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

中文翻译：

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Presheaf 拓扑中的紧凑 Hausdorff 语言环境

我们证明，对于任何小类别\({\mathcal {C}}\) ，紧凑 Hausdorff 语言环境的类别\(\textbf{KHausLoc}_{\hat{{\mathcal {C}}}}\) presheaf topos \(\hat{{\mathcal {C}}}\)相当于函子的范畴\({\mathcal {C}} \rightarrow \textbf{KHausLoc}\)。