Proposition. Let $T$ be a monad on $C$ and consider the forgetful functor $$ R^T \colon C^T \to C $$ from the category of Eilenberg-Moore algebras to $C$. This functor

  • creates limits;
  • creates colimits which are preserved by $T$ and $T^2$.

I have an example in mind of a functor which is created: direct limits (aka. colimits under directed diagrams) by the forgetful functor $$ U \colon \operatorname{Ring} \to \operatorname{Set}. $$

Question: are direct limits in Set preserved by

  1. the monad associated to the adjunction with Rings and its square,
  2. every monad and its square?

More generally, can I expect to prove that any direct limit (aka. colimit under directed diagam) is created by the forgetful functor from any Eilenberg-Moore algebra via the above proposition?

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    $\begingroup$ Without having thought too much about the problem, my initial strategy for investigation would be: first ask if $M$ is a monoid, then does the forgetful functor from $M$-sets to sets create directed limits? If so, can I recast the proof into categorical language which would then generalize to arbitrary monads? $\endgroup$ – Daniel Schepler Jan 23 at 23:48
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    $\begingroup$ Hmm, but on the other hand, the underlying set function from compact Hausdorff topological spaces to sets is monadic, yet a direct limit of compact $T_2$ spaces is not necessarily compact. $\endgroup$ – Daniel Schepler Jan 23 at 23:52
  • $\begingroup$ @DanielSchepler Ah, you got my example right before me! $\endgroup$ – Kevin Arlin Jan 23 at 23:59
  • $\begingroup$ Side note for the case of rings: consider reading the chapter on algebraic theories in Borceux "Handbook of categorical algebra 2". $\endgroup$ – piombino May 30 at 23:24

Yes $U:\mathrm{Ring}\to\mathrm{Set}$ preserves directed colimits. The directed colimit of the underlying set of a directed family $(R_i)$ of rings is endowed with a ring structure in which the ring operations on a sequence $(r_1,...,r_k)$ of elements are defined by mapping all the $r_i$ into some ring $R_k$ in which they are all represented, then applying the operations of $R_k$. It is straightforward to verify that this ring satisfies the universal property of the direct limit, and that the same argument applies to any category of algebras and finitary operations, and to filtered colimits as well as directed colimits.

However, the same does not hold true for arbitrary monads. For instance, the monad $\beta$ which sends a set $A$ to the set of ultrafilters on $A$ has as category of algebras the category of compact Hausdorff spaces, and the forgetful functor from $\mathrm{CompHaus}$ to $\mathrm{Set}$ does not preserve directed colimits. For instance, the colimit of the sequence of discrete spaces $\{1,...,n\}$ in $\mathrm{CompHaus}$ is the Stone-Cech compactification of $\mathbb N$, a rare natural example of a space with cardinality greater than that of the continuum, while the colimit of the underlying sets is good old $\mathbb{N}$.

Any monad for algebras with infinitary operations, such as the monad for lattices equipped with countable suprema, will similarly fail to preserve directed colimits.

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  • $\begingroup$ Is the one-point compactification of $\mathbb{N}$ really the directed limit? If you consider the compatible maps $\{ 1, \ldots, n \} \to \{ \pm 1 \}$, $i \mapsto (-1)^i$, where does the induced map send $\infty$? $\endgroup$ – Daniel Schepler Jan 24 at 0:07
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    $\begingroup$ In fact, if $\beta^{top} : \mathrm{Set} \to \mathrm{CompHaus}$ is the left adjoint, then $\beta^{top} \{ 1, \ldots, n \}$ is the discrete space $\{ 1, \ldots, n \}$ and it must preserve colimits, so I would think the colimit must be $\beta^{top} \mathbb{N}$. $\endgroup$ – Daniel Schepler Jan 24 at 0:19
  • $\begingroup$ Ah, thanks, silly me. $\endgroup$ – Kevin Arlin Jan 24 at 1:56
  • $\begingroup$ Thanks for the counterexamples! Regarding the case of Rings, I feel a bit twisted about how one ends up proving that $U$ creates directed colimits, ie. giving a set theoretic description and showing that "it works". I hoped to use a categorical argument to say that the directed colimits in Set has rings operations induced by colimits property making it a directed colimit in Ring, in a similar way to the first point of the proposition. Yet, assuming that $U$ preserves directed colimits then its clear that $T = UF$ preserves them, so you gave an answer. Thanks for the heads up on finitary algs. $\endgroup$ – piombino Jan 24 at 7:42
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    $\begingroup$ @piombino Sure! I don't think it would be hard to argue from the universal property in Set, but it would be a bit uglier. A key reason why directed colimits are so popular is the simplicity of their explicit construction in Set. $\endgroup$ – Kevin Arlin Jan 24 at 14:45

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