Image (category Theory)

Image (category Theory)

Given a category C and a morphism in C, the image of f is a monomorphism satisfying the following universal property:

1. There exists a morphism such that f = hg.
2. For any object Z with a morphism and a monomorphism such that f = lk, there exists a unique morphism 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.