There is no good definition of what quantization is. It usually means that we start with a commutative structure $A$, a 1st order (non commutative) deformation $A\otimes k[h]/(h^2)$ of this structure and we want to lift it to an actual (or formal) deformation $A_h$ so that $A_h/h^2A_h = A\otimes_k k[h]/(h^2)$.

In some cases we know that the lifting exists but it is never unique. Instead, in the formal setting, there is a group that acts simply transitively on the set of quantizations. So the set of "quantizations" is a torsor under this group.

For example, quantum groups theory is a theory of quantization of semi-simple groups or more precisely of their Hopf algebra. Drinfeld showed that, given the enveloping algebra $A = Ug$ of a Lie algebra $g$ (it is co-commutative Hopf algebra) over a field of caracteristic 0, and an ad-invariant symetric tensor $t\in Sym^2(g)^g$ (corresponding to a first order deformation), there exists a quantization of $Ug$, namely a quasi-triangular quasi-Hopf algebra $A_h \simeq Ug[[h]]$ reducing to these data mod $h^2$. The set of (universal) quantizations is a pro-algebraic variety $Assoc$ of "associators" and it is a torsor under the Grothendieck-Teichmuller group $GT$ which is an extension of $G_m$ (because we have an action $\varphi:h\mapsto \lambda h$) by a pro-unipotent group (because we can filter it by $\varphi \equiv id \mod h^n$).

The theorem of Kontsevich on deformation quantization of Poisson varieties shows a similar pattern. Here the structure is associative algebras. A 1st order deformation of a commutative algebra $A$ is a Poisson bracket. And we try to lift it to a formal deformation $A_h$.

In both cases, things are functorial. But we only consider "universal" quantizations. If we consider a single object (Hopf algebra or Poisson algebra) it may have some deformations that are not given by the universal recipe. Also note that in both cases, we just ask for an isorphism $A_h/h^2A_h = A\otimes k[h]/(h^2)$, we don't ask for a section $a\mapsto \hat a_h$ (a quantization rule like symetric ordering). I think such a requirement would always break functoriality.

There is no good definition of what quantization is. It usually means that we start with a commutative structure $A$, a 1st order (non commutative) deformation $A\otimes k[h]/(h^2)$ of this structure and we want to lift it to an actual (or formal) deformation $A_h$ so that $A_h/h^2A_h = A\otimes_k k[h]/(h^2)$.

In some cases we know that the lifting exists but it is never unique. Instead, in the formal setting, there is a group that acts simply transitively on the set of quantizations. So the set of "quantizations" is a torsor under this group.

For example, quantum groups theory is a theory of quantization of semi-simple groups or more precisely of their Hopf algebra. Drinfeld showed that, given the enveloping algebra $A = Ug$ of a Lie algebra $g$ (it is co-commutative Hopf algebra) over a field of caracteristic 0, and an ad-invariant symetric tensor $t\in Sym^2(g)^g$ (corresponding to a first order deformation), there exists a quantization of $Ug$, namely a quasi-triangular quasi-Hopf algebra $A_h \simeq Ug[[h]]$ reducing to these data mod $h^2$. The set of (universal) quantizations is a pro-algebraic variety $Assoc$ of "associators" and it is a torsor under the Grothendieck-Teichmuller group $GT$ which is an extension of $G_m$ (because we have an action $\varphi:h\mapsto \lambda h$) by a pro-unipotent group (because we can filter it by $\varphi \equiv id \mod h^n$).

The theorem of Kontsevich on deformation quantization of Poisson varieties shows a similar pattern. Here the structure is associative algebra (with some additional properties). A 1st order deformation of a commutative algebra $A$ is a Poisson bracket. And we try to lift it to a formal deformation $A_h$.

In both cases, I think the choice of an associator gives you a functor. But a) we only consider "universal" quantizations. If we consider a single object (Hopf algebra or Poisson algebra) it may have some deformations that are not given by the universal recipe. b) in both cases, we just ask for an isorphism $A_h/h^2A_h = A\otimes k[h]/(h^2)$. We do not ask for a section $a\mapsto \hat a_h$ (a quantization rule like symetric ordering). I think such a requirement would always break functoriality.

