  Notwithstanding known obstructions to this idea, we formulate an attempt to
turn quantization into a functorial procedure. We define a category PO of
Poisson manifolds, whose objects are integrable Poisson manifolds and whose
arrows are isomorphism classes of regular Weinstein dual pairs; it follows that
identity arrows are symplectic groupoids, and that two objects are isomorphic
in PO iff they are Morita equivalent in the sense of P. Xu. It has a
subcategory LPO that has duals of integrable Lie algebroids as objects and
cotangent bundles as arrows. We argue that naive C*-algebraic quantization
should be functorial from LPO to the well-known category KK, whose objects are
separable C*-algebras and whose arrows are Kasparov's KK-groups. This limited
functoriality of quantization would already imply the Atiyah-Singer index
theorem, as well as its far-reaching generalizations developed by Connes and
others. In the category KK, isomorphism of objects implies isomorphism of
K-theory groups, so that the functoriality of quantization on all of PO would
imply that Morita equivalent Poisson algebras are quantized by C*-algebras with
isomorphic K-theories. Finally, we argue that the correct codomain for the
possible functoriality of quantization is the category RKK(I), which takes the
deformation aspect of quantization into account.
