开广群的Quantales
Quantales of open groupoids
摘要：
众所周知，逆半群与Etale群胚密切相关。特别地，最近已经证明了局部的Etale群胚与完全的和无限分配的逆半群（抽象的完全伪群）之间存在（非函子的）等价。这种对应是由一类量子介导的，称为逆量子框架，它是从逆半群通过一个简单的连接完成，产生一个等价的范畴。因此，我们可以把抽象的完备伪群看作本质上与逆量子框架“相同”，在本文中，我们利用这一事实，在开群胚不一定是tale的情况下，找到逆半群的合适替代品。这种推广的兴趣在于开广群在算子代数、微分几何和拓扑理论等领域的重要性和普遍性，我们通过一类称为开量子框架的量子来实现它，它推广了逆量子框架，我们详细研究了它的性质。量子和开群胚之间的对应关系并不是以前关于étale群胚的结果的直接推广，它在很大程度上依赖于所涉及的量子的局部对分的逆半群的存在性。
英文摘要：
It is well known that inverse semigroups are closely related to étale groupoids. In particular, it has recently been shown that there is a (non-functorial) equivalence between localic étale groupoids, on one hand, and complete and infinitely distributive inverse semigroups (abstract complete pseudogroups), on the other. This correspondence is mediated by a class of quantales, known as inverse quantal frames, that are obtained from the inverse semigroups by a simple join completion that yields an equivalence of categories. Hence, we can regard abstract complete pseudogroups as being essentially ``the same'' as inverse quantal frames, and in this paper we exploit this fact in order to find a suitable replacement for inverse semigroups in the context of open groupoids that are not necessarily étale. The interest of such a generalization lies in the importance and ubiquity of open groupoids in areas such as operator algebras, differential geometry and topos theory, and we achieve it by means of a class of quantales, called open quantal frames, which generalize inverse quantal frames and whose properties we study in detail. The resulting correspondence between quantales and open groupoids is not a straightforward generalization of the previous results concerning étale groupoids, and it depends heavily on the existence of inverse semigroups of local bisections of the quantales involved.
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