There is no good definition of what quantization is. It usually means that we start with a commutative structure $A$, a 1st order (non commutative) deformation $A\otimes k[h]/(h^2)$ of this structure and we want to lift it to an actual (or formal) deformation $A_h$ so that $A_h/h^2A_h = A\otimes_k k[h]/(h^2)$.

In some cases we know that the lifting exists but it is never unique. Instead, in the formal setting, there is a group that acts simply transitively on the set of quantizations. So the set of "quantizations" is a torsor under this group.

For example, quantum groups theory is a theory of quantization of semi-simple groups or more precisely of their Hopf algebra. Drinfeld showed that, given the enveloping algebra $A = Ug$ of a Lie algebra $g$ (it is co-commutative Hopf algebra) over a field of caracteristic 0, and an ad-invariant symetric tensor $t\in Sym^2(g)^g$ (corresponding to a first order deformation), there exists a quantization of $Ug$, namely a quasi-triangular quasi-Hopf algebra $A_h \simeq Ug[[h]]$ reducing to these data mod $h^2$. The set of (universal) quantizations is a pro-algebraic variety $Assoc$ of "associators" and it is a torsor under the Grothendieck-Teichmuller group $GT$ which is an extension of $G_m$ (because we have an action $h\mapsto \lambda h$) by a pro-unipotent group (because we can filter it by $g \equiv 1 \mod h^n$).

The theorem of Kontsevich on deformation quantization of Poisson varieties shows a similar pattern. Here the structure is associative algebras. A 1st order deformation of a commutative algebra $A$ is a Poisson bracket. And we try to lift it to a formal deformation $A_h$.

In both cases, things are functorial. But we only consider "universal" quantizations. If we consider a single object (Hopf algebra or Poisson algebra) it may have some deformations that are not given by the universal recipe. Also note that in both cases, we just ask for an isorphism $A_h/h^2A_h = A\otimes k[h]/(h^2)$, we don't ask for a section $a\mapsto \hat a_h$ (a quantization rule like symetric ordering). I think such a requirement would always break functoriality.

There is no good definition of what quantization is. It usually means that we start with a commutative structure $A$, a 1st order (non commutative) deformation $A\otimes k[h]/(h^2)$ of this structure and we want to lift it to an actual (or formal) deformation $A_h$ so that $A_h/h^2A_h = A\otimes_k k[h]/(h^2)$.

In some cases we know that the lifting exists but it is never unique. Instead, in the formal setting, there is a group that acts simply transitively on the set of quantizations. So the set of "quantizations" is a torsor under this group.

For example, quantum groups theory is a theory of quantization of semi-simple groups or more precisely of their Hopf algebra. Drinfeld showed that, given the enveloping algebra $A = Ug$ of a Lie algebra $g$ (it is co-commutative Hopf algebra) over a field of caracteristic 0, and an ad-invariant symetric tensor $t\in Sym^2(g)^g$ (corresponding to a first order deformation), there exists a quantization of $Ug$, namely a quasi-triangular quasi-Hopf algebra $A_h \simeq Ug[[h]]$ reducing to these data mod $h^2$. The set of (universal) quantizations is a pro-algebraic variety $Assoc$ of "associators" and it is a torsor under the Grothendieck-Teichmuller group $GT$ which is an extension of $G_m$ (because we have an action $\varphi:h\mapsto \lambda h$) by a pro-unipotent group (because we can filter it by $\varphi \equiv id \mod h^n$).

The theorem of Kontsevich on deformation quantization of Poisson varieties shows a similar pattern. Here the structure is associative algebras. A 1st order deformation of a commutative algebra $A$ is a Poisson bracket. And we try to lift it to a formal deformation $A_h$.

In both cases, things are functorial. But we only consider "universal" quantizations. If we consider a single object (Hopf algebra or Poisson algebra) it may have some deformations that are not given by the universal recipe. Also note that in both cases, we just ask for an isorphism $A_h/h^2A_h = A\otimes k[h]/(h^2)$, we don't ask for a section $a\mapsto \hat a_h$ (a quantization rule like symetric ordering). I think such a requirement would always break functoriality.