Coalgebraic semantics for positive modal logic

Positive Modal Logic is the restriction of the modal local consequence relation defined by the class of all Kripke models to the propositional negation-free modal language. The class of positive modal algebras is the one canonically associated with PML according to the theory of the algebraization of logics. In [4], a Priestley-style duality is established between the category of positive modal algebras and the category of K⁺-spaces. In this paper, we establish a categorical equivalence between the category K⁺ of K⁺-spaces and the category Coalg(V) of coalgebras of a suitable endofunctor V on the category of Priestley spaces.