# 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
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.