For purposes of this page, a fork (some might say a “cofork”) in a category is a diagram of the form
such that . A split coequalizer is a fork together with morphisms and such that , , and . This is equivalent to saying that the morphism has a section in the arrow category of .
Coequalizers and absolute coequalizers
The name “split coequalizer” is appropriate, because in any split coequalizer diagram, the morphism is necessarily a coequalizer of and . For given any such that , the composite provides a factorization of through , since , and such a factorization is unique since is (split) epic. In fact, a split coequalizer is not just a coequalizer but an absolute coequalizer: one preserved by all functors.
On the other hand, suppose we are given only and such that and (which is certainly the case in any split coequalizer, since ). Such a situation is sometimes called a contractible pair. In this case, any coequalizer of and is split, for if is a coequalizer of and , then the equation implies, by the universal property of , a unique morphism such that , whence and so since is epic.
Similarly, if splits the idempotent with section , so that and , then we have
and the other identities are obvious; thus is a split coequalizer of and .
Dually, if is a split epimorphism, with a splitting , say, then is a split coequalizer of , the morphism being the identity.
Moreover, is also the split coequalizer of its kernel pair, if the latter exists. For if is this kernel pair, then the two maps satisfy , and hence induce a map such that and .
Beck coequalizer for algebras over a monad
The “ur-example” of a split coequalizer is the following. Let be an algebra for the monad on the category , with structure map . Then the diagram
called the canonical presentation of the algebra , is a split coequalizer in (and a reflexive coequalizer in the category of -algebras). The splittings are given by and . (Here and are the multiplication and unit of the monad .)
This split coequalizer figures prominently in Beck’s monadicity theorem, whence also called the Beck coequalizer.
See also at Eilenberg-Moore category – As a colimit completion of the Kleisli category.