Extremally disconnected spaces as {{u->a,b<-v}-->{u->a=b<-v}}^l, and being proper as ({{o}-->{o->c}}^r_<4)^lr

TL;DR

This paper explores the characterization of extremally disconnected spaces and proper maps in topology through the lens of Quillen lifting properties, revealing underlying preorders and categorical structures.

Contribution

It introduces a novel categorical framework for understanding extremally disconnected spaces and proper maps using Quillen lifting properties and weak factorization systems.

Findings

Extremally disconnected spaces are characterized as projective objects in a categorical setting.

Proper maps can be described via lifting properties related to finite topological spaces.

The work interprets Gleason's theorem within a categorical and preorders context.

Abstract

We observe that the notions of a topological space being extremally disconnected, and of a continuous map of compact Hausdorff spaces being proper, and being surjective proper, can each be defined in terms of the Quillen lifting property with respect to a surjective proper morphism of finite topological spaces, i.e. in terms of a monotone map of finite preorders. This reveals the preorders implicit in the statement of the Gleason theorem that extremally disconnected spaces are projective in the category of compact Hausdorff topological spaces, and interprets it as an instance of a weak factorisation system generated by an explicitly given morphism.