The answers to this question do a good job of exploring, at a heuristic level, what "quantization" should be. From my perspective, quantization involves replacing a (commutative) Poisson algebra by some related noncommutative associative algebra. Poisson algebras arise naturally especially as algebras of functions in geometry and physics. Noncommutative algebras arise naturally as algebras of operators on linear spaces.

I've often heard it said that "quantization is not a functor". I'm wondering what a precise statement of this is.

For example, I could imagine statements of the following form.

1. There is no functor from the category of Poisson manifolds (and Poisson maps?) to the (opposite of the) category of associative algebras satisfying some nice property.
2. There is no functor from the category of symplectic manifolds (and Poisson maps?) to the (opposite of the) category of associative algebras satisfying some nice property.
3. Recall that for any smooth manifold, its cotangent bundle is naturally symplectic. There is no functor from the category of smooth manifolds to the category of associative algebras that quantizes the cotangent bundle.
4. Recall that the dual to the universal enveloping algebra of a Lie bialgebra is naturally Poisson Hopf. There is no functor from the category of Lie bialgebras to the category of Hopf algebras satisfying some nice property.

Actually, 4. is false. Indeed, Etingof and Khazdan constructed a functor from bialgebras to Hopf algebras satisfying a host of properties, and Enriquez classified all the ones with nice properties. Note that Kontsevich does give a quantization of any Poisson manifold, but perhaps his isn't functorial?

The answers to this question do a good job of exploring, at a heuristic level, what "quantization" should be. From my perspective, quantization involves replacing a (commutative) Poisson algebra by some related noncommutative associative algebra. Poisson algebras arise naturally especially as algebras of functions in geometry and physics. Noncommutative algebras arise naturally as algebras of operators on linear spaces.

I've often heard it said that "quantization is not a functor". I'm wondering what a precise statement of this is.

For example, I could imagine statements of the following form.

1. There is no functor from the category of Poisson manifolds (and Poisson maps?) to the (opposite of the) category of associative algebras satisfying some nice property.
2. There is no functor from the category of symplectic manifolds (and Poisson maps?) to the (opposite of the) category of associative algebras satisfying some nice property.
3. Recall that for any smooth manifold, its cotangent bundle is naturally symplectic. There is no functor from the category of smooth manifolds to the category of associative algebras that quantizes the cotangent bundle.
4. Recall that the dual to the universal enveloping algebra of a Lie bialgebra is naturally Poisson Hopf. There is no functor from the category of Lie bialgebras to the category of Hopf algebras satisfying some nice property.

Actually, 4. is false. Indeed, Etingof and Khazdan constructed a functor from bialgebras to Hopf algebras satisfying a host of properties, and Enriquez classified all the ones with nice properties. Note that Kontsevich does give a quantization of any Poisson manifold, but perhaps his isn't functorial?