Synopsis: Domain-theoretic categories are axiomatised by means of categorical non-order-theoretic requirements on a cartesian closed category equipped with a dominance and a lifting. We show 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 D has a full and faithful representation in Cpo[Dop, Set], the category of cpos and continuous functions in the presheaf topos [Dop, Set]. Our analysis suggests more liberal notions of domains. In particular, we present a category where the path preorder 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.
ECS-LFCS-95-331, October 1995.
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