Bohrification is a natural way to construct a quantum "phase space" (with some nice insights on foundational problems like non-contextuality through Kochen-Specker etc). I was wondering, since we get quantum kinematics from Bohr topos is there a link with quantization (geometric or deformation)?

More precisely: since the internal algebra $\underline{A} \in Bohr(A)$ is commutative, the Gelfand duality identifies it with a classical phase space $X$.

Is there a link ("geometric" morphism?) between the quantization of this classical phase space $X$ and the topos-theoretic quantum phase space?