Last time, we concluded with a mysterious observation. There is a theory, that of unital inflationary topological semi-lattices, which plays a fundamental role in the study of the Hoare powerspace. On the one hand, H(X) is the free sober such thing. On the other hand, the algebras of the H monad are exactly those things that are sober. We shall investigate that by looking at theories themselves, and show how those constructions arise from a logical perspective. In the end, this turned out to be more complicated than what I had thought initially… Read more.