Final coalgebras for functors on measurable spaces

Abstract

We prove that every functor on the category Meas of measurable spaces built from the identity and constant functors using products, coproducts, and the probability measure functor Δ has a final coalgebra. Our work builds on the construction of the universal Harsanyi type spaces by Heifetz and Samet and papers by Rößiger and Jacobs on coalgebraic modal logic. We construct logical languages, probabilistic logics of transition systems, and interpret them on coalgebras. The final coalgebra is carried by the set of descriptions of all points in all coalgebras. For the category Set , we work with the functor D of discrete probability measures. We prove that every functor on Set built from D and the expected functors has a final coalgebra. The work for Set differs from the work for Meas : negation in needed for final coalgebras on Set but not for Meas .