There is a "partial category" of Lagrangian relations, otherwise known as canonical relations, whose objects are symplectic vector spaces and whose morphisms are Lagrangian submanifolds.
By changing the objects to symplectic vector spaces, and the morphisms to linear Lagrangian subspaces, one obtains a category of linear lagrangian relations which I am much more comfortable with.
The "odd dimensional" version of symplectic manifolds are apparently contact manifolds: here the analogue of a Lagrangian subspace is a Legendrian subspace.
Here is my question:
- First, is there any interesting notion of "contact vector space"?
- Second, if this is true, is there a category of "linear Legendrian relations"?
