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Dear All:

I would like some refs and/or thoughts on the following two related questions:

1) If I am not mistaken, there is a " Groupoid Convolution Algebra" (GCA) contravariant functor from the category of Compact Topological Groupoids into the one of C*-algebras and their morphisms

GCA: KTopGroupoid -----------------------> C*-Alg^{op}

which is essentially a generalization of the standard complex matrix algebra.

Is it known whether GCA has adjoints, either between the two categories above or some suitable restrictions/replacements thereof?

2) If one starts from a topological groupoid G together with its associated groupoid convolution algebra GCA(G), one should be able to construct its classifying topos, together with an internal C*-alg object (it would be the equivariant sheaf of algebras obtained by GCA(G))..

I am a bit familiar with the notion of ringed topoi, but I do not know whether there is a name for a gadget of this sort, so in temporary lack of the proper label, I''ll call it a C*-topos.

Maps between such objects should be geometric morphisms "respecting" their internal *-algebras, forming a large category.

Is there some ref that studies this cat (or similar ones) as a candidate for a "topos theoretical version" of non commutative topology a' la Connes?

Thanks

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About the second question, I remember there is a short comment about this in a paper by Cartier (see the end of Section 6 in library.msri.org/books/sga/from_grothendieck.pdf, where Cartier refer's to an old paper of Tapia) –  DamienC May 22 '11 at 17:37
    
Actually you can take the codomain of the functor you describe to be locally compact groupoids equipped with a Haar measure. I then I would presume that the functors between them would have to respect this measure in some way. One could also consider 'measured groupoids', which are slightly more general, in that they don't necessarily arise from a topological groupoid (IIRC source and target maps, for example, are only measurable, not necessarily continuous). –  David Roberts May 22 '11 at 18:45
    
@ Damien: thanks, a friend of mine pointed to the same (excellent) article, and indeed it is along the same lines of my questions. Do you know if there are more recent refs following up Tapia? @David: thanks, yes, indeed locally compact groupoids should be enough. The question remains, though: is there some way to "invert" the arrow from groupoids to C* algebras? I believe there is, perhaps by giving the structure of groupoid to the set if irriducible reps of the algebra. –  Mirco Mannucci May 22 '11 at 20:45
    
@Mirco: I don't know. I think you can directly send an e-mail to Tapia in order to ask him. I also vaguely remember that Moerdijk has some work about topoi associated to Lie groupoids, together with Mrčcun (maybe even a book). –  DamienC May 22 '11 at 20:52
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Two comments: first, that construction mentioned in the article by Heunen is the "Bohr topos" construction (ncatlab.org/nlab/show/Bohr%20topos); it remembers only the Jordan algebra structure of the C-star algebra. Second, a characterization of groupoid convolution algebras (aka "Hopf algebroids"!) of at least etale Lie groupoids is here: ncatlab.org/nlab/show/… –  Urs Schreiber Apr 2 '13 at 17:00
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