Weak bisimulation for coalgebras over order enriched monads

The paper introduces the notion of a weak bisimulation for coalgebras whose type is a monad satisfying some extra properties. In the first part of the paper we argue that systems with silent moves should be modelled coalgebraically as coalgebras whose type is a monad. We show that the visible and invisible part of the functor can be handled internally inside a monadic structure. In the second part we introduce the notion of an ordered saturation monad, study its properties, and show that it allows us to present two approaches towards defining weak bisimulation for coalgebras and compare them. We support the framework presented in this paper by two main examples of models: labelled transition systems and simple Segala systems.