A Coalgebraic Approach to Dualities for Neighborhood Frames

We develop a uniform coalgebraic approach to J\'onsson-Tarski and Thomason type dualities for various classes of neighborhood frames algebras. In the first part paper we construct an endofunctor on category complete atomic Boolean algebras that is dual double powerset functor $\mathsf{Set}$. This allows us show duality can be viewed as algebra-coalgebra duality. generalize this any class presented by one-step axioms in language infinitary modal logic. As consequence, obtain frames, including monotone pretopological spaces, topological spaces. second J\'{o}nsson-Tarski descriptive frames. introduce analogue Vietoris Stone spaces are isomorphic coalgebras endofunctor. proof between Using finitary logic, restrict other studied literature, contingency conclude connecting two types via canonical extensions, discuss when these extensions functorial.