Vietoris bisimulations

Building on the fact that descriptive frames are coalgebras for the Vietoris functor on the category of Stone spaces, we introduce and study the concept of a Vietoris bisimulation between two descriptive modal models, together with the associated notion of bisimilarity. We prove that our notion of bisimilarity, which is defined in terms of relation lifting, coincides with Kripke bisimilarity (with respect to the underlying Kripke models), with behavioural equivalence, and with modal equivalence, but not with Aczel-Mendler bisimilarity. As a corollary, we obtain that the Vietoris functor does not preserve weak pullbacks. Comparing Vietoris bisimulations between descriptive models to Kripke bisimulations on the underlying Kripke models, we prove that the closure of such a Kripke bisimulation is a Vietoris bisimulation. As a corollary, we show that the collection of Vietoris bisimulations between two descriptive models forms a complete lattice. Finally, we provide a game-theoretic characterization of Vietoris bisimilarity.