A topos for algebraic quantum theory

Motivated by Bohr's idea that the empirical content of quantum physics is accessible only through classical physics, we show how a C*-algebra A induces a topos in which the amalgamation of all its commutative subalgebras comprises a single //commutative// C*-algebra. According to the constructive Gelfand duality theorem of Banaschewski and Mulvey, the latter has an internal spectrum X in the topos, which plays the role of a quantum phase space of the system. States on A become probability integrals on X, and self-adjoint elements of A define functions from X to the pertinent internal real numbers (the interval domain), allowing for a state-proposition pairing. Thus the quantum theory defined by A is turned into a classical theory by restriction to its associated topos.