We can show that the category of profinite algebras, cofiltered limits of finite algebras, is monadic over Stone spaces as follows. So, I wonder if there are any other examples.

In case that I was mistaken, a proof is sketched as follows.

Proposition. Let $A$ be an algebraic theory and $\mathbf{StoneA}$ the category of topological $A$-algebras with continuous homomorphisms. Then, the category of profinite $A$-algebras is monadic over $\mathbf{Stone}$.

Proof sketch. First we observe that the full subcategory $Pro\mathbf{A}_f$ on profinite $A$-algebras is the pro-completion of the category $\mathbf{A}_f$ of finite $A$-algebras (it can found in a Remark of Johnstone's Stone Spaces on profiniteness). Also, profinite algebras are closed under limits in $\mathbf{StoneA}$ and limits in $\mathbf{StoneA}$ are created in $\mathbf{Stone}$, so the forgetful functor $U\colon Pro\mathbf{A}_f \to \mathbf{Stone}$ preserves all limits. By Adjoint Functor Theorem between (co-)LFP categories, this forgetful functor has a left adjoint. Now, given a $U$-absolute pair $f, g\colon X \rightrightarrows Y$, i.e. $f$ and $g$ has an absolute coequaliser $e\colon Y \twoheadrightarrow Z$ in $\mathbf{Stone}$, we can define a topological $A$-algebra on $Z$ so that $Z$ is a quotient algebra of $Y$. Then, a quotient of a profinite algebra is again profinite (by, for example, Theorem 4.3 of http://arxiv.org/abs/1309.2422). Therefore, by Beck's Monadicity Theorem, $U$ is monadic. $\square$

Except the last step (a quotient of a profinite object remains profinite), other steps are fairly standard. Hence, there might be other examples such as:

Question Is the category of profinite $T$-algebra for a monad $T$ on $\mathbf{Stone}$ monadic over $\mathbf{Stone}$? Or, is there any other similar example?


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