Barr-coexactness for metric compact Hausdorff spaces

A metric compact Hausdorff space is a Lawvere metric space equipped with a compatible compact Hausdorff topology (which does not need to be the induced topology).These spaces maintain many important features of compact metric spaces, but the resulting category is much better behaved.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 to X by their kernel metrics, which are characterised as the continuous metrics below the metric on X.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.Contents 1 Introduction 196 2 Separated metric compact Hausdorff spaces 198 3 Quotient objects and continuous submetrics 203 4 Coregularity for separated metric compact Hausdorff spaces 206 5 Barr-coexactness for separated metric compact Hausdorff spaces 214 6 The symmetric and the ordered cases 222