A coequalizer of a pair of maps in a category is a map such that

1. , where denotes composition.

2. For any other map with the same property, there is exactly one map such that 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

and by the canonical map , where is the minimal equivalence relation on that identifies and for all .

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

The dual notion of the coequalizer is the equalizer.

This entry contributed by Margherita Barile

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.

CITE THIS AS:

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