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Topological Groupoid Quantales

A. Palmigiano, R. Re
2010 Studia Logica: An International Journal for Symbolic Logic  
We associate a canonical unital involutive quantale to a topological groupoid.  ...  When the groupoid is alsoétale, this association is compatible with but independent from the theory of localicétale groupoids and their quantales [19] of P. Resende.  ...  Topological groupoid quantales In this section we will define a general procedure that associates a unital involutive quantale with any topological groupoid G.  ... 
doi:10.1007/s11225-010-9251-9 fatcat:2dmnzfpeifgibmpup47xmw6nxy

Quantales as geometric objects: symmetry beyond groupoids? [article]

Pedro Resende
2005 arXiv   pre-print
Its purpose is to highlight some aspects of the interplay between quantales, inverse semigroups, and groupoids.  ...  The topology of a topological open groupoid G is, similarly to that of anétale groupoid, closed under pointwise multiplication of open sets, and thus it is a quantale.  ...  This makes Ω(G 1 ) a unital (and involutive) quantale. A topological groupoid isétale precisely if its topology has this property [16, Th. 5.18].  ... 
arXiv:math/0506451v1 fatcat:kq4jx2fz2rcgfbybgkupikuhue

Groupoid quantales: A non-étale setting

Alessandra Palmigiano, Riccardo Re
2011 Journal of Pure and Applied Algebra  
groupoid quantales Q(G, S); in Section 4 we introduce the SGF-quantales: these quantales are to groupoid quantales what locales are to topologies.  ...  Correspondingly, their associated groupoids do not have a topological (or localic) structure on their spaces of arrows G 1 .  ...  Groupoid quantales Definition 3.8.  ... 
doi:10.1016/j.jpaa.2010.11.005 fatcat:4jfrojxibjglbmpqpzdqws5hwy

Groupoid Quantales: a non étale setting [article]

Alessandra Palmigiano, Riccardo Re
2010 arXiv   pre-print
This topology has a natural structure of unital involutive quantale. We present the analogous construction for any non \'etale groupoid with sober unit space G_0.  ...  It is well known that if G is an \'etale topological groupoid then its topology can be recovered as the sup-lattice generated by G-sets, i.e. by the images of local bisections.  ...  groupoid quantales Q(G, S); in Section 4 we introduce the SGF-quantales: these quantales are to groupoid quantales what locales are to topologies.  ... 
arXiv:1006.5128v1 fatcat:7t7qgbqqgjafdgnndhop457ury

Étale groupoids and their quantales

Pedro Resende
2007 Advances in Mathematics  
In particular, to each etale groupoid, either localic or topological, there is associated a unital involutive quantale.  ...  and localic etale groupoids that generalizes more classical results concerning inverse semigroups and topological etale groupoids.  ...  In other words, we have: Quantal frames We have already remarked that the topologies of certain topological groupoids are quantales.  ... 
doi:10.1016/j.aim.2006.02.004 fatcat:3yesgi3lovdaxiu6khtznzpuuu

Effective equivalence relations and principal quantales [article]

Juan Pablo Quijano, Pedro Resende
2019 arXiv   pre-print
A notion of principal quantale is introduced which, in the case of groupoid quantales, corresponds precisely to effective equivalence relations.  ...  Stably supported quantales generalize pseudogroups and provide an algebraic context in which to study the correspondences between inverse semigroups and \'etale groupoids.  ...  are the topological groupoids whose topologies are monoids.  ... 
arXiv:1807.08979v2 fatcat:o5m23xfv55arppm6qsgkhwzvna

Functoriality of groupoid quantales. I

Pedro Resende
2015 Journal of Pure and Applied Algebra  
functors whose morphisms, in the context of topological groupoids, have been studied by Lawson and Lenz.  ...  We provide three functorial extensions of the equivalence between localic etale groupoids and their quantales.  ...  Concretely, the quantale of a topological groupoid G (with open domain map) is the topology of the arrow space G 1 equipped with pointwise operations of multiplication and involution.  ... 
doi:10.1016/j.jpaa.2014.10.004 fatcat:ww3ltenfmfhyxi53baubfjevci

The many groupoids of a stably Gelfand quantale

Pedro Resende
2018 Journal of Algebra  
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  ...  We study the projections of an arbitrary stably Gelfand quantale Q and show that each projection determines a pseudogroup S⊂ Q (and a corresponding localic étale groupoid G) together with a map of involutive  ...  Then the locale L ∨ (I b (Q)) is order-isomorphic to the topology of (the space of arrows of) the groupoid G b (Q).  ... 
doi:10.1016/j.jalgebra.2017.11.042 fatcat:ia354ubgaja6ppclryi5rm236q

