Observationally-induced Effect Monads: Upper and Lower Powerspace Constructions

Alex Simpson has suggested to use an observationally-induced approach towards modelling computational effects in denotational semantics. The principal idea is that a single observation algebra is used for defining the computational type structure. He advocates that besides giving algebraic structure this approach also allows the characterisation of the monadic types concretely. We show that free observationally-induced algebras exist in the category of continuous maps between topological spaces for arbitrary pre-chosen observation algebras. Moreover, we use this approach to give a lower and an upper powerdomain construction on general topological spaces, both of which generalise the classical characterisations on continuous dcpos. Our lower powerdomain construction is for all topological spaces given by the space of non-empty closed subsets with the lower Vietoris topology. Dually, our upper powerdomain construction is for a wide class of topological spaces given by the space of proper open filters of its topology with the upper Vietoris topology. We also give a counterexample showing that this characterisation does not hold for all topological spaces.