Coimage

Template:Short description In algebra, the coimage of a homomorphism

$f:A\to B$

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

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

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