The Synthetic Plotkin Powerdomain

Plotkin [1976] introduced a powerdomain construction on domains in order to give semantics to a non-deterministic binary choice constructor, and later [1979] characterised it as the free semilattice. Smyth [1983] and Winskel [1985] showed that it could be interpreted in terms of modal predicate transformers and Robinson [1986] recognised it as a special case of Johnstone’s [1982] Vietoris construction, which itself generalises the Hausdorff metric on the set of closed subsets of a metric space. The domain construction involves a curious order relation known as the Egli-Milner order. In this paper we relate the powerdomain directly to the free semilattice, which in a topos is simply the finite powerset, i.e. the object of (Kuratowski-)finite subobjects of an object. We show that the Egli-Milner order coincides (up to “¬¬”) with the intrinsic order induced by a family of “observable predicates.” This problem originally arose in the context of the Effective topos, in which the observable predicates are the recursively enumerable subsets. However we find that the results of this paper hold for any elementary topos, and so by considering a (pre)sheaf topos (which the Effective topos is not) we may compare them with the classical approach. Important Note: Much of the credit for the work in this paper is due to Wesley Phoa and Martin Hyland, but I take the blame for its presentation. Comments on it are most welcome. When it is finished it will be submitted as a joint paper with Wesley Phoa, and an announcement will be made on types.