Bisimulation on Markov Processes over Arbitrary Measurable Spaces

We introduce a notion of bisimulation on labelled Markov Processes over generic measurable spaces in terms of arbitrary binary relations. Our notion of bisimulation is proven to coincide with the coalgebraic definition of Aczel and Mendler in terms of the Giry functor, which associates with a measurable space its collection of (sub)probability measures. This coalgebraic formulation allows one to relate the concepts of bisimulation and event bisimulation of Danos et al. (i.e., cocongruence) by means of a formal adjunction between the category of bisimulations and a (full sub)category of cocongruences, which gives new insights about the real categorical nature of their results. As a corollary, we obtain sufficient conditions under which state and event bisimilarity coincide.