Poset-enriched pretoposes and compact ordered spaces

Abstract

We provide a characterisation of the category $\mathsf{KOrd}$ of Nachbin's compact ordered spaces as a poset-enriched category. Up to equivalence, $\mathsf{KOrd}$ is the only non-degenerate poset-enriched pretopos whose terminal object is a (discrete) generator and in which every object is covered by an order-filtral object. Order-filtral objects satisfy an appropriate form of compactness and separation. Throughout, we make extensive use of the internal language of poset-enriched pretoposes.