re: duality

re: duality

[jpradines]

%%%%%%%%%%%%%
Kirill wrote:
.................
Something on this has been done.

Duality for vector bundle objects in the category of Lie groupoids
was done by Jean Pradines in 1988, and is part of the fundamental
work on symplectic groupoids. The cotangent bundle $T^*G$ of any
Lie groupoid $G$ has a groupoid structure with base the dual of
$AG$, the Lie algebroid of $G$, and Pradines' construction
realizes this as the dual of the tangent prolongation $TG$ of $G$.
...............
Kirill Mackenzie
%%%%%%%%%%%%%%%%%%%

Since Kirill alluded to my note  entitled "Remarque sur le groupoide cotangent de Weinstein-Dazord", published in CRAS, 306, 557-560 (1988), which I did not intend to do, I think I've better add some more comments.
The origin of this Note is an attempt to understand (and categorify!) the discovery by Alan Weinstein, and (more or less independantly) by various authors, that, given a smooth groupoid  G (in the sense of Ehresmann) with base B, its cotangent bundle T*G owns a canonical groupoid structure (apart from the abelian group bundle structure of course), which, when G is a Lie group, reduces to the action groupoid describing the coadjoint action (the base of this groupoid is then the dual of the Lie algebra; in the general case it is the dual vector bundle of what I introduced under the name of Lie algebroid).
The construction of these authors made an extensive use of the symplectic structure of the cotangent bundle, and of symplectic duality and orthogonality, describing the graph of the groupoid composition law by means of Lagrangian submanifolds. Actually they prove more, since they show that this groupoid is a symplectic groupoid (in a sense that has also certainly to be better categorified, since it is not easily described as an internal groupoid in some known category, but that's another story, which I don't want to tackle here).
I was amazed by this result, since, while it is obvious, by (covariant!) functoriality of T, that the tangent bundle TG has a groupoid structure with base TB (which describes the action of G on "vertical vectors", more precisely on the Lie algebroid), one would better expect for the cotangent bundle some kind of co-groupoid structure.
Finally I realized that the result has nothing to do with the symplectic structure, and extends in a natural way for what is called in my Note "vector bundle groupoids", which means internal groupoids in the category of vector bundles (without any given extra structure ). Such a vector bundle groupoid has a canonically defined dual object in the same category.  
The construction is made "by hand", and I am not really satisfied with it (though it oversimplifies the genuine one). The point is of course that the"duality" in the category of vector bundles is easy to define and handle only when the base of the bundles remains fixed, while the construction has to cope with general vector bundle morphisms, with at least three different bases: B, G and the set of composable arrows.
Notably I was not able to discover some simple general duality relation between the nerves of the groupoid and of its dual, as one would like.
So I am still amazed with the result, which should become "obvious" in a better adapted framework to be discovered.
Best regards.
Jean Pradines