Image (category theory)
- 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 h = lm.
- 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.
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 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
This holds especially in abelian categories.