Observationally-induced lower and upper powerspace constructions

Abstract

Computational effects can be modelled by observationally-induced algebras, which are algebras whose structure is completely determined by a chosen computational prototype. We show that the category of continuous maps between topological spaces supports a free observationally-induced algebra construction for arbitrary pre-chosen prototype, and give a characterisation of the free algebras as subalgebras of certain powers of these prototypes.Moreover, we present observationally-induced lower and upper powerspace constructions in the category of topological spaces. Our lower powerspace construction is for all topological spaces given by its non-empty closed subsets equipped 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 equipped with the upper Vietoris topology. Thus, both constructions generalise the classical construction on continuous dcpos, and unify abstract and concrete characterisations of powerdomains on a broader scale.Finally, we show that with a small adjustment of the definitions, observationally-induced algebras form the smallest full reflective subcategory of the category of algebras for the corresponding signature, which contains the computational prototype and is complete and closed under isomorphism.