  We introduce a general categorical framework for the definition of weak
behavioural equivalences, building on and extending recent results in the
field. This framework is based on parametrized saturation categories, i.e.
categories whose hom-sets are endowed with complete orders and a suitable
iteration operators; this structure allows us to provide the abstract
definitions of various (weak) behavioural equivalence. We show that the Kleisli
categories of many common monads are categories of this kind. This allows us to
readily instantiate the abstract definitions to a wide range of existing
systems (weighted LTS, Segala systems, calculi with names, etc.), recovering
the corresponding notions of weak behavioural equivalences. Moreover, we can
provide neatly new weak behavioural equivalences for more complex behaviours,
like those definable on topological spaces, measurable spaces, etc.
