Unification of Modal Logic via Topological Categories

In this paper we provide a unifying description of different types of semantics of modal logic found in the literature via the framework of topological categories. In the style of categorical logic, we establish an exact correspondence between various syntactic extensions of modal logic on one hand, including modal dependence, group agent structures, and logical dynamic, and semantic structures in topological categories on the other hand. This framework provides us a uniform treatment of interpreting these syntactic extensions in all different types of semantics of modal logic, and it deepens our conceptual understanding of the abstract structure of modal logic.