Barr-coexactness for metric compact Hausdorff spaces
Marco Abbadini, Dirk Hofmann
TL;DR
The paper studies the category $MetCH_sep$ of separated metric compact Hausdorff spaces with continuous non-expansive maps, establishing that regular monomorphisms are embeddings and epimorphisms are surjections. It introduces kernel metrics $kappa_f$ and continuous submetrics $gamma$, proving a dual equivalence between quotient data and internal submetrics, thereby encoding quotients on objects via internal structures. A central result is that the dual category $MetCH_sep^{op}$ is Barr-exact, by showing pushouts preserve regular monos and that all internal equivalence relations are effective; the work also analyzes symmetric and ordered variants, showing Barr-coexactness in these settings and clarifying their Mal'tsev properties. Collectively, these results provide a robust algebraic duality framework for metric-domain theory and extend Stone-type dualities to the metric-compact Hausdorff setting, with systematic treatment of quotients, submetrics, and exactness. The framework unifies metric and ordered topologies, enabling a principled approach to quotient construction and dualization in a complete and cocomplete setting.
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.