From T-Coalgebras to Filter Structures and Transition Systems

Abstract

For any set-endofunctor \(T : {\mathcal S}et \rightarrow {\mathcal S}et\) there exists a largest sub-cartesian transformation μ to the filter functor \({\mathbb F}: {\mathcal S}et \rightarrow {\mathcal S}et\). Thus we can associate with every T-coalgebra A a certain filter-coalgebra \(A_{\mathbb F}\).

Precisely, when T (weakly) preserves preimages, μ is natural, and when T (weakly) preserves intersections, μ factors through the covariant powerset functor \({\mathbb P}\), thus providing for every T-coalgebra A a Kripke structure \(A_{\mathbb P}\).

We characterize preservation of preimages, preservation of intersections, and preservation of both preimages and intersections via the existence of natural, sub-cartesian or cartesian transformations from T to either \({\mathbb F}\) or \({\mathbb P}\).

Moreover, we define for arbitrary T-coalgebras \({\mathcal A}\) a next-time operator \(\bigcirc_{\mathcal A}\) with associated modal operators □ and \(\lozenge\) and relate their properties to weak limit preservation properties of T. In particular, for any T-coalgebra \({\mathcal A}\) there is a transition system \({\mathcal K}\) with \(\bigcirc_{A} = \bigcirc_{K}\) if and only if T preserves intersections.