Pushout in category of Stone spaces

Question

I suspect that the pushout in the category of Stone spaces and continuous maps exists and is the same as in the category of all topological spaces, but I have not found it in literature yet. I am looking for a reference.

Answer 1

As mentioned in the comments, existence is immediate by Stone duality: pushouts of Stone spaces are just dual to pullbacks of Boolean algebras. Alternatively, Stone spaces have all colimits because they are a reflective subcategory of topological spaces, so you can compute a colimit of a diagram of Stone spaces by first taking its colimit as topological spaces and then applying the reflector to Stone spaces. Explicitly, the reflector sends a space $X$ to the closure of the image of the evaluation map $X\to \{0,1\}^I$ where $I$ is the set of all continuous maps $X\to\{0,1\}$. (This is closely analogous to the construction of the Stone-Cech compactification, i.e. the reflector to compact Hausdorff spaces, which uses $[0,1]$ instead of $\{0,1\}$.)

Pushouts of Stone spaces do not coincide with pushouts of topological spaces in general. For instance, in this answer to a closely related question, I gave an example of a pair of maps $$f,g:\beta\mathbb{N}\to K$$ (here $K$ is the Cantor set) whose coequalizer in the category of topological spaces is $[0,1]$. Since $[0,1]$ is not a Stone space, the coequalizer in Stone spaces must obviously be different, and in fact is just a point (the reflection of $[0,1]$ in Stone spaces). You can turn this into an example with pushouts instead of coequalizers by just taking the diagram $$K\stackrel{(f,1)}\leftarrow \beta\mathbb{N}\coprod K\stackrel{(g,1)}\to K$$ where $(f,1)$ is the map that is $f$ on $\beta\mathbb{N}$ and the identity on $K$, and similarly for $(g,1)$. A pushout of this diagram is the same thing as a coequalizer of $f$ and $g$, so the pushout in topological spaces is $[0,1]$ while the pushout in Stone spaces is a point.