Re: dualities

[K C H Mackenzie <K.Mackenzie@sheffield.ac.uk>]

Quoting Peter Freyd <pjf@saul.cis.upenn.edu>:

> On the subject of favorite dualities:
>
> Surely the most important are the self-dualities and the most
> important of these (so important we stop noticing it as we age) is the
> category of finite-dimensional vector spaces over a given field.

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 $G$.

A double vector bundle (in the sense of Ehresmann) is a particular
instance of a vector bundle in the category of Lie groupoids.
Pradines' duality can be applied to such a structure in two ways,
and these do not commute. If $D$ is a double vector bundle over
vector bundles $A$ and $B$, each of which is a vector bundle over
a manifold $M$, then $D$ can be dualized over $A$ and over $B$.
These dualization operations generate the dihedral group of order 6.
See `Duality and triple structures', pp455--481 of `The breadth of
symplectic and Poisson geometry', (Weinstein Festschrift), Progr.
Math., Birkh\"auser Boston, 2005.

Alfonso Gracia-Saz and I are preparing a paper on the duality
of $n$-fold vector bundles.

Details and references for the double case can be found in my
`General Theory of Lie groupoids and Lie algebroids', Cambridge,
2005, Chapter 9.

Whether categlorification would add anything to this I do not know.

Kirill Mackenzie