Duality for some categories of coalgebras

Abstract. A contravariant duality is constructed between the category of coalgebras of a given signature, and a category of Boolean algebras with operators, including modal operators corresponding to state transitions in coalgebras, and distinguished elements abstracting the sets of states defined by observable equations.¶This duality is used to give a new proof that a class of coalgebras is definable by Boolean combinations of observable equations if it is closed under disjoint unions, domains and images of coalgebraic morphisms, and ultrafilter enlargements. The proof reduces the problem to a direct application of Birkhoff's variety theorem characterising equational classes of algebras.