2024/11/20 by Matteo Viale, Gehrke, Mai, Pozzan, Elena +1
Computer Science · #Advanced Algebra and Logic #Category Theory (math.CT) #FOS: Mathematics #General Topology (math.GN) #Logic (math.LO)
paper · pdf · doi:10.48550/arxiv.2411.13482
openalex publication_date 2024/11/20 · openalex created_date 2024/11/24 · openalex updated_date 2026/07/28
We present a contravariant adjunction between compact T1-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 T1-spaces, rather than sober spaces, and to identify points in them with minimal prime filters on some base for a T1-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 T1-spaces with arrows given by closed continuous map as a reflective subcategory of a category SbfL whose objects are the bounded distributive lattices isomorphic to some base of a T1-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 T2-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-Čech compactification theorem which we have not been able to trace elsewhere in the literature.