Coimage

In algebra, the coimage of a homomorphism

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

is the quotient

${\displaystyle {\text{coim}}f=A/\ker(f)}${\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 ${\displaystyle f:X\rightarrow Y}${\displaystyle f:X\rightarrow Y}, then a coimage of ${\displaystyle f}${\displaystyle f} (if it exists) is an epimorphism ${\displaystyle c:X\rightarrow C}${\displaystyle c:X\rightarrow C} such that

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

• Quotient object
• Cokernel

• Abstract algebra
• Isomorphism theorems
• Category theory
• Linear algebra stubs

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