Saturated Kripke Structures as Vietoris Coalgebras

TL;DR

This paper establishes an isomorphism between the category of Vietoris coalgebras on topological spaces and modally saturated Kripke structures, extending previous results and exploring their topological properties.

Contribution

It demonstrates the categorical equivalence between Vietoris coalgebras and modally saturated Kripke structures, including their closure properties and existence of a terminal object.

Findings

Vietoris subcoalgebras and bisimulations admit topological closure

The category of Vietoris coalgebras has a terminal object

Establishment of an isomorphism between coalgebras and Kripke structures

Abstract

We show that the category of coalgebras for the compact Vietoris endofunctor $\mathbb{V}$ on the category Top of topological spaces and continuous mappings is isomorphic to the category of all modally saturated Kripke structures. Extending a result of Bezhanishvili, Fontaine and Venema, we also show that Vietoris subcoalgebras as well as bisimulations admit topological closure and that the category of Vietoris coalgebras has a terminal object.