Compact Hausdorff Locales in presheaf toposes

TL;DR

This paper establishes an equivalence between the category of compact Hausdorff locales in a presheaf topos and functor categories, extending the understanding of locale theory in topos-theoretic contexts.

Contribution

It proves a new equivalence relating compact Hausdorff locales in presheaf toposes to functor categories, generalizing previous locale theory results.

Findings

Category of compact Hausdorff locales in presheaf topos is equivalent to functor category

Extends locale theory to presheaf toposes

Provides a new framework for studying locales in topos theory

Abstract

We prove that for any small category $\mathcal{C}$, the category $\mathbf{KHausLoc}_{\hat{\mathcal{C}}}$ of compact Hausdorff locales in the presheaf topos $\hat{\mathcal{C}}$, is equivalent to the category of functors $\mathcal{C} \to \mathbf{KHausLoc}$.