A Non-commutative Generalization of Stone Duality

We prove that the category of boolean inverse monoids is dually equivalent to the category of boolean groupoids. This generalizes the classical Stone duality between boolean algebras and boolean spaces. As an instance of this duality, we show that the boolean inverse monoid Cn associated with the Cuntz groupoid Gn is the strong orthogonal completion of the polycyclic (or Cuntz) monoid Pn. The group of units of Cn is the Thompson group Vn,1. 2000 Mathematics Subject Classification: 20M18,18B40,06E15. 1. Statement of the theorem The importance of partial, as opposed to global, symmetries in mathematics is well-established. The question is how to describe them mathematically. One approach, advocated in [16], is to use inverse semigroups; these are direct generalizations of groups and are ultimately descended from the pseuodgroups of transformations used in differential geometry. Another approach is to use topological groupoids such as in the recent work of Hughes [8, 9]. Although the surface structure of these two approaches looks very different, they are in fact closely related. Classically, pseudogroups of transformations give rise to topological groupoids of germs. More generally, Paterson [24] used ideas from functional-analysis to construct topological groupoids from inverse semigroups and Renault [25] constructed inverse semigroups from topological groupoids using bisections. This work has been developed in a number of directions [4, 5, 6, 13, 14, 22, 24, 26, 27, 28], to name but a few. The goal of our paper is to set up an exact correspondence between a class of inverse monoids, we call boolean monoids, and a class of topological groupoids, we call boolean groupoids. As the terminology suggests, our correspondence can be seen as a natural generalization of the Stone duality between boolean algebras and boolean spaces. For background on inverse semigroups see [16], for groupoids [7] and for topological groupoids [5, 26, 24, 25]. Although our theorem appears to link semigroups and groupoids in reality it is linking two different kinds of groupoid. Boolean inverse monoids are semigroups but they are also special kinds of ordered groupoids by virtue of the Ehresmann–Schein– Nambooripad Theorem [16]. It follows that our theorem could also be viewed as providing a duality between a class of ordered groupoids on the one hand and a class of topological groupoids on the other. The advantages of being able to re-encode algebraic structures as topological ones cannot be overstated. In the remainder of This research was supported by an EPSRC grant (EP/F004184, EP/F014945, EP/F005881), the Fundação para a Ciência e a Tecnologia, courtesy of Pedro Resende under the grant PPCDT/MAT/55958/2004, Groupoids and quantales in geometry and analysis, at the Instituto Superior Técnico, Lisbon, and by Prof Stuart Margolis of Bar-Ilan University, Israel.