Coimage (original) (raw)

Concept in category theory (in mathematics)

In algebra, the coimage of a homomorphism

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

is the quotient

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

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

See also

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References

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