A duality for the class of compact \(T_1\)-spaces.

par Elena Pozzan (Turin)

jeudi 9 janv. 2025, 11:00 → 12:00 Europe/Paris

We present a contravariant adjunction between compact \( T_1 \)-spaces and a class of distributive lattices (i.e. subfit lattices), which encompasses key aspects of both Stone's duality and Omega-point duality in its instantiations.  Focusing on \( T_1 \)-spaces, rather than sober spaces, we identify points in these spaces with minimal prime filters on the topology, in spite of completely prime filters (which is what Omega-point duality does in the case of sober spaces).

In particular,  this adjunction can be restricted to a duality between the category of compact \( T_1 \)-spaces with continuous closed maps and the subcategory of compact, complete, and subfit lattices, where morphisms are given by set-like morphisms (a natural class of morphisms defined by a first-order expressible constraint). Moreover, when restricted on the topological side to compact Hausdorff spaces with arbitrary continuous maps, this duality coincides with the (restriction of the) Omega-point duality. A noteworthy by-product of these results is a lattice-theoretic reformulation of the Stone-\v{C}ech compactification theorem for compact \( T_1 \)-spaces. This is joint work with Matteo Viale.