Image (category theory)
- 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 .
- such a factorization does not necessarily exist
- g is unique by definition of monic
- m is monic.
- h=lm already implies that m is unique.
The image of f is often denoted by im f or Im(f).
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.
- im f = ker coker f
In an abelian category (which is in particular binormal), if f is a monomorphism then f = ker coker f, and so f = im f.