Elena Pozzan (Università degli Studi di Torino) will give a talk on
Functoriality of the Wallman Compactification
Abstract:
The Wallman compactification of a T_1 space can be described as a compact T_1 space whose points are the minimal prime filters on the lattice of open sets, endowed with a Stone-type topology. A natural question, originally posed by Herrlich, is whether this construction can be made functorial and, in particular, epireflective. Harris showed that a positive answer can be obtained once one works with suitable classes of continuous maps rather than with all continuous maps. In this talk, I will focus on two such classes: WO-maps, characterized by a simple topological condition, and WC-maps, namely maps admitting a unique closed extension to Wallman compactifications. I will show that these classes have natural algebraic counterparts among morphisms of distributive lattices, giving rise to contravariant adjunctions between T_1 spaces and suitable lattice categories. These adjunctions restrict to dualities for compact T_1 spaces, closely related to the usual duality between spatial frames and sober spaces, but with points represented by minimal prime filters rather than completely prime filters. In particular, this yields an internal characterization of maps admitting a unique closed Wallman extension, solving a problem left open since the 1970s.