Domain Theory in Topical Form

In his short story " Pierre Menard Author of the Quixote " , Jorge Luis Borges tells the story of a French author who sets out to compose Don Quixote – not, you understand, as a mechanical transcription or copy of Cervantes' original, but as a recreation , word for word and line for line, of fragments of it. In 1999 I had a similar experience with my paper " Topical Categories of Domains " [3], based on some results from Samson's thesis [1] that also appeared in his " Domain Theory in Logical Form " [2]. My aim was to give a presentation of Samson's results that recreated, as closely as possible, Samson's own presentation. Needless to say, it was not exactly the same, but why should such a recreation have been a worthy aim? The answer is one of foundations: I had an idea for refounding the work using toposes, technically by replacing categories of domains by toposes classifying them (or their compact bases). One aim from this was to use the topos theory to give canonical answers to questions of continuity. When solving domain equations D ∼ = F (D), F needs certain continuity properties that have been formulated in a special purpose way in domain theory. The canonical answer from topos theory would be to require F to be represented by a geometric morphism. As an unexpected bonus, the toposes also recreate the trick of " embedding-projection pairs " , introduced in domain theory to deal with the fact that some important constructions F are not functorial with respect to Scott continuous maps. They reappear – in the case of SFP domains – as homomorphisms between the domains as models of a geometric theory. Topos machinery can be heavy and untransparent, and Samson for one was not persuaded of the benefits. Why would anyone put themselves to the trouble of using toposes? Seeking answers to such objections was the start of my Menardian quest to recreate parts of his thesis: to leave the essential mathematics of his presentation undisturbed, but by logical means have it reinterpreted in terms of the toposes. The measure of success was to be the similarity to what Samson actually wrote. With regard to the logical means (using geometric logic), I was by then beginning to understand the topos-theoretic techniques better – particularly through some collaboration with Peter Johnstone. However, it still …