Equalizers and Coequalizers

Equalizers

Equalizer

In any category , given parallel morphisms , an equalizer of and consists of and , universal such that:. That is, given with there is a unique with all as in the diagram:

Proposition

If is an equalizer of some pair of morphisms, then is monic.

Proof Suppose is the equalizer of . Then for all object with morphisms and , we have Hence there exists unique with , that is .

e.g.

• In , given functions , their equalizer is the injection into of the equationally defined subset . Since if for some , then for all . Then there is a unique with such that as is monic.
• Given a set , every subset is an equalizer for some pair of functions. In fact, Let set . Then consider the characteristic function for any subset :Then we have . So the following diagram makes it an equalizer:
  
  Moreover, for every function we can form the equational subset: as an equalizer, in the same way.

Prop Write as power set of , we have in .
Proof Inherit the notation from the above proof, clearly we can derive that and are mutually inverse. Then indeed forms an isomorphism. Thus .

Prop In the category of abelian groups, given , the arbitrary homomorphism forms an equalizer:

Coequalizers

Coequalizer

For any parallel morphisms in a category , a coequalizer consists of and , universal with the property , as in

That is, given any and , if , then there exists a unique such that .

Proposition

If is a coequalizer of some pair of morphisms, then is epic.

Proof Observe by duality, we know that such a coequalizer in a category is an equalizer in , hence monic by the proposition, and so is epic in .

Proposition

The coequalizer of a pair of morphisms is unique up to isomorphism.

Proof We use epicness and the universal property:

e.g.

• Let be an equivalence relation on a set with are the two projections of the inclusion . The canonical projection defined by is then a coequalizer:
  In fact, the coequalizer in of an arbitrary parallel pair of functions can be constructed by quotienting by the equivalence relation generated by the equations for all .
• The quotient map in is a coequalizer of the two projections of the equivalence relation . In fact, all coequalizers in are quotient maps.
• Every split epimorphism is a coequalizer of some pair of morphisms. In fact, if is a section of , then is the coequalizer of and .

Prop For every monoid there are sets and and a coequalizer diagram, with and free, thus .