Abstract: This thesis describes some new categories of domains. These categories have two novel, and important, features:
In the next few chapters, we study domains in the effective topos. Following Scott and Rosolini, we base the theory on the notion of r.e. subobject; we use it to define an intrinsic preorder on every object of the topos. Our predomains are the complete Sigma-spaces: the objects for which this preorder is a chain-complete partial order. Every map between two complete Sigma-spaces is automatically monotone and preserves suprema of chains; they form a complete full subcategory for the modest sets.
There is a natural notion of lifting for predomains, namely the `computable partial map classifier'; this defines a monad on the category, whose algebras are the complete Sigma-spaces with a bottom element - the domains. The categories of predomains and domains have many other attractive properties.
In the last couple of chapters, we find categories of domains in other realizability toposes: those based on the Plotkin/Scott graph models, and on the term model of the untyped lambda-calculus in which two terms are identified if they have the same Böhm tree. The methods used are rather different, and the results here are still a little mysterious.
PhD Thesis - Price £8.50
LFCS report LFCS-91-171 (also published as CST-82-91)Previous | Index | Next