A reflexive pair is a parallel pair having a common section, i.e. a map such that . A reflexive coequalizer is a coequalizer of a reflexive pair. A category has reflexive coequalizers if it has coequalizers of all reflexive pairs.
Dually, a reflexive coequalizer in the opposite category is called a coreflexive equalizer in .
Reflexive coequalizers should not be confused with split coequalizers, a distinct concept.
If is a monad on a cocomplete category , then the category of Eilenberg Moore algebras is cocomplete if and only if it has reflexive coequalizers. This is the case particularly if preserves reflexive coequalizers.
This is due to (Linton).
Suppose has reflexive coequalizers. Then certainly has coproducts, because if is a collection of -algebras, then we can form the coequalizer in of the reflexive pair
using the fact that the displayed coproducts exist because, for example,
since the left adjoint preserves coproducts, assumed to exist in . That this reflexive coequalizer is the coproduct in is routine.
Finally, a category with coproducts and reflexive coequalizers is cocomplete. It suffices that general coequalizers exist, but it is easily seen that if
is a parallel pair, then the coequalizer of the reflexive pair
(note both maps are retracts of the inclusion ) also exists, and gives the coequalizer of the first pair. This completes the proof.
If is a functor of two variables which preserves reflexive coequalizers in each variable separately (that is, and preserve reflexive coequalizers for all and ), then preserves reflexive coequalizers in both variables together.
This is emphatically not the case for arbitrary coequalizers.
of proposition 1 Suppose given two reflexive coequalizers
and let denote so that we have a diagram
in which all rows and columns are reflexive coequalizers (using preservation of reflexive coequalizers in separate variables), and all squares are serially commutative. According to Toposes, Triples, Theories, lemma 4.2 page 248, the diagonal is also a (reflexive) coequalizer, as claimed. (See also the lemma on page 1 of Johnstone’s Topos Theory.)
Proposition 1 is particularly interesting when is the tensor product of a cocomplete closed monoidal category . In this case it implies that the free monoid monad on such a category preserves reflexive coequalizers, and thus (by Linton’s theorem) the category of monoid objects in is cocomplete.
the -fold product functors preserve reflexive coequalizers.
Of course, the diagonal functor , being left adjoint to the product functor, preserves reflexive coequalizers; therefore the composite
also preserves reflexive coequalizers.
This has a further consequence which is technically very convenient:
If is a finitary monad on , then preserves reflexive coequalizers.
We have a coend formula for :
and since this is a colimit of functors which preserve reflexive coequalizers, must also preserve reflexive coequalizers.
Since finitary monads preserve reflexive coequalizers, it follows that the monadic functor reflects reflexive coequalizers, and so since has reflexive coequalizers, must as well. Therefore, by proposition 1, is cocomplete. This is actually true for infinitary monads on as well, at least if we assume the axiom of choice (see here for a proof), but the argument just given is a choice-free proof for the case of finitary monads.