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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.

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    $\begingroup$ Stone spaces are dually equivalent to boolean algebras, and the category of boolean algebras has pullbacks, hence the category of stone spaces has pushouts: existence is easy $\endgroup$ – Maxime Ramzi Mar 31 '18 at 14:45
  • $\begingroup$ Do you know an explicit description of the pullback of Boolean algebras? Maybe that helps.. $\endgroup$ – Math Student 020 Apr 1 '18 at 18:35
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    $\begingroup$ Well pullbacks are limits and the forgetful functor to $Set$ preserves limits and we know how to describe pullbacks in $Set$ so yes, we can describe it easily. However, the description of the equivalence uses ultrafilters on the boolean algebras so the description one would get for the pushout would just be weird. But now that you know a pushout exists, you can try describing it, for instance (I don't know if that works at all) you can try to see if by any chance the pushout in $Top$ is still a Stone space $\endgroup$ – Maxime Ramzi Apr 1 '18 at 19:00
  • $\begingroup$ I wouldn't expect that pushout of Stone spaces, taken in $Top$, would always produce a Stone space again. At least, this is not true for compact Hausdorff spaces, with a classical example of $[-1,1]\ \hookleftarrow\ [-1,1]\setminus\{0\}\ \hookrightarrow\ [-1,1]$ $\endgroup$ – Berci Apr 1 '18 at 23:37
  • $\begingroup$ @Berci But $[-1,1]\setminus\{0\}$ is not compact. The pushout of two continuous injections of compact Hausdorff spaces is still compact Hausdorff. $\endgroup$ – Idéophage Jul 30 '20 at 3:12
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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.

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