Non-Artin Gluing in Recursion Theory and Lifting in Abstract Stone Duality
Paul J. Taylor · 2003
Stone duality is a radical reformulation of general topology, in which the topology on a space X is not considered as a set carrying an infinitary lattice structure, but as another space that’s the exponential Σ in the category. In this way it restricts attention to recursive unions, in place of arbitrary ones, and thereby provides a natural formulation of computably based locally compact spaces [E], into which traditional arguments involving open and compact spaces nevertheless lift very easily. As ASD axiomatises spaces and continuous functions directly, arguments involving points and non-continuous functions cannot be imported directly into it. In fact, “underlying sets” may be added as a further axiom, with the result that the theory is equivalent to the standard one for locally compact locales over an elementary topos [G], but the recursive character is thereby lost. The difficulty over arbitrary unions cannot be defined away by substituting recursive ones. They are used very heavily in traditional topology, often in the guise of the right adjoints to which they give rise, the most familiar being the interior of a subspace. The categorically more sophisticated treatment in locale theory shows this very clearly, and introduces further ideas that depend on such right adjoints, notably the nucleus of a subspace. We cannot, therefore, expect every topological idea to survive the unification with recursion theory. ASD does not completely deprive us of tools besides the continuous functions X → Y themselves, as we may also use continuous functions between the spaces Σ and Σ . In the traditional formulation, these spaces carry the Scott topology, and “Scott-continuous” functions are those between the infinitary lattices that preserve directed joins. All frame homomorphisms are Scottcontinuous, but Scott-continuous functions in general need not preserve finite meets and joins. Continuous functions between the spaces that the frames represent in locale theory correspond, of course, to frame homomorphisms in the opposite direction. The general plan for translating a theorem of topology into abstract Stone duality is therefore to massage the localic formulation in such a way that only Scott-continuous functions between frames are used. In particular, direct image maps (f∗) and Heyting implications have to be eliminated. Except, that is, in the special situations in which they are Scott continuous, such as !∗ for ! : K → 1, where K is compact. It is in this fashion that we formulate a particular technique in topology in Sections 1–5 of this paper, before adapting it to abstract Stone duality. Thus you can appreciate the topological ideas without first studying the abstract setting. Our argument then “factors through” the interpretation of abstract Stone duality, in the sense that, for example, what we say concretely about Scott continuous functions between frames is replaced by use of abstract morphisms Σ → Σ in the category axiomatised by ASD. You have to read the paper twice: once wearing localic spectacles, then again in terms of abstract Stone duality; on each reading, certain parts may be transparent (i.e. vacuous) and others opaque (not soundly defined). The strictly recursion-theoretic point of view is confined to Example 4.3. The particular topological question considered in this paper is the recovery of a space Γ from an open subspace U and its complementary closed subspace C. (We also need some information about how they fit together.) Michael Artin showed that the frame of open subsets of Γ may be expressed as a comma square that involves the frames corresponding to U and C and a functor linking them. Artin actually studied this problem for Grothendieck toposes [AGV64, Expose IV,