Stone coalgebras

In this paper 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 the category of modal algebras and that of coalgebras over 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 the category of T-sorted Boolean algebras with operators and the category of Stone coalgebras over T. Since the unit of this adjunction is an isomorphism, this shows that Coalg(T)op is a full re?ective subcategory of BAOT . Applications include a general theorem providing ?nal coalgebras in the category of T-coalgebras.