Abstract By restricting to a class of localic open groupoids G which, similarly to Lie groupoids, possess appropriate covers G ˆ → G by etale groupoids, we extend results about groupoid actions and quantales that were previously proved for etale groupoids but do not seem to work for arbitrary open groupoids. In particular we obtain a characterization of the category of G-actions as a category of quantale modules on O ( G ˆ ) that satisfy a condition related to the quantale O ( G ) . This leads to a simple description of G-sheaves and the classifying topos of G in terms of Hilbert O ( G ˆ ) -modules. The bicategory whose 1-cells are the groupoid bi-actions is bi-equivalent to a corresponding bicategory of quantales and bimodules.

中文翻译：

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étale 覆盖的 groupoids 的动作

摘要 通过将一类局部开群群 G 限制为与 Lie groupoids 类似，具有适当的覆盖 G ˆ → G by etale groupoids，我们扩展了先前为 etale groupoids 证明但似乎不起作用的 groupoid action 和 quantales 的结果对于任意开放的 groupoids。特别地，我们将 G-actions 的类别表征为 O ( G ˆ ) 上满足与 quantale O ( G ) 相关的条件的 quantale 模块的类别。这导致对 G 层的简单描述和 G 根据 Hilbert O ( G ^ ) 模的分类拓扑。其 1-cells 是 groupoid bi-actions 的双类别是双等价于相应的 quantale 和双模模块的双类别。