Maps and Monads for Modal Frames

The category-theoretic nature of general frames for modal logic is explored. A new notion of “modal map” between frames is defined, generalizing the usual notion of bounded morphism/p-morphism. The category Fm of all frames and modal maps has reflective subcategories CHFm of compact Hausdorff frames, DFm of descriptive frames, and UEFm of ultrafilter enlargements of frames. All three subcategories are equivalent, and are dual to the category of modal algebras and their homomorphisms. The ultrafilter enlargement of a frame A is shown to be the free compact Hausdorff frame generated by A relative to Fm. The monad E of the resulting adjunction is examined and its Eilenberg-Moore category is shown to be isomorphic to CHFm. A categorical equivalence between the Kleisli category of E and UEFm is defined from a construction that assigns to each frame A a frame A∗ that is “image-closed” in the sense that every point-image {b : aRb} in A is topologically closed. A∗ is the unique image-closed frame having the same ultrafilter enlargement as A. These ideas are connected to a category W shown by S. K. Thomason to be dual to the category of complete and atomic modal algebras and their homomorphisms. W is the full subcategory of the Kleisli category of E based on the Kripke frames.