On the parallel between the suplattice and preframe approaches to locale theory

AbstractThis paper uses the locale theory approach to topology. Two descriptions are given of all locale limits, the first description using suplattice constructions and the second preframe constructions. The symmetries between these two approaches to locale theory are explored. Given an informal assumption that open locale maps are parallel to proper maps (an assumption hinted at by the underlying finitary symmetry of the lattice theory but not formally proved) we argue that various pairs of locale theory results are ‘parallel’, that is, identical in structure but prove facts about proper maps on one side of the pair and about open maps on the other. The pairs of results are: pullback stability of proper/open maps, regularity of the category of compact Hausdorff/discrete locales, and theorems on information systems. Some remarks are included on a possible formalization of this parallel as a duality