一句话：用余代数统一刻画邻域框架与邻域代数的 Jónsson-Tarski 与 Thomason 型对偶。
论文针对邻域框架与邻域代数的多种对偶性缺乏统一范畴论解释的问题，发展了余代数方法。第一部分在完全原子布尔代数范畴上构造与 Set 上双幂集函子对偶的自函子，把 Thomason 对偶解释为代数-余代数对偶，并借助无穷模态逻辑的一步公理推广到单调邻域框架、预拓扑空间和拓扑空间等类。第二部分在 Stone 空间范畴上引入 Vietoris 自函子的类似物，证明描述性邻域框架同构于其余代数，从而给出 Jónsson-Tarski 对偶的余代数证明，并用有穷模态逻辑一步公理限制到单调模态代数、偶然性代数等。最后通过规范扩张连接两类对偶并讨论其函子性。
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.
@article{paperbot1685,
title = {A Coalgebraic Approach to Dualities for Neighborhood Frames},
author = {Guram Bezhanishvili and Nick Bezhanishvili and Jim de Groot},
journal = {Logical Methods in Computer Science},
volume = {Volume 18, Issue 3},
year = {2022},
doi = {10.46298/lmcs-18(3:4)2022}
}