Generating the algebraic theory of $C(X)$: the case of partially ordered compact spaces

TL;DR

This paper characterizes the algebraic theory dual to categories of partially ordered compact spaces, extending classical duality results and describing the algebraic structure and copresentability within this context.

Contribution

It demonstrates that the dual of the category of partially ordered compact spaces is an a1-ary quasivariety and extends these results to Vietoris coalgebras.

Findings

Dual category is an a1-ary quasivariety.

Characterization of a1-copresentable spaces.

Extension of duality results to Vietoris coalgebras.

Abstract

It is known since the late 1960's that the dual of the category of compact Hausdorff spaces and continuous maps is a variety -- not finitary, but bounded by $\aleph_1$. In this note we show that the dual of the category of partially ordered compact spaces and monotone continuous maps is a $\aleph_1$-ary quasivariety, and describe partially its algebraic theory. Based on this description, we extend these results to categories of Vietoris coalgebras and homomorphisms. We also characterise the $\aleph_1$-copresentable partially ordered compact spaces.