A class of Partial Equivalence Relations (PERs) is described such that the resulting full subcategory of the realizability universe has the expected properties of a good category of CPOs; that is, it is a Cartesian closed category and every endomorphism has a canonical fixed point. Moreover, the reflection functor into the subcategory of strict maps (usually called the “lifting operation”) yields a good notion of “partial map,” a necessary condition if one wishes to maintain a connection between strict maps and partial maps. It is shown that all functors that arise in practice have canonical invariant objects; hence a host of domain equations are guaranteed to have solutions.

A search is conducted for a class of PERs (partial equivalence relations on the natural numbers) such that the resulting full subcategory has the expected properties of any good category of CPOs: it should be a CCC (Cartesian closed category) and every endomorphism should have a canonical fixed point. Moreover the reflection functor (usually called the lifting operation) should yield a good notion of partial map. The following topics are discussed: conventions, partial-map classifiers, ExPERS, ExPERS as domains, reflectivity of strict maps, multicorreflectivity of strict maps, the extensional natural numbers, domain equations, and intrinsic descriptions.>