Coimage Explained

In algebra, the coimage of a homomorphism

f:A → B

coimf=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

(if it exists) is an epimorphism

c:X → C

such that

f_c:C → Y

with

f=f_c\circc

z:X → Z

for which there is a map

f_z:Z → Y

with

f=f_z\circz

, there is a unique map

h:Z → C

such that both

c=h\circz

and

f_z=f_c\circh

See also