Coimage

Coimage is a construction in category theory, dual to the notion of image. For a morphism f: A → B in a category that has kernels and cokernels, the coimage Coim(f) is defined as the cokernel of the kernel of f: Coim(f) = coker(ker(f) → A). The universal property yields a canonical morphism p: Coim(f) → B such that p composed with the canonical map A → Coim(f) equals f. Dually, the image Im(f) is defined as the kernel of the cokernel of f, i.e., Im(f) = ker(coker(f) → B).

The morphism f factors through both constructions as A → Coim(f) → B, where A → Coim(f) is the cokernel of ker(f) and hence an epimorphism, and Coim(f) → B is the morphism induced by f. The image factorization of f is obtained by a similar process using the kernel of the cokernel, resulting in a mono Im(f) → B.

In abelian categories (and, more generally, exact categories), the canonical comparison map Coim(f) → Im(f) is an isomorphism, so Coim(f) ≅ Im(f). This leads to the standard epi–mono factorization of f: f factors as an epimorphism A → Im(f) followed by a monomorphism Im(f) → B. In categories that are not abelian, Coim(f) and Im(f) may differ, and the comparison map need not be an isomorphism.

Notes: The coimage concept relies on the existence of kernels and cokernels, and it plays a key role in discussions of exactness, factorization systems, and the relationship between kernels, cokernels, and morphism factorizations.