# Coequalizers in an Eilenberg-Moore category

Last month I proved that some category $\mathbf C$ that I happen to care about is isomorphic to the Eilenberg-Moore category for a monad on the category of bounded posets $\mathbf{BPos}$.

I know from other results that $\mathbf C$ is cocomplete. Coproducts in $\mathbf C$ are easy to describe, but I am struggling to find an explicit description of coequalizers in $\mathbf C$. I need this to prove a hypothesis that a certain complicated explicit construction in $\mathbf C$ is, in fact, just a colimit.

Is it possible to give an explicit description of coequalizers in an Eilenberg-Moore category over a concrete category, such as $\mathbf{BPos}$?

Any pointers to relevant papers are appreciated.

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Proposition: If $C$ is cocomplete and a monad $T$ on $C$ preserves reflexive coequalizers, then the category of algebras $C^T$ is cocomplete.

Indeed, the forgetful functor $U: C^T \to C$ preserves and reflects any class of colimits that $T$ preserves, so that if $T$ preserves reflexive coequalizers and $C$ has them, then so will $C^T$. As Zhen said, we can get general coequalizers in $C^T$ if $C^T$ has binary coproducts and reflexive coequalizers, but it turns out that this follows from $C^T$ having reflexive coequalizers and $C$ having binary coproducts. See the arguments presented here for details. See particularly theorem 1, and the second corollary below it.

On the off-chance that your monad is finitary (preserves filtered colimits), this might come in handy:

Proposition: If $C$ is complete and cocomplete and $T$ is a finitary monad, then $C^T$ has coequalizers (and therefore is also complete and cocomplete).

See Barr and Wells, Toposes, Theories, and Triples, p. 267 (theorem 3.9) for a somewhat sharper statement.

For example, if products in $C$ distribute over colimits (as they do in the category of bounded posets), and your monad came from a finitary Lawvere theory $T$, this proposition would apply. See also this MO answer and this page from the ncatlab written in support of that answer.

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Todd, an enriched version of this question was asked very recently (Reference request on the bottom of the related question list) and comments to that refer to Linton's paper and give a reference for your first proposition, which is II.7.4 in math.uchicago.edu/~may/BOOKS/EKMM.pdf. – Peter May Jul 30 '13 at 1:24
@PeterMay: thanks for the pointer, and also for the reference for the proposition (which must be written down in a number of places, and probably rediscovered many times). – Todd Trimble Jul 30 '13 at 2:19

This question is addressed by Linton [1969, Coequalizers in categories of algebras]. The first step is to notice that, in the presence of binary coproducts, coequalisers exist if and only if reflexive coequalisers exist.

One option (which always works when the base category is $\mathbf{Set}$) is to assume that the underlying endofunctor of the monad preserves (mono, regular epi) factorisations and that the base is a complete well-powered effective regular category. Under these hypotheses, the coequaliser of a congruence can be computed in the base category, and moreover any parallel pair generates a "smallest" congruence, whose coequaliser will also be the coequaliser of the original parallel pair. Linton describes a more general version of this argument where the factorisation system need not be (mono, regular epi); perhaps that will be applicable in your context.

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