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Coequalizer


Coequalizer

A coequalizer of a pair of maps f,g:X->Y in a category is a map c:Y->C such that

1. c degreesf=c degreesg, where  degrees denotes composition.

2. For any other map c^':Y->C^' with the same property, there is exactly one map gamma:C->C^' such that c^'=gamma degreesc, i.e., one has the above commutative diagram.

It can be shown that the coequalizer is an epimorphism and that, moreover, it is unique up to isomorphism.

In the category of sets, the coequalizer is given by the quotient set

 C=Y/∼,

and by the canonical map c:Y->C, where ∼ is the minimal equivalence relation on Y that identifies f(x) and g(x) for all x in X.

The same construction is valid in the categories of additive groups, modules, and vector spaces. In these cases, the cokernel of a morphism f can be viewed, in a more abstract categorical setting, as the coequalizer C of f and the zero map.

The dual notion of the coequalizer is the equalizer.


See also

Cokernel, Commutative Diagram, Equalizer

This entry contributed by Margherita Barile

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References

Herrlich, H. and Strecker, G. E. "Equalizers and Coequalizers." §16 in Category Theory: An Introduction. Berlin: Heldermann Verlag, pp. 100-107, 1979.Schubert, H. "Coequalizers." §8.2 in Categories. Berlin: Springer-Verlag, pp. 64-65, 1972.

Referenced on Wolfram|Alpha

Coequalizer

Cite this as:

Barile, Margherita. "Coequalizer." From MathWorld--A Wolfram Web Resource, created by Eric W. Weisstein. https://mathworld.wolfram.com/Coequalizer.html

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