An Enrichment Theorem for an Axiomatisation of Categories of Domains and Continuous Functions

Domain-theoretic categories are axiomatised by means of categorical non-order-theoretic requirements on a cartesian closed category equipped with a commutative monad. An enrichment theorem showing that every axiomatic domain-theoretic category can be endowed with an intensional notion of approximation, the path relation, with respect to which the category Cpo-enriches is proved. Our analysis suggests more liberal notions of domains. In particular, a category where the path order is not !-complete, but in which the constructions of domain theory (as, for example, the existence of uniform xed-point operators and the solution of domain equations) are available is presented.