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Etale groupoids and their quantales
Category Theory · Submitted 23 Dec 2004
abstract compiled · 2026-09-23 · no language-model review
The authors' main claim 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.
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The abstract, sentence by sentence
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.
background contribution method result interpretation main claim
| # | sentence | role | the authors' own work | states the main conclusion | strength |
|---|---|---|---|---|---|
| s0 | 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. | contribution 0.78 | 0.95 | 0.83 | asserted 1.00 |
| s1 | In particular, to each etale groupoid, either localic or topological, there is associated a unital involutive quantale. | result 0.92 | 0.88 | 0.17 | asserted 1.00 |
| s2 | 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. | result 0.94 | 0.96 | 0.23 | asserted 1.00 |
| s3 | 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. | result 0.95 | 0.95 | 0.18 | asserted 1.00 |
| s4 | This generalization is entirely algebraic and it is valid in an arbitrary topos. | result 0.54 | 0.79 | 0.08 | asserted 1.00 |
| s5 | 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. | result 0.94 | 0.89 | 0.13 | asserted 1.00 |
| s6 | 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. | interpretation 0.78 | 0.72 | 0.06 | asserted 0.91 |
| s7 | 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. | contribution 0.60 | 0.94 | 0.07 | asserted 0.80 |
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Original abstract
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.
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How the literature uses it
4 arXiv papers cite this in 8 sentences
Papers citing it · 6 sentences without a confident label
Cites it as background 2 papers
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See [ref] for a study of inverse quantal frames.
Spectral C*-categories and Fell bundles with path-lifting · Groupoids · 0.96
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A localic groupoid is defined in [ref] as a groupoid internal in Loc (recall that a topological groupoid is a groupoid internal in Top).
Two partial monoid structures on a set · Applications and discussion · 0.99
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