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