Topological duality for intuitionistic modal algebras

Abstract

This paper describes a generalisation of the adjunction and duality between topological spaces and frames, inspired by the Kripke semantics of modal logic. A relational space is defined as a topological space with a binary relation on the points, and a modal frame is defined as a frame with two operations □ and ◊ satisfying various axioms. The frame of open sets of a relational space has a natural modal structure; with appropriate morphisms, this extends to a contravariant functor from the category of relational spaces to the category of modal frames. This paper defines an adjoint to this functor, and shows that the adjunction restricts to a duality between the subcategories in the image of the adjunction. The paper goes on to consider those modal frames which are freely generated from modal distributive lattices, and in particular, those arising from the Lindenbaum algebras of intuitionistic propositional modal languages. These spectral modal frames are constructed by defining a modal structure on the frame of ideals, and characterised (up to isomorphism) by exactness and compactness conditions of the modal connectives. Finally, they are shown to be equivalent to the frames of open sets of relational spaces; from this a completeness theorem for intuitionistic modal logic follows.