Coalgebraic Geometric Logic: Basic Theory

Abstract

Using the theory of coalgebra, we introduce a uniform framework for addingmodalities to the language of propositional geometric logic. Models for thislogic are based on coalgebras for an endofunctor on some full subcategory ofthe category of topological spaces and continuous functions. We investigatederivation systems, soundness and completeness for such geometric modal logics,and we specify a method of lifting an endofunctor on Set, accompanied by acollection of predicate liftings, to an endofunctor on the category oftopological spaces, again accompanied by a collection of (open) predicateliftings. Furthermore, we compare the notions of modal equivalence, behaviouralequivalence and bisimulation on the resulting class of models, and we provide afinal object for the corresponding category