We argue that the category of Stone spaces forms an interesting base category for coalgebras, in particular, if one considers the Vietoris functor as an analogue to the power set functor on the category of sets.
We prove that the so-called descriptive general frames, which play a fundamental role in the semantics of modal logics, can be seen as Stone coalgebras in a natural way. This yields a duality between modal algebras and coalgebras for the Vietoris functor.
Building on this idea, we introduce the notion of a Vietoris polynomial functor over the category of Stone spaces. For each such functor T we provide an adjunction between T-sorted Boolean algebras with operators and the Stone coalgebras for T. We also identify the subcategory of algebras on which the adjunction restricts to an equivalence and show that the final T-coalgebra is the dual of the initial T-BAO.
C. Kupke, A. Kurz, and Y. Venema, “Stone coalgebras,” Theoretical Computer Science, vol. 327, no. 1–2, pp. 109–134, Oct. 2004. DOI: 10.1016/j.tcs.2004.07.023