## Étale groupoids and their quantales (2004)

Citations: | 16 - 7 self |

### BibTeX

@TECHREPORT{Resende04étalegroupoids,

author = {Pedro Resende},

title = {Étale groupoids and their quantales},

institution = {},

year = {2004}

}

### OpenURL

### Abstract

We establish a close and previously unknown relation between quantales and groupoids, in terms of which the notion of étale groupoid is subsumed in a natural way by that of quantale. In particular, to each étale groupoid, either localic or topological, there is associated a unital involutive quantale. We obtain a bijective correspondence between localic étale groupoids and their quantales, which are given a rather simple characterization and are here called inverse quantal

### Citations

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Citation Context ...rse semigroups, and groupoids. There are close and well known connections between the latter two concepts, whose importance in algebra, geometry, and analysis is by now firmly established (see, e.g., =-=[2, 3, 5, 12, 13, 14, 17, 27, 28]-=-), and in this paper we intend to provide evidence showing that also the theory of quantales may have a natural role to play in this context, indeed providing useful new insights and techniques. The h... |

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Citation Context ...s are injective when restricted to U. (We remark that this terminology is unfortunate because it colides with the standard usage of “G-set” for a set equipped with an action by a group G — see, e.g., =-=[16]-=-.) This is an inverse monoid with unit G0. The Wagner-Preston theorem asserts that every inverse semigroup is concretely representable as a pseudogroup. The natural order of an inverse semigroup S is ... |

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Citation Context ...rse semigroups, and groupoids. There are close and well known connections between the latter two concepts, whose importance in algebra, geometry, and analysis is by now firmly established (see, e.g., =-=[2, 3, 5, 12, 13, 16, 26, 27]-=-), and in this paper we intend to provide evidence showing that also the theory of quantales may have a natural role to play in this context, indeed providing useful new insights and techniques. The h... |

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Citation Context ...rse semigroups, and groupoids. There are close and well known connections between the latter two concepts, whose importance in algebra, geometry, and analysis is by now firmly established (see, e.g., =-=[2, 3, 5, 12, 13, 14, 17, 27, 28]-=-), and in this paper we intend to provide evidence showing that also the theory of quantales may have a natural role to play in this context, indeed providing useful new insights and techniques. The h... |

150 | Sketches of an Elephant: a Topos Theory Compendium - Johnstone - 2002 |

113 |
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Citation Context ...e sense of [27]), etc. Similarly, any localic groupoid whose multiplication map is at least semiopen gives rise to a quantale. For instance, any Grothendieck topos arises from such a localic groupoid =-=[9]-=-. We shall see that under reasonable assumptions the whole groupoid structure can be recovered from its “quantal topology”, just as in the discrete case, in addition obtaining a precise characterizati... |

68 |
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Citation Context ...rse semigroups, and groupoids. There are close and well known connections between the latter two concepts, whose importance in algebra, geometry, and analysis is by now firmly established (see, e.g., =-=[2, 3, 5, 12, 13, 14, 17, 27, 28]-=-), and in this paper we intend to provide evidence showing that also the theory of quantales may have a natural role to play in this context, indeed providing useful new insights and techniques. The h... |

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29 | A homology theory for étale groupoids
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Citation Context ...oupoids in Joyal and Tierney’s fundamental representation theorem for Grothendieck toposes [9], by the existence of homology and cohomology theories directly associated to étale groupoids (see, e.g., =-=[4]-=-), by a wealth of applications of Lie groupoids in differential geometry [2, 3, 14, 17], etc. In addition, specific topics worth looking at are: the étale groupoid of Penrose tilings (in [25] the topo... |

26 |
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Citation Context ...e latter, for instance proving early on that right sided quantales classify postliminary C*-algebras but not general C*-algebras [1]. More recently, advances in the representation theory of quantales =-=[23]-=- have led to an improvement of this, namely enabling one to show that the quantale valued functor Max (a noncommutative analogue of the maximal spectrum of commutative C*-algebras [19, 24]) is a compl... |

20 |
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Citation Context ...hilosophical matter, a theorem proved constructively is in fact many theorems (an equivariant version, a version for bundles, etc.), and so constructivity is also a matter of practical relevance (cf. =-=[21]-=-). An important aspect of these results is that they suggest ways of generalizing the duality of inverse semigroups and étale groupoids beyond étale groupoids; the natural generalization of the notion... |

18 |
Inverse Semigroups: the Theory
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Citation Context ...dy quantales, inverse semigroups, and groupoids. There are close and well known connections between the latter two concepts, whose importance in algebra, geometry, and analysis is widely acknowledged =-=[2, 9, 21, 22]-=-, and in this paper we intend to provide evidence showing that also the theory of quantales has a very natural role to play in this context, indeed providing useful new insights and techniques. The hi... |

13 | 2002): Sketches of an Elephant: A Topos Theory - Johnstone |

12 |
On the Quantization of Spaces
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Citation Context ...ry of quantales [23] have led to an improvement of this, namely enabling one to show that the quantale valued functor Max (a noncommutative analogue of the maximal spectrum of commutative C*-algebras =-=[19, 24]-=-) is a complete invariant of unital C*-algebras [11]. Although this is a positive (and to some extent surprising) result, it is still unsatisfactory because Max does not have the functorial properties... |

9 | Sup-lattice 2-forms and quantales
- Resende
(Show Context)
Citation Context ...attice L [22] (this is isomorphic to ℘(X × X) if L ∼ = ℘(X)) — L is both a right and a left module over the quantale. 4. More generally, the quantale of endomorphisms of a symmetric suplattice 2-form =-=[31]-=-. 135. The quantale Sub(R) of additive subgroups of a unital involutive ring R. The set of additive subgroups Sub(M) of any left R-module M is a left module over Sub(R). 6. The quantale Max A of norm... |

8 |
Hazewinkel (Ed.), The Encyclopaedia of Mathematics, third supplement
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Citation Context ...es. The historical motivation for this is clear. Quantales (short for “quantum locales” [18]) are algebraic structures that can be regarded as point-free noncommutative spaces (see, e.g., the surveys =-=[20, 26]-=-), and it is therefore natural to investigate the extent to which they relate to basic notions in noncommutative geometry such as groupoids and operator algebras. Several papers have been devoted to t... |

8 | A noncommutative theory of Penrose tilings
- Mulvey, Resende
(Show Context)
Citation Context ...tion, the first direct connection between a quantale and a (principal) groupoid, again via representation theory, appeared in a paper where the space of Penrose tilings has been modeled as a quantale =-=[25]-=-. The present paper is inspired by this but it is very different because (besides being concerned with more than the single example of Penrose tilings) it does not use representations, it also address... |

6 |
On quantales and spectra
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(Show Context)
Citation Context ...me extent surprising) result, it is still unsatisfactory because Max does not have the functorial properties that would be appropriate for making it a noncommutative generalization of Gelfand duality =-=[7]-=-, the quest for which is one of the guiding ideas behind the theory. This suggests that more specific examples should be looked at, such as C*algebras obtained from locally compact groupoids [21, 22],... |

5 |
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Citation Context ...G, or, in 2 G , of its characteristic function. 4Such quantales are said to be unital and involutive, and examples of them are well known, namely the quantale ℘(X × X) of binary relations on a set X =-=[22]-=-, or, for a group G, the quantale ℘(G), which is the image of G under a left adjoint functor from involutive monoids to unital involutive quantales whose right adjoint is the forgetful functor, and wh... |

4 |
den Bossche
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Citation Context ...groupoids and operator algebras. Several papers have been devoted to the latter, for instance proving early on that right sided quantales classify postliminary C*-algebras but not general C*-algebras =-=[1]-=-. More recently, advances in the representation theory of quantales [23] have led to an improvement of this, namely enabling one to show that the quantale valued functor Max (a noncommutative analogue... |

4 |
On quantales that classify C*-algebras, Cahiers de Top. et Geom
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(Show Context)
Citation Context ... namely enabling one to show that the quantale valued functor Max (a noncommutative analogue of the maximal spectrum of commutative C*-algebras [19, 24]) is a complete invariant of unital C*-algebras =-=[11]-=-. Although this is a positive (and to some extent surprising) result, it is still unsatisfactory because Max does not have the functorial properties that would be appropriate for making it a noncommut... |

3 | A note on infinitely distributive inverse semigroups
- Resende
(Show Context)
Citation Context ...ty sets, but the proof applies equally to joins of any subsets. An analogous and related property of inverse semigroups concerns distributivity of binary meets over joins: 24Proposition 2.7 (Resende =-=[30]-=-) Let S be an infinitely distributive inverse semigroup, let x ∈ S, and let (yi) be a family of elements of S. Assume that the join ∨ i yi exists, and that the meet x ∧ ∨ i yi exists. Then, for all i ... |

2 |
Two applications of the dual groupoid of a C*-algebra, In: Operator Algebras and Their Connections with Topology and Ergodic Theory, Lect
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Citation Context ... been ignored); non-étale groupoids, groupoid representations, and Morita equivalence; Lie groupoids and applications to differential geometry; the notion of dual groupoid of a C*-algebra of [3] (see =-=[23]-=-), and its relation to the functor Max; in a totally different area, applications to logic via interpretations of modal logics in supported quantales that generalize in a natural way the usual Kripke ... |

1 |
Categories and Groupoids, Reprints in Theory and Applications of Categories
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Sur le théorème de Riesz dans les algèbres stellaires, Thèse de 3ème cycle
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Citation Context ...enrose tilings (in [25] the topology of the groupoid has been ignored); constructions of groupoids from C*-algebras, such as AF-groupoids from AFalgebras [28], or the dual groupoid of a C*-algebra of =-=[6]-=- (see [29]), along with their relations both to the kind of quantales studied in this paper and to the functor Max; also applications in rather different areas, for instance in logic where supported q... |

1 |
398–404. 57 A. Cannas da
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(Show Context)
Citation Context ...groupoids and operator algebras. Several papers have been devoted to the latter, for instance proving early on that right sided quantales classify postliminary C*-algebras but not general C*-algebras =-=[1]-=-. More recently, advances in the representation theory of quantales [22] have led to an improvement of this, namely enabling one to show that the quantale valued functor Max (a noncommutative analogue... |