Bi-coalgebraic view of Fitting's Heyting-valued modal logic

Fitting's Heyting-valued modal logic and Heyting-valued logic have previously been examined from an algebraic perspective. Topological duality theorems have been developed in addition to algebraic axiomatizations with the completeness of Fitting's logic and modal logic. Recently, bitopological techniques have been used to study duality for Heyting-valued logic. But the development of duality for Heyting-valued modal logic noticeably lacks bitopology and biVietoris-coalgebra techniques. We are trying to bridge this gap in this paper. We establish a bitopological duality for algebras of Fitting's Heyting-valued modal logic. We build a bi-Vietoris functor on the category of Heyting-valued pairwise Boolean spaces, denoted by $PBS_{\mathcal{L}}$. In the end, we derive a dual equivalence between algebras of Fitting's Heyting-valued modal logic and categories of bi-Vietoris coalgebras. We thus conclude that, with respect to the coalgebras of a bi-Vietoris functor, Fitting's many-valued modal logic is sound and complete.