Quantales of open groupoids

Abstract

It is well known that inverse semigroups are closely related toétale groupoids. In particular, it has recently been shown that there is a (non-functorial) equivalence between localicétale 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 necessarilyétale. 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 concerningétale groupoids, and it depends heavily on the existence of inverse semigroups of local bisections of the quantales involved.

Effective equivalence relations and principal quantales

“…These are called open groupoids. Such semigroups encode plenty of information about open groupoids [13,17], hence yielding algebraic tools with which to study geometric structures that give rise to them. One particular class of examples which is worthy of special mention is that ofétale groupoids (those such that d is a local homeomorphism), for which u : G 0 → G 1 is an open map (so, for topological groups, beingétale means being discrete).…”

“…The algebraic characterization of the class of quantales that corresponds to open groupoids given in [13] is a direct generalization of inverse quantal frames and no corresponding generalization of supported quantales is provided. The purpose of this paper is to address the correspondence between quantales and non-étale groupoids in a way that recovers some of the algebraic simplicity and convenience of supported quantales.…”

Quantales and Fell bundles

“…These have been, so far, pairwise relations: one is the very prolific interplay between C*-algebras and locally compact groupoids [9,33,36], which pervades much of the modern literature on operator algebras and noncommutative geometry and, in the case ofétale groupoids, has led to fruitful notions of "diagonal" for C*-algebras along with a geometric understanding of them in terms ofétale groupoids, inverse semigroups, and Fell bundles [5-7, 12, 23, 24, 36, 37]; another is the relation between groupoids and quantales [35,40], which in particular yields a biequivalence between the bicategories of localicétale groupoids and inverse quantal frames [42], and also a representation ofétendues by inverse quantal frames [41] which is an instance of the general representation of Grothendieck toposes by Grothendieck quantales meanwhile developed in [14][15][16]; finally, each C*-algebra A has an associated quantale Max A [30][31][32] that can be regarded as the "noncommutative spectrum" of A and, if A is unital, classifies A up to a * -isomorphism [22]. It is fair to say that the relation between quantales and C*-algebras, which is the one that motivated the terminology "quantale" in the first place [29], is still the least well understood: in particular, despite the fact that the functor Max is a complete invariant of unital C*-algebras, it is not full and it is not clear how to obtain from it an equivalence of categories that generalizes Gelfand duality.…”

Functoriality of groupoid quantales. I

“…There is more than one way in which such ideas can be carried over to more general groupoids. For arbitrary open groupoids [32] a definition of morphism G → H can of course be based on a homomorphism of involutive quantales O(G) → O(H), but additional requirements are needed, in particular due to the absence of multiplicative units. Besides, for groupoids equipped with non-trivial additional structure, such as non-étale Lie groupoids, a homomorphism of quantales only takes the structure of topological groupoid into account.…”

“…The idea that some quantales can be regarded as generalized, and C*-algebra related, point-free spaces has been around since the term "quantale" was coined [2,16,18,[27][28][29]38], and there is a particularly good interplay between quantales and groupoids [30,32,34]. Concretely, the quantale of a topological groupoid G (with open domain map) is the topology of the arrow space G 1 equipped with pointwise operations of multiplication and involution.…”