Coimage

In algebra, the coimage of a homomorphism

${\displaystyle f:A\to B}$

is the quotient

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

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

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

See also

• Quotient object
• Cokernel

References

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

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