Reflecting compact $T_1$-spaces into bounded distributive lattices

Abstract

We present a contravariant reflection of the compact $T_1$-spaces with arrows given by closed continuous functions into the category of bounded distributive lattices with arrows given by closed subfit morphisms. This reflection extends both Stone duality and Isbell's duality between frames and sober spaces for those compact $T_1$-spaces that fall within each of these dualities, that is, respectively, zero-dimensional compact Hausdorff spaces, and compact sober $T_1$-spaces. On the topological side, we allow all compact $T_1$-spaces rather than just sober ones and we identify points in these with minimal prime filters on some base. On the lattice side, the shift goes from the notion of frame homomorphism to that of closed subfit morphism between bounded distributive lattices (closed subfit morphisms are defined by a natural and first order expressible constraint). The reflection becomes a duality when one restricts on the algebraic side to the complete and compact subfit lattices (i.e. compact subfit frames). Furthermore, restricting our duality on the topological side to the subcategory of compact $T_2$-spaces with all continuous maps, we obtain a duality for these with the category of complete, compact and normal lattices, thus recovering a classical result of Cornish. We also relate our adjunction to the duality introduced by Maruyama between $T_1$-spaces with continuous maps and a category having as objects a particular type of subfit frames and as arrows a certain type of morphism, of which we give an alternative and explicit algebraic characterization.