Image (category theory)

Given a category "C" and a morphism $f:X\; ightarrow\; Y$ in "C", the image of "f" is a monomorphism $h:I\; ightarrow\; Y$ satisfying the following universal property:
#There exists a morphism $g:X\; ightarrow\; I$ such that "f" = "hg".
#For any object Z with a morphism $k:X\; ightarrow\; Z$ and a monomorphism $l:Z\; ightarrow\; Y$ such that "f" = "lk", there exists a unique morphism $m:I\; ightarrow\; Z$ 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".

Examples

In the category of sets the image of a morphism $f\; :\; X\; o\; Y$ is the inclusion from the ordinary image $\{f(x)\; ~|~\; x\; in\; X\}$ 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.

See also

*subobject
*coimage
*image (mathematics)