Steve Vickers - Topical Categories of Domains
The talk concerns a rather thoroughgoing topologization of domain theory in which not only are the domains themselves treated as spaces in the familiar way (more precisely, as locales), but the class of domains itself is treated as a generalized space in Grothendieck's sense - that is to say as a topos. Technically it is the classifying topos for the theory of information systems, so its points are the information systems. When the same is done for the maps between domains (or, rather, the approximable mappings between information systems) one gets an internal category in the category Top of toposes and geometric morphisms, a "topical category".
The advantage of this approach is that the characteristic fixpoint constructions, both within and amongst domains, then come directly out of the generalized topological structure by a very general result: that local toposes are "algebraically complete" in a natural sense. In addition, the domain-theoretic trick of using embedding-projection pairs is rationally reconstructed through the natural appearance of homomorphisms between strongly algebraic (SFP) information systems.
An apparent disadvantage of the approach is its reliance on heavy topos machinery; yet I shall how this does not obtrude at all provided one adheres to a "geometrically" constructivist mathematical discipline. Toposes and geometric morphisms can then be discussed very much as though they were classes and functions.
The expositional tendency is that in a certain sense sets or classes can no longer be considered the fundamental notion of collection - sets are a particularly discrete kind of collection, and in general (and this happens when one wants to exponentiate or take powersets) collections are topologized and must be considered as toposes. I shall conclude with some foundational conjectures based on this, involving Joyal's arithmetic universes.