A Coalgebraic Approach to Dualities for Neighborhood Frames

TL;DR

This paper introduces a unified coalgebraic framework for dualities in neighborhood frames and algebras, generalizing existing dualities and applying to various classes including topological and monotone structures.

Contribution

It develops a coalgebraic approach to Jönsson-Tarski and Thomason dualities, extending them to a broad class of neighborhood structures via one-step axioms.

Findings

Unified coalgebraic duality framework for neighborhood frames and algebras.

Representation of Thomason duality as algebra-coalgebra duality.

Application to classes like monotone neighborhood frames and topological spaces.

Abstract

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.