We develop a uniform coalgebraic approach to Thomason and J\'{o}nsson-Tarski 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.

翻译：我们对Thomason和J\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\-Tarski 型的煤矿介质。我们在论文的第一部分中为邻区框架和邻区代代体的不同类别, 开发一种统一的煤基的二元化方法。 在本文的第二部分中,我们将这种方法推广到任何一类的代代数, 由一阶级的直系代数和邻代的内代数 。 因此, 我们获得一种统一的方法来研究各种邻系的双元区框架, 包括单级邻代区框架、 前代区框架、 前代体空间和表的双代代体。