  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.
