Version 3 contains 16 additional pages (now the total is 75) and completes the characterization of etale groupoid quantales; in particular it is proved that every inverse quantal frame is multiplicative
We establish a close and previously unknown relation between quantales and groupoids, in terms of which the notion of etale groupoid is subsumed in a natural way by that of quantale. In particular, to each etale groupoid, either localic or topological, there is associated a unital involutive quantale. We obtain a bijective correspondence between localic etale groupoids and their quantales, which are given a rather simple characterization and are here called inverse quantal frames. We show that the category of inverse quantal frames is equivalent to the category of complete and infinitely distributive inverse monoids, and as a consequence we obtain a correspondence between these and localic etale groupoids that generalizes more classical results concerning inverse semigroups and topological etale groupoids. This generalization is entirely algebraic and it is valid in an arbitrary topos. As a consequence of these results we see that a localic groupoid is etale if and only if its sublocale of units is open and its multiplication map is semiopen, and an analogue of this holds for topological groupoids. In practice we are provided with new tools for constructing localic and topological etale groupoids, as well as inverse semigroups, for instance via presentations of quantales by generators and relations. The characterization of inverse quantal frames is to a large extent based on a new quantale operation, here called a support, whose properties are thoroughly investigated, and which may be of independent interest.
We study the projections of an arbitrary stably Gelfand quantale $Q$ and show that each projection determines a pseudogroup $S\subset Q$ (and a corresponding localic étale groupoid $G$) together with a map of involutive quantales $p:Q\to\mathcal L^{\bigvee}(S)\ [=\mathcal O(G)]$. As an application we obtain a simplified axiomatization of inverse quantal frames (= quantales of étale groupoids) whereby such a quantale is shown to be the same as a unital stably Gelfand quantal frame whose partial units cover $Q$.
For an arbitrary localic etale groupoid G we provide simple descriptions, in terms of modules over the quantale O(G) of the groupoid, of the continuous actions of G, including actions on open maps and sheaves. The category of G-actions is isomorphic to a corresponding category of O(G)-modules, and as a corollary we obtain a new quantale based representation of etendues.
Taking advantage of the quantale-theoretic description of étale groupoids we study principal bundles, Hilsum-Skandalis maps, and Morita equivalence in terms of modules on inverse quantal frames. The Hilbert module description of quantale sheaves leads naturally to a formulation of Morita equivalence in terms of bimodules that resemble imprimitivity bimodules of C*-algebras.
Stably supported quantales generalize pseudogroups and provide an algebraic context in which to study the correspondences between inverse semigroups and étale groupoids. Here we study a further generalization where a non-unital version of supported quantale carries the algebraic content of such correspondences to the setting of open groupoids. A notion of principal quantale is introduced which, in the case of groupoid quantales, corresponds precisely to effective equivalence relations.
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…
This is a short survey paper, partly meant as a research announcement. Its purpose is to highlight some aspects of the interplay between quantales, inverse semigroups, and groupoids. Many of the results mentioned have not yet been presented and will appear in detail elsewhere.
We provide three functorial extensions of the equivalence between localic etale groupoids and their quantales. The main result is a biequivalence between the bicategory of localic etale groupoids, with bi-actions as 1-cells, and a bicategory of inverse quantal frames whose 1-cells are bimodules. As a consequence, the category InvQuF of inverse quantale frames, whose morphisms are the (necessarily involutive) homomorphisms of unital quantales, is equivalent to a category of localic etale groupoids whose arrows are the algebraic morphisms in the sense of Buneci and Stachura. We also show that th…
Taking into account the correspondence between open groupoids and their quantales, we establish a bijective correspondence between the principal $G$-bundles whose left projection is an open surjection and the principal $\opens(G)$-locales for any open groupoid $G$ and its groupoid quantale $\opens(G)$, the latter requires a generalization of the supported modules for groupoid quantales.
