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

Weak bisimulation for coalgebras over order enriched monads