Recall that a Stonean space is a compact, Hausdorff extremally disconnected topological space. It is known that under the Stone duality they correspond to the complete Boolean algebras.
Now, in nLab webpage, it is said that they form a category if we take open continuous maps, which is a bit strange, since they also form a category if one takes just continuous maps. My guess is that it is said this may be because open continuous maps correspond to continuous homomorphism between complete Boolean algebras, and not just homomorphisms of Boolean algebras. Is that right? There is also other stronger reasons?
On the other hand, I am not sure if the category of Stonean spaces have fibre products, or more general finite limits. May be they have only if one takes open continuous maps as morfisms?