By restricting to a class of localic open groupoids $G$ which, similarly to Lie groupoids, possess appropriate covers $\widehat G\to G$ by étale groupoids, we extend results about groupoid actions and quantales that were previously proved for étale groupoids but do not seem to work for arbitrary open groupoids. In particular we obtain a characterization of the category of $G$-actions as a category of quantale modules on $\mathcal O(\widehat G)$ that satisfy a condition related to the quantale $\mathcal O(G)$. This leads to a simple description of $G$-sheaves and the classifying topos of $G$ in…
Several notions of sheaf on various types of quantale have been proposed and studied in the last twenty five years. It is fairly standard that for an involutive quantale Q satisfying mild algebraic properties the sheaves on Q can be defined to be the idempotent self-adjoint Q-valued matrices. These can be thought of as Q-valued equivalence relations, and, accordingly, the morphisms of sheaves are the Q-valued functional relations. Few concrete examples of such sheaves are known, however, and in this paper we provide a new one by showing that the category of equivariant sheaves on a localic eta…
We examine a semigroup analogue of the Kumjian-Renault representation of C*-algebras with Cartan subalgebras on twisted groupoids. Specifically, we show how to represent semigroups with distinguished normal subsemigroups as `slice-sections' of groupoid bundles.
Paterson showed how to construct an etale groupoid from an inverse semigroup using ideas from functional analysis. This construction was later simplified by Lenz. We show that Lenz's construction can itself be further simplified by using filters: the topological groupoid associated with an inverse semigroup is precisely a groupoid of filters. In addition, idempotent filters are closed inverse subsemigroups and so determine transitive representations by means of partial bijections. This connection between filters and representations by partial bijections is exploited to show how linear represen…
The study of Haeflier suggests that it is natural to regard a pseudogroup as an etale groupoid. We show that any etale groupoid corresponds to a pseudogroup sheaf, a new generalization of a pseudogroup. This correspondence is an analog of the equivalence of the two definitions of a sheaf: as an etale space and as a contravariant functor.
We prove a generalisation of the correspondence, due to Resende and Lawson--Lenz, between étale groupoids---which are topological groupoids whose source map is a local homeomorphisms---and complete pseudogroups---which are inverse monoids equipped with a particularly nice representation on a topological space. Our generalisation improves on the existing functorial correspondence in four ways. Firstly, we enlarge the classes of maps appearing to each side. Secondly, we generalise on one side from inverse monoids to inverse categories, and on the other side, from étale groupoids to what we call …
This paper studies simplicity, primitivity and semiprimitivity of algebras associated to étale groupoids. Applications to inverse semigroup algebras are presented. The results also recover the semiprimitivity of Leavitt path algebras and can be used to recover the known primitivity criterion for Leavitt path algebras.
The author has previously associated to each commutative ring with unit $\Bbbk$ and étale groupoid $\mathscr G$ with locally compact, Hausdorff, totally disconnected unit space a $\Bbbk$-algebra $\Bbbk\mathscr G$. The algebra $\Bbbk\mathscr G$ need not be unital, but it always has local units. The class of groupoid algebras includes group algebras, inverse semigroup algebras and Leavitt path algebras. In this paper we show that the category of unitary $\Bbbk\mathscr G$-modules is equivalent to the category of sheaves of $\Bbbk$-modules over $\mathscr G$. As a consequence we obtain a new proof …
A pseudogroup is a complete infinitely distributive inverse monoid. Such inverse monoids bear the same relationship to classical pseudogroups of transformations as frames do to topological spaces. The goal of this paper is to develop the theory of pseudogroups motivated by applications to group theory, C*-algebras and aperiodic tilings. Our starting point is an adjunction between a category of pseudogroups and a category of etale groupoids from which we are able to set up a duality between spatial pseudogroups and sober etale groupoids. As a corollary to this duality, we deduce a non-commutati…
Under non-commutative Stone duality, there is a correspondence between second countable Hausdorff étale groupoids which have a Cantor space of identities and what we call Tarski inverse monoids: that is, countable Boolean inverse $\wedge$-monoids with semilattices of idempotents which are countable and atomless. Tarski inverse monoids are therefore the algebraic counterparts of the étale groupoids studied by Matui and provide a natural setting for many of his calculations. Under this duality, we prove that natural properties of the étale groupoid correspond to natural algebraic properties of t…
In this article we will study semigroupoids, and more specifically inverse semigroupoids. These are a common generalization to both inverse semigroups and groupoids, and provide a natural language on which several types of dynamical structures may be described. Moreover, this theory allows us to precisely compare and simultaneously generalize aspects of both the theories of inverse semigroups and groupoids. We begin by comparing and settling the differences between two notions of semigroupoids which appear in the literature (one by Tilson and another by Exel). We specialize this study to inver…
