A duality for the class of compact $T_1$-spaces

Abstract: We present a contravariant adjunction between compact $T_1$-spaces and a class of distributive lattices which recomprises key portions of Stone's duality and of Isbell's duality among its instantiations. This brings us to focus on $T_1$-spaces, rather than sober spaces, and to identify points in them with minimal prime filters on some base for a $T_1$-topology (which is what Stone's duality does on the base of clopen sets of compact $0$-dimensional spaces), in spite of completely prime filters on the topology (which is what Isbell's duality does on a sober space). More precisely our contravariant adjunction produces a contravariant, faithful and full embedding of the category of compact $T_1$-spaces with arrows given by closed continuous map as a reflective subcategory of a category $\mathsf{SbfL} $ whose objects are the bounded distributive lattices isomorphic to some base of a $T_1$-topological space (e.g. subfits, when the lattices are frames) and whose arrows are given by (what we call) set-like-morphisms (a natural class of morphisms characterized by a first order expressible constraint). Furthermore this contravariant adjunction becomes a duality when one restricts on the topological side to the category of compact $T_2$-spaces with arbitrary continuous maps, and on the lattice-theoretic side to the category of compact, complete, and normal lattices. A nice by-product of the above results is a lattice-theoretic reformulation of the Stone-\v{C}ech compactification theorem which we have not been able to trace elsewhere in the literature.