The aim of this paper is to relate algebraic quantum mechanics to topos theory, so as to construct new foundations for quantum logic and quantum spaces. Motivated by Bohr’s idea that the empirical content of quantum physics is accessible only through classical physics, we show how a noncommutative C*-algebra of observables A induces a topos T (A) in which the amalgamation of all of its commutative subalgebras comprises a single commutative C*-algebra A. According to the constructive Gelfand duality theorem of Banaschewski and Mulvey, the latter has an internal spectrum �(A) in T (A), which in our approach plays the role of the quantum phase space of the system. Thus we associate a locale (which is the topos-theoretical notion of a space and which intrinsically carries the intuitionistic logical structure of a Heyting algebra) to a C*-algebra (which is the noncommutative notion of a space). In this setting, states on A become probability measures (more precisely, valuations) on �, and self-adjoint elements of A define continuous functions (more precisely, locale maps) from � to Scott’s interval domain. Noting that open subsets of �(A) correspond to propositions about the system, the pairing map that assigns a (generalized) truth value to a state and a proposition assumes an extremely simple categorical form. Formulated in this way, the quantum theory defined by A is essentially turned into a classical theory, internal to the topos T (A). These results were inspired by the topos-theoretic approach to quantum physics proposed by Butterfield and Isham, as recently generalized by Döring and Isham.

This paper is the third in a series whose goal is to develop a fundamentally new way of viewing theories of physics. Our basic contention is that constructing a theory of physics is equivalent to finding a representation in a topos of a certain formal language that is attached to the system. In paper II, we studied the topos representations of the propositional language PL(S) for the case of quantum theory, and in the present paper we do the same thing for the, more extensive, local language L(S). One of the main achievements is to find a topos representation for self-adjoint operators. This involves showing that, for any physical quantity A, there is an arrow ˘ δo (A) : Σ → IR ≽ , where IR ≽ is the quantity-value object for this theory. The construction of ˘ δo ( Â) is an extension of the daseinisation of projection operators that was discussed in paper II., of The object IR ≽ is a monoid-object only in the topos, τφ = SetsV(H)op the theory, and to enhance the applicability of the formalism, we apply to IR ≽ a topos analogue of the Grothendieck extension of a monoid to a group. The resulting object, k(IR ≽), is an abelian group-object in τφ. We also discuss another candidate, IR ↔ , for the quantity-value object. In this presheaf, both inner and outer daseinisation are used in a symmetric way. Finally, there is a brief discussion of the role of unitary operators in the quantum topos scheme.