A coalgebraic view on 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 algebrization of logics [12]. A Priestley-style duality is established between the category of positive modal algebras and the category of K+-spaces in [4]. 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.