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.