A characterisation of the category of compact Hausdorff spaces

TL;DR

This paper characterizes the category of compact Hausdorff spaces using categorical properties, introducing filtrality for coherent categories, and establishing a unique categorical characterization of KH.

Contribution

It provides a novel categorical characterization of KH, the category of compact Hausdorff spaces, through filtrality and properties of pretoposes.

Findings

KH is the unique non-trivial well-pointed pretopos that is filtral.

KH admits all set-indexed copowers of its terminal object.

The paper introduces the notion of filtrality for coherent categories.

Abstract

We provide a characterisation of the category KH of compact Hausdorff spaces and continuous maps by means of categorical properties only. To this aim we introduce a notion of filtrality for coherent categories, relating certain lattices of subobjects to their Boolean centers. Our main result reads as follows: Up to equivalence, KH is the unique non-trivial well-pointed pretopos which is filtral and admits all set-indexed copowers of its terminal object.