On the geometry of physical measurements: topological and algebraic aspects [article]

Pedro Resende
2022 arXiv   pre-print
Two important classes of measurement spaces are obtained, respectively from C*-algebras and from second-countable locally compact open sober topological groupoids.  ...  In this paper the topological and algebraic foundations of the theory are developed.  ...  Since the multiplication map of an open groupoid is open, the topology Ω(G 1 ) is a quantale under pointwise multiplication (see [27] ).  ... 
arXiv:2005.00933v4 fatcat:tv7wwiyccbdzfdppxnxq5ndce4

Quantales of open groupoids

M. Clarence Protin, Pedro Resende
2012 Journal of Noncommutative Geometry  
The resulting correspondence between quantales and open groupoids is not a straightforward generalization of the previous results concerning \'etale groupoids, and it depends heavily on the existence of  ...  of quantales, called open quantal frames, which generalize inverse quantal frames and whose properties we study in detail.  ...  For topological étale groupoids something analogous holds, with the quantale being simply the topology of G with product given by pointwise multiplication of open sets.  ... 
doi:10.4171/jncg/90 fatcat:wd7kfe2xzfgwdkrescpqeurlla

Quantales and Fell bundles

Pedro Resende
2018 Advances in Mathematics  
We study Fell bundles on groupoids from the viewpoint of quantale theory.  ...  a map of involutive quantales p:Max A→Ω(G), where Max A consists of the closed linear subspaces of A and Ω(G) is the topology of G.  ...  Example 2. 9 . 9 Let L be a locale and Y a topological space, regarded as a groupoid G = G 0 = Y .  ... 
doi:10.1016/j.aim.2017.12.001 fatcat:bg5opd752ffwnb4w22sy24x5ia

Groupoid sheaves as quantale sheaves

Pedro Resende
2012 Journal of Pure and Applied Algebra  
frame O(G) by placing it in the wider context of stably supported quantales, on one hand, and in the wider context of a module theoretic description of arbitrary actions of \'etale groupoids, both of  ...  of G) is equivalent to the category of sheaves on its involutive quantale O(G).  ...  Each étale groupoid G has an associated involutive quantale O(G) [65] (for a topological groupoid this quantale is just the topology equipped with pointwise multiplication of groupoid arrows), and the  ... 
doi:10.1016/j.jpaa.2011.05.002 fatcat:s2vu2hhpungexjeiciktjoyoqi

Relational Representation of Groupoid Quantales

Alessandra Palmigiano, Riccardo Re
2011 Order  
be representable as sub unital involutive quantales of quantales arising from set groupoids.  ...  Keywords Unital involutive quantale · Strongly Gelfand quantale · Set groupoid · Representation theorem Mathematics Subject Classifications (2010) 06D05 · 06D22 · 06D50 · 06F07 · 18B40 · 20L05 · 22A22  ...  Every topological space X can be seen as a groupoid by setting G 1 = G 0 = X and identity structure maps. In this case, G 1 × G0 G 1 = {(x, x) | x ∈ X} and xx = x for every x ∈ X. 4.  ... 
doi:10.1007/s11083-011-9227-z fatcat:rfcfhtoecneq5ab7hzjbbkqi3y

Functoriality of groupoid quantales. II [article]

Juan Pablo Quijano, Pedro Resende
2018 arXiv   pre-print
Taking advantage of the quantale-theoretic description of \'etale 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.  ...  For a topological groupoid G the associated unital involutive quantale is the topology Ω(G 1 ) equipped with pointwise multiplication and inversion of open sets, and with e = u(G 0 ).  ... 
arXiv:1803.01075v2 fatcat:q6ombfpe5rahphgzhgs3tm5gly

Algebras with compatible quantale-enriched structures

Bin Zhao, Min Liu
2014 Science China Information Sciences  
We mainly study groupoids and semigroups with compatible quantale-enriched structures from this viewpoint.  ...  Some basic concepts such as ideals, homomorphisms, residuated quantale-enriched groupoids are developed and some examples of them are given.  ...  For example, frames are closely related to topology, Girard quantales have some applications in linear logic and the quantale of the extended non-negative reals is closely related to metric theory.  ... 
doi:10.1007/s11432-014-5141-5 fatcat:6nu33bedbjb4dogysnu44rfoyq
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