Enrichment and Representation Theorems for Categories of Domains and Continuous Functions Marcelo P. Fiore Department of Computer Science Laboratory for Foundations of Computer Science University of Edinburgh, The King's Buildings Edinburgh EH8 3JZ, Scotland February 1996 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 omega-complete, but in which the constructions of domain theory (as, for example, the existence of uniform fixed-point operators and the solution of domain equations) are possible.