On a purely categorical framework for coalgebraic modal logic

A category CoLog of distributive laws is introduced to unify different approaches to modal logic for coalgebras, based merely on the presence of a contravariant functor P that maps a state space to its collection of predicates. We show that categorical constructions, including colimits, limits, and compositions of distributive laws as a tensor product, in CoLog generalise and extend existing constructions given for Set coalgebraic logics and that the framework does not depend on any particular propositional logic or state space. In the case that P establishes a dual adjunction with its dual functor S , we show that a canonically defined coalgebraic logic exists for any type of coalgebras. We further restrict our discussion to finitary algebraic logics and study equational coalgebraic logics. Objects of predicate liftings are used to characterise equational coalgebraic logics. The expressiveness problem is studied via the mate correspondence, which gives an isomorphism between CoLog and the comma category from the pre-composition to the post-composition with S . Then, the modularity of the expressiveness is studied in the comma category via the notion of factorisation system. Dedicated to my parents