An adjunction between a category of complete restriction monoids and a category of étale categories

Abstract

We prove that there is an adjunction between what we call étale topological categories and restriction quantal frames that leads to an adjunction with a category of complete restriction monoids. This generalizes the adjunction between étale groupoids and pseudogroups.Denote the category of topological spaces and continuous functions by Top. We have therefore defined a functor Ω from Top to Locale.We shall define a functor going in the opposite direction, but before we do that we shall motivate our definitions.Definition. Let X be a topological space and x ∈ X. Define O x to be the set of all open sets that contain x.Observe that O x ⊆ Ω(X) is a completely prime filter [7, Page 41].Definition. If F is a frame, denote by pt(F ) the set of all completely prime filters in F ; we think of these as the 'points' of F .