Fitting's Heyting-valued logic and Heyting-valued modal logic have already been studied from an algebraic viewpoint. In addition to algebraic axiomatizations with the completeness of Fitting's Heyting-valued logic and Heyting-valued modal logic, both topological and coalgebraic dualities have also been developed for algebras of Fitting's Heyting-valued modal logic. Bitopological methods have recently been employed to investigate duality for Fitting's Heyting-valued logic. However, the concepts of bitopology and biVietoris coalgebras are conspicuously absent from the development of dualities for Fitting's many-valued modal logic. With this study, we try to bridge that gap. We develop a bitopological duality for algebras of Fitting's Heyting-valued modal logic. We construct a bi-Vietoris functor on the category $PBS_{\mathcal{L}}$ of $\mathcal{L}$-valued ($\mathcal{L}$ is a Heyting algebra) pairwise Boolean spaces. Finally, we obtain a dual equivalence between categories of biVietoris coalgebras and algebras of Fitting's Heyting-valued modal logic. As a result, we conclude that Fitting's many-valued modal logic is sound and complete with respect to the coalgebras of a biVietoris functor. We discuss the application of this coalgebraic approach to bitopological duality.

翻译：Fitting的海廷值逻辑与海廷值模态逻辑已从代数视角得到研究。除Fitting海廷值逻辑与海廷值模态逻辑的代数公理化及完备性外，针对Fitting海廷值模态逻辑代数的拓扑对偶性与余代数对偶性亦已建立。近期双拓扑方法已被用于探究Fitting海廷值逻辑的对偶性。然而，在Fitting多值模态逻辑对偶性的发展中，双拓扑与双Vietoris余代数的概念明显缺失。本研究试图弥合这一空白。我们为Fitting海廷值模态逻辑代数建立双拓扑对偶性，在$\mathcal{L}$值（$\mathcal{L}$为海廷代数）成对布尔空间范畴$PBS_{\mathcal{L}}$上构造双Vietoris函子，最终获得双Vietoris余代数范畴与Fitting海廷值模态逻辑代数范畴间的对偶等价。由此我们得出结论：Fitting多值模态逻辑相对于双Vietoris函子的余代数是可靠且完备的。本文进一步探讨该余代数方法在双拓扑对偶性中的应用。