Étale groupoids and their quantales

By Pedro Resende. 2004

162 citations recorded by the source corpus.

Abstract We establish close and previously unknown relations between quantales and groupoids. 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 here are 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 (non-functorial) 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.

Subfields: Immunology and Allergy, Molecular Biology, Oncology

View Paper: https://www.semanticscholar.org/paper/d6f425449d52a18836e9ab59e472ab9776b3b49e

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Immunology and Allergy

1. Ben Steinberg
2. John L. Rhodes
3. Daniel Lenz
4. Mark V. Lawson

In Molecular Biology

1. John L. Rhodes
2. Ben Steinberg
3. Daniel Lenz
4. Mark V. Lawson

In Oncology

1. Ben Steinberg
2. John L. Rhodes
3. Daniel Lenz
4. Mark V. Lawson

Citations in the corpus

• A Groupoid Approach to Discrete Inverse Semigroup Algebras (2009, methodology): “There is a well-known construction assigning to each non-degenerate action φ : S → IX of an inverse semigroup S on a locally compact Hausdorff space X an étale groupoid, which we denote S ⋉φ X, known as the groupoid of germs of the action [6,17,20].”
• Simplicity, primitivity and semiprimitivity of etale groupoid algebras with applications to inverse semigroup algebras (2014, unknown): “Details can be found in [9, 19,22].”
• The q-theory of Finite Semigroups (2008, background): “It turns out that certain involutive quantales are closely connected to inverse semigroups and étale groupoids [255]; this should be explored further.”
• Pseudogroups and their étale groupoids (2011, methodology): “Thus a complete abstract pseudogroup, to use the terminology of Resende [26, 27], is a complete, infinitely distributive inverse monoid; we shall explain what this definition means later in this section.”
• A NONCOMMUTATIVE GENERALIZATION OF STONE DUALITY (2009, background): “This work has been developed in a number of directions [4, 5, 12, 13, 21, 23, 25, 26, 27], to name but a few.”