Enrichment and Representation Theorems for Categories of Domains and Continuous Functions

Synopsis Domain-theoretic categories are axiomatised by means of categorical non-order-theoretic requirements on a cartesian closed category equipped with a commutative monad. We prove 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. Subsequently, we provide a representation theorem of the form: every small domain-theoretic category (with a lifting monad) has a full and faithful representation in a domain-theoretic category of cpos and continuous functions (with a lifting monad) in a suitable intuitionistic set theory. Our analysis suggests more liberal notions of domains. In particular, we present 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 possible.