Studying repleteness in the category of cpos

We consider the notion of replete object in the category of directed complete partial orders and Scott-continuous functions, and show that, contrary to previous expectations, there are non-replete objects. The same happens in the case of ω-complete posets. Synthetic Domain Theory developed from an idea of Dana Scott: it is consistent with intuitionistic set theory that all functions between domains are continuous. He never wrote about this point of view explicitly, though he presented his ideas in many lectures also suggesting that the model offered by Kleene’s realizability was appropriate, and influenced various thesis works, e.g. [10,13,11,8,12], see also [14]. SDT can now be recognized as defining the “good properties” required on a category C (usually, a topos with a dominance t: 1 Σ) in order to develop domain theory within a theory of sets. One of the problems addressed early in the theory was the identification of the sets to be considered as the Scott domains. As one would expect in a synthetic approach, the collection of these should be determined by the “good properties” of the universe, in an intrinsic way. The best suggestion so far for such a collection comes from [6,15,5] and is that of repleteness. It is an orthogonality condition, see [2], and determines the replete objects of C as those which are completely recoverable from their properties detected by Σ. Say that A is replete (wrt. Σ) if it is orthogonal to all f :X Y in 1 Research supported by NSERC Canada and FCAR Quebec, and completed during a visit to the University of Genoa, supported by Italian C.N.R. 2 Research supported by MURST 40%. c ©2004 Published by Elsevier Science B. V.