# Bohr topos and quantization

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?

• There is some initial work link.aip.org/link/doi/10.1063/1.4813960 with a lot of room for improvement. – Chris Heunen Aug 3 '13 at 13:55
• Incidentally, your question would probably do better with some improvement, too: mathoverflow.net/help/how-to-ask – Chris Heunen Aug 3 '13 at 13:55
• Thanks for your answer and comment Chris, I tried to make the question more precise. – Issam Ibnouhsein Aug 3 '13 at 14:13
• Several standard constructions on obervables immediately prolong to the Bohr toposes. For instance maps from some subalgebra of classical observables into the quantum observables. This is what Nakayama writes down. For a genuine quantization in terms of Bohr toposes one would probably wish to relate the sheaf topos on the classical phase space to the Bohr toposes of one of its C*-algebraic deformation quantizations. If one understands the latter as a field of C*-algebras over $[0,1)$, then again one can try to prolong this to a "field" of Bohr toposes. – Urs Schreiber Aug 4 '13 at 16:45