Coimage

In algebra, the coimage of a homomorphism

f: A → B

is the quotient

coim f = A/ker f

of the domain by the kernel. The coimage is canonically isomorphic to the image by the first isomorphism theorem, when that theorem applies.

More generally, in category theory, the coimage of a morphism is the dual notion of the image of a morphism. If f : X → Y, then a coimage of f (if it exists) is an epimorphism c : X → C such that

1. there is a map f_c : C → Y with f = f_c ∘ c,
2. for any epimorphism z : X → Z for which there is a map f_z  : Z → Y with f = f_z ∘ z, there is a unique map π : Z → C such that both c = π ∘ z and f_z = f_c ∘ π.

See also

• Quotient object
• Cokernel

References

• Mitchell, Barry (1965). Theory of categories. Pure and applied mathematics. Vol. 17. Academic Press. ISBN 978-0-124-99250-4. MR 0202787.