Proximity Frames and Regularization

It is well known that the category KHaus of compact Hausdorff spaces is dually equivalent to the category KRFrm of compact regular frames. By de Vries duality, KHaus is also dually equivalent to the category DeV of de Vries algebras, and so DeV is equivalent to KRFrm, where the latter equivalence can be described constructively through Booleanization. Our purpose here is to lift this circle of equivalences and dual equivalences to the setting of stably compact spaces. The dual equivalence of KHaus and KRFrm has a well-known generalization to a dual equivalence of the categories StKSp of stably compact spaces and StKFrm of stably compact frames. Here we give a common generalization of de Vries algebras and stably compact frames we call proximity frames. For the category PrFrm of proximity frames we introduce the notion of regularization that extends that of Booleanization. This yields the category RPrFrm of regular proximity frames. We show there are equivalences and dual equivalences among PrFrm, its subcategories StKFrm and RPrFrm, and StKSp. Restricting to the compact Hausdorff setting, the equivalences and dual equivalences among StKFrm, RPrFrm, and StKSp yield the known ones among KRFrm, DeV, and KHaus. The restriction of PrFrm to this setting provides a new category StrInc whose objects are frames with strong inclusions and whose morphisms and composition are generalizations of those in DeV. Both KRFrm and DeV are subcategories of StrInc that are equivalent to StrInc. For a compact Hausdorff space X, the category StrInc not only contains both the frame of open sets of X and the de Vries algebra of regular open sets of X, these two objects are isomorphic in StrInc, with the second being the regularization of the first. The restrictions of these categories are considered also in the setting of spectral spaces, Stone spaces, and extremally disconnected spaces.