Source
Döring & Isham (2007; J. Math. Phys. 49, 2008), “A topos foundation for theories of physics: III”, LIT-299, read in full in NOTE-290. The merged text, Döring & Isham (2008), “‘What is a Thing?’”, LIT-378, §§7–9, read in full in NOTE-326, restates it with corrections and adds the value in a pseudo-state.
What was actually shown
Setting. Contexts are the unital abelian von Neumann subalgebras V of B(H), ordered by inclusion; the topos is presheaves over them; the state object Σ assigns each V its Gel’fand spectrum. Operators are bounded.
Daseinisation of quantities (Def 2.4). δ^o(Â)_V is the least element of V above  in the spectral order, δ^i(Â)_V the greatest below (2.21, 2.18). Both equal  if  ∈ V, and sp(δ^o(Â)_V) ⊆ sp(Â) (2.23). Both maps are non-linear and are not Borel functions of Â.
Not the real-number object (C3). The real-number object is the constant presheaf ℝ. The Gel’fand transforms of δ^o(Â)_V grow as V shrinks (3.3)–(3.4), and an arrow into a constant presheaf would need equality where there is strict growth.
The representation (Thm 3.1). For each Â, λ ↦ (V′ ↦ λ(δ^o(Â)_{V′})) is a natural transformation δ̆^o(A) : Σ → ℝ^⪰, where ℝ^⪰_V is the set of order-reversing functions on the subcontexts of V. The check is one line (3.15). With inner daseinisation, δ̆(A) : Σ → ℝ^↔ pairs an order-preserving and an order-reversing function. The map  ↦ δ̆^o(A) is injective (Appendix), resting on de Groote’s bijection between operators and observable functions.
The spread (3.47)–(3.52). At a point λ of Σ_V the value is the interval [λ(δ^i(Â){V′}), λ(δ^o(Â){V′})] over subcontexts V′, widening as V′ shrinks. It is state-independent, as a classical function on phase space is. LIT-378 §8.5 adds the value in a pseudo-state: if ψ is an eigenvector of  and  ∈ V, both components equal the eigenvalue at V.
Algebra (§3.5, §4; LIT-378 §9). ℝ^⪰ and ℝ^↔ are commutative monoid objects. δ^o(Â) + δ^o(B̂) ≠ δ^o(Â + B̂) in general (fn. 19). The group completion k(Γℝ^⪰) is the functions of bounded variation on the context poset (one direction written; NOTE-290 checked it). Squares exist only for classes [λ, 0] (4.19), which makes an “intrinsic dispersion” definable and nothing more.
What could have come out otherwise. Daseinised operators could have been constant along restrictions, making the reals adequate, or the monotone-function arrows could have failed to be natural. Neither held.
What this does not say
- Which value object is right. ℝ^⪰, ℝ^⪯, ℝ^↔ and their group completions are all offered, with no criterion; LIT-