It is easy to find references stating that the category of compact Hausdorff spaces $\mathbf{CompHaus}$ is equivalent to the category of algebras for the ultrafilter monad, $\mathbf{\beta Alg}$. After doing some digging, the $\mathbf{CompHaus}\to \mathbf{\beta Alg}$ half of the equivalence is simple enough, but I haven't been able to find a description of the $\mathbf{\beta Alg}\to \mathbf{CompHaus}$ half of this equivalence. I have tried to work it out, but I have little experience with topological spaces and am not sure what the associated topology ought to look like.
I'm wondering if anyone has a good reference that describes the $\mathbf{\beta Alg}\to \mathbf{CompHaus}$ half of the equivalence, or can describe it here.
Best Answer
The other half of the equivalence is described on the nLab page on ultrafilters.
Given an algebra structure $\xi\colon \beta X \to X$, we define the topology on $X$ by declaring that a subset $U\subseteq X$ is open if and only if for every point $x\in U$ and every ultrafilter $F\in \beta X$ such that $\xi(F) = x$, we have $U\in F$.