Barr-coexactness for metric compact Hausdorff spaces

TL;DR

This paper studies the category of metric compact Hausdorff spaces, characterizing morphisms and showing that its dual category is Barr-exact, thus revealing algebraic properties and internal representations of quotients.

Contribution

It characterizes regular monomorphisms and epimorphisms in this category and proves the dual category is Barr-exact, highlighting its algebraic structure.

Findings

Regular monomorphisms are embeddings.

Epimorphisms are surjective morphisms.

Dual category is Barr-exact.

Abstract

Compact metric spaces form an important class of metric spaces, but the category that they define lacks many important properties such as completeness and cocompleteness. In recent studies of "metric domain theory" and Stone-type dualities, the more general notion of a (separated) metric compact Hausdorff space emerged as a metric counterpart of Nachbin's compact ordered spaces. Roughly speaking, a metric compact Hausdorff space is a metric space equipped with a \emph{compatible} compact Hausdorff topology (which does not need to be the induced topology). These spaces maintain many important features of compact metric spaces, and, notably, the resulting category is much better behaved. Moreover, one can use inspiration from the theory of Nachbin's compact ordered spaces to solve problems for metric structures. In this paper we continue this line of research: in the category of separated metric compact Hausdorff spaces we characterise the regular monomorphisms as the embeddings and the epimorphisms as the surjective morphisms. Moreover, we show that epimorphisms out of an object $X$ can be encoded internally on $X$ by their kernel metrics, which are characterised as the continuous metrics below the metric on $X$; this gives a convenient way to represent quotient objects. Finally, as the main result, we prove that its dual category has an algebraic flavour: it is Barr-exact. While we show that it cannot be a variety of finitary algebras, it remains open whether it is an infinitary variety.