A Classifying groupoid for compact Hausdorff locales

Abstract

We construct a localic groupoid $\mathbb{G}_{KH}$ such that for any locale $X$ the category of compact Hausdorff locales in the topos of sheaves over $X$ is equivalent to a category whose objects are principal $\mathbb{G}_{KH}$-bundles over $X$ and whose morphisms are $\mathbb{S}$-homotopies (where $\mathbb{S}$ is the Sierpi\'{n}ski locale). This result can be intuitively viewed as the compact Hausdorff dual of the well known result from topos theory that there is an object classifier.