a monomorphism $h:I\rightarrow Y$ satisfying the following universal property:

There exists a morphism $g:X\rightarrow I$ such that f = hg.
For any object Z with a morphism $k:X\rightarrow Z$ and a monomorphism $l:Z\rightarrow Y$ such that f = lk, there exists a unique morphism $m:I\rightarrow Z$ 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.

In the category of sets the image of a morphism

f : X \to Y

is the inclusion from the ordinary image

\{f(x) ~|~ x \in X\}

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.

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