Coalgebraic Weak Bisimulation from Recursive Equations over Monads

Strong bisimulation for labelled transition systems is one ofthe most fundamental equivalences in process algebra, and has beengeneralised to numerous classes of systems that exhibit richer transi-tion behaviour. Nearly all of the ensuing notions are instances of themore general notion of coalgebraic bisimulation. Weak bisimulation, how-ever, has so far been much less amenable to a coalgebraic treatment.Here we attempt to close this gap by giving a coalgebraic treatment of(parametrized) weak equivalences, including weak bisimulation. Our anal-ysis requires that the functor defining the transition type of the systemis based on a suitable order-enriched monad, which allows us to captureweak equivalences by least fixpoints of recursive equations. Our notionis in agreement with existing notions of weak bisimulations for labelledtransition systems, probabilistic and weighted systems, and simple Segalasystems.