## First Steps on the Representations of Domains (Extended
Abstract)

**Marcelo P Fiore**
*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**[*D*^{op}, **Set**],
the category of cpos and continuous functions in the presheaf topos
[*D*^{op}, **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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