Grothendieck's approach to algebraic geometry in particular tells us to treat all rings as rings of functions
on some sort of space. This can also be applied outside of scheme theory (e.g., Gelfand-Neumark theorem
says that the category of measurable spaces is contravariantly equivalent to the category
of commutative von Neumann algebras). Even though we do not have a complete geometric description
for the noncommutative case, we can still use geometric intuition from the commutative case effectively.
A generalization of this idea is given by Grothendieck's relative point of view, which says that a morphism of rings f: A → B
should be regarded geometrically as a bundle of spaces with the total space Spec B fibered over the base space Spec A
and all notions defined for individual spaces should be generalized to such bundles fiberwise.
For example, for von Neumann algebras we have operator valued weights, relative L^p-spaces etc.,
which generalize the usual notions of weight, noncommutative L^p-space etc.
In noncommutative geometry this point of view is further generalized to bimodules.
A morphism f: A → B can be interpreted as an A-B-bimodule B, with the right action of B given by the multiplication
and the left action of A given by f.
Geometrically, an A-B-bimodule is like a vector bundle over the product of Spec A and Spec B.
If a bimodule comes from a morphism f, then it looks like a trivial line bundle with the support being
equal to the graph of f.
In particular, the identity morphism corresponds to the trivial line bundle over the diagonal.
For the case of commutative von Neumann algebras all of the above can be made fully rigorous
using an appropriate monoidal category of von Neumann algebras.
This bimodule point of view is extremely fruitful in noncommutative geometry (think of Jones' index,
Connes' correspondences etc.)
However, I have never seen bimodules in other branches of geometry (scheme theory,
smooth manifolds, holomorphic manifolds, topology etc.) used to the same extent as they are used
in noncommutative geometry.
Can anybody state some interesting theorems (or theories) involving bimodules in such a setting?
Or just give some references to interesting papers?
Or if the above sentences refer to the empty set, provide an explanation of this fact?
Best Answer
In "commutative geometry," I think bimodules tend to be a little concealed. People are more likely to talk about "correspondences" which are the space version of bimodules: A correspondence between spaces X and Y is a space Z with maps to X and Y.
When you think in this language, there are lots of examples you're missing. For example, the right notion of a morphism between two symplectic manifolds is a Lagrangian subvariety of their product, or even a manifold mapping to their product with Lagrangian image (maybe not embedded). See, for example, Wehrheim and Woodward's Functoriality for Lagrangian correspondences in Floer homology.
Similarly, correspondences are incredibly important in geometric representation theory. See, for example, the work of Nakajima on quiver varieties.
The theory of stacks also is at least partially founded on taking correspondences seriously as objects, and in particular being able to quotients by any (flat) correspondence.
This same philosophy also underlies groupoidification as studied by the Baez school (they tend to use the word "span" instead of "correspondence" but it's the same thing).