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