// Workers AI · dad joke modeWhy was coimage sad? It was feeling a little flat.

In algebra, the coimage of a homomorphism

$f:A\rightarrow B$

is the quotient

${\text{coim}}f=A/\ker(f)$

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 $f:X\rightarrow Y$, then a coimage of $f$ (if it exists) is an epimorphism $c:X\rightarrow C$ such that

1. there is a map $f_{c}:C\rightarrow Y$ with $f=f_{c}\circ c$,
2. for any epimorphism $z:X\rightarrow Z$ for which there is a map $f_{z}:Z\rightarrow Y$ with $f=f_{z}\circ z$, there is a unique map $h:Z\rightarrow C$ such that both $c=h\circ z$ and $f_{z}=f_{c}\circ h$

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.