## Domains in H

by
Marcelo P. Fiore
,
Giuseppe Rosolini

by
Marcelo P. Fiore
,
Giuseppe Rosolini

@MISC{Fiore_domainsin,

author = {Marcelo P. Fiore and Giuseppe Rosolini},

title = {Domains in H},

year = {}

}

We give various internal descriptions of the category !-Cpo of !-complete posets and !-continuous functions in the model H of Synthetic Domain Theory introduced in [8]. It follows that the !-cpos lie between the two extreme synthetic notions of domain given by repleteness and well-completeness. Introduction Synthetic Domain Theory aims at giving a few simple axioms to be added to an intuitionistic set theory in order to obtain domain-like sets. The idea at the core of this study was proposed by Dana Scott in the late 70's: domains should be certain "sets" in a mathematical universe where domain theory would be available. In particular, domains would come with intrinsic notions of approximation and passage to the limit with respect to which all functions will be continuous. Various suggestions for the notion of domain (typically within a set-theoretic universe given by an elementary topos with natural numbers object [17]) appeared in the literature, e.g. in [11, 26, 10, 23, 20, 16]. A...

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