Functoriality of Groupoid Quantales. II

Abstract

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.

Actions of étale-covered groupoids

“…The purpose of this section is to recall some concepts and to fix terminology and notation, mostly following [12,13,16,17]. For general references on sup-lattices, locales, quantales, or groupoids see [4,5,15,20].…”

“…The following theorem gives a useful formula for computing the inner products of quantale sheaves for inverse quantal frames (see [12,Th. 3.6]).…”

“…Remark 5.3 Let G be an étale-covered groupoid. Any principally covered G-sheaf X is such that X, X ∈ I(O( G)) [12,Lemma 5.3]. Therefore the principally covered G-sheaves provide an example of a class of G-actions which does not satisfy the descent condition.…”

“…This problem is circumvented by considering bicategories and functoriality in the form of a bi-equivalence where the morphisms (1-cells) are groupoid bi-actions and quantale bimodules [18], which furthermore enables one to 'explain' why other notions of morphism between groupoids relate well to quantale homomorphisms -just as they are known to relate well to * -homomorphisms of C*-algebras [1] and to homomorphisms of inverse semigroups [2]. This study of functoriality was continued in [12], where principal bundles, Hilsum-Skandalis maps and Morita equivalence are described in the language of modules on inverse quantal frames.…”

“…The second application is an extension of the functoriality results of [18], ultimately yielding a biequivalence between the bicategory of étale-covered groupoids and the bicategory of inverse-embedded quantal frames. Extensions of later functoriality results, namely those of [12] regarding Hilsum-Skandalis maps and Morita equivalence, face additional difficulties and will not be addressed in this paper.…”