Is direct limit created by the forgetful functor from an Eilenberg-Moore algebra

Question

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?

Answer 1

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.