# Image (category theory)

Jump to navigation
Jump to search

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 .
- For any object Z with a morphism and a monomorphism such that , there exists a unique morphism such that .

Remarks:

- such a factorization does not necessarily exist
*g*is unique by definition of monic (= left invertible, abstraction of injectivity)*m*is monic.*h*=*lm*already implies that*m*is unique.*k*=*mg*

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

## References

- Section I.10 of Template:Mitchell TOC