Factorizing the Top-Loc adjunction through positive topologies

Abstract

We characterize the category of Sambin's positive topologies as a fibration over the category of locales Loc. The fibration is obtained by applying the Grothendieck construction to a doctrine over Loc. We then construct an adjunction between the category of positive topologies and that of topological spaces Top, and show that the well-known adjunction between Top and Loc factors through the newly constructed adjunction.