Localic Relations with Open Cones

Abstract

Localic relations are relations internal to the category of locales, forming the point-free analogues of set-theoretic relations, and providing the general backdrop of localic order theory. This work studies 'open cone' localic relations, whose source and target maps are open, and provides a frame-theoretic description via point-free up and down closure operators, called 'cones'. The cones arising from open cone localic relations form join-preserving and 'parallel' pairs of maps on the underlying frame. Axiomatising this structure, a frame equipped with such a pair of cones is called a 'conic frame'. The main construction shows that, conversely, any conic frame induces a localic relation with open cones, whose cones are exactly the given ones. The main result is an adjunction with identity counit between the category of locales equipped with open cone localic relations, and the opposite of the category of conic frames. The unit gives a strongly dense inclusion of an open cone localic relation into the relation induced by its own cones. Fixed points of the adjunction are those relations recovered by their cones, and include kernel pairs of open maps and all weakly closed localic relations with open cones. As a special case we recover Kock's Godement theorem for locales. Moreover, for fixed points, internal reflexivity and transitivity are completely characterised in terms of the cones.