Butz and Moerdijk famously showed that every (Grothendieck) topos with enough points is equivalent to the category of sheaves on some topological groupoid. We give an alternative, more algebraic construction in the special case of a topos of presheaves on an arbitrary monoid. If the monoid is embeddable in a group, the resulting topological groupoid is the action groupoid for a discrete group acting on a topological space. For these monoids, we show how to compute the points of the associated topos.

中文翻译：

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表示幺半群上预层拓扑的拓扑群

Butz 和 Moerdijk 著名地表明，每个 (Grothendieck) 具有足够点的拓扑等价于某个拓扑群上的滑轮类别。在任意幺半群上的预层拓扑的特殊情况下，我们给出了另一种更代数的构造。如果幺半群可以嵌入到一个群中，那么得到的拓扑群群就是作用在拓扑空间上的离散群的动作群群。对于这些幺半群，我们展示了如何计算相关拓扑的点。