It is well known that étale groupoids are closely related to inverse semigroups. In particular, it has recently been shown that there is a non-functorial equivalence between localic étale groupoids, on the one hand, and complete and infinitely distributive inverse semigroups (abstract complete pseudogroups), on the other, which 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 noncommutative geometry, operator algebras, differential geometry, topos theory, etc., 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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M. Clarence Protin, Pedro Resende, Quantales of open groupoids. J. Noncommut. Geom. 6 (2012), no. 2, pp. 199–247DOI 10.4171/JNCG/90