ABSTRACT
It is well known (among aficionados of continuous lattices, at least) that the Vietoris space construction defines a monad on the category of compact Hausdorff spaces, and that the algebras for this monad are exactly the compact Lawson semilattices. Now when one is working constructively (inside a topos, if you wish) it is by now well established [2,4,5] that various constructions and theorems that are classically applied to compact Hausdorff spaces (and that classically involve nonconstructive principles such as the prime ideal theorem) must be considered instead in the context of compact regular locales; that is, we must forget about the points of the spaces and concentrate on their open-set lattices. Accordingly, in seeking to understand how one might define ’’Lawson semi lattice s’’ in a topos, I began by investigating whether the Vietoris-space monad admitted a constructive generalization to the category of compact regular locales. [A different approach was indicated by Porta and Wyler [11], who (implicitly) defined Lawson semilattices in a topos as algebras for the filter monad on the topos itself; but since the resulting category does not admit a forgetful functor to compact regular locales, this approach seems to me unsatisfactory.]
