邻域框架对偶性的余代数方法
A Coalgebraic Approach to Dualities for Neighborhood Frames
AI总结：
本文提出统一的余代数框架，通过构造对偶函子与一步公理，统一处理邻域框架的Thomason与Jónsson-Tarski对偶性，并推广至多种邻域代数类。
摘要：
我们发展了一种统一的余代数方法，用于处理各类邻域框架和邻域代数的Jónsson-Tarski型与Thomason型对偶性。在论文的第一部分，我们在完全原子布尔代数范畴上构造了一个自函子，它与集合范畴上的双幂集函子相对偶。这使我们能够证明邻域框架的Thomason对偶性可以被视为代数-余代数对偶性。我们将此方法推广到由无穷模态逻辑语言中的一步公理呈现的自函子的任何代数类。作为结果，我们获得了对各类邻域框架（包括单调邻域框架、预拓扑空间和拓扑空间）对偶性的统一方法。在论文的第二部分，我们发展了一种针对邻域代数和描述性邻域框架的Jónsson-Tarski对偶性的余代数方法。我们在Stone空间范畴上引入了Vietoris自函子的类似物，并证明了描述性邻域框架同构于该自函子的余代数。这使我们能够获得描述性邻域框架与邻域代数之间对偶性的余代数证明。利用有限模态逻辑语言中的一步公理，我们将此对偶性限制到文献中研究的其他邻域代数类，包括单调模态代数和偶然性代数。最后，我们通过典范扩张将这两种类型的对偶性联系起来，并讨论了这些扩张何时是函子性的。
英文摘要：
We develop a uniform coalgebraic approach to Jónsson-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 Jónsson-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.
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