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 as below
such that , , and . This is equivalent to saying that the morphism has a section in the arrow category of .
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. This is meant in the sense in which any functor can “preserve a coequalizer”, namely that is a coequalizer of and for any functor . Of course this happens because also preserves the splitting morphisms and , but those are not part of the coequalizer diagram that is “preserved” in this statement.
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 .
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.
See also split equalizer.
Last revised on February 23, 2024 at 22:17:31. See the history of this page for a list of all contributions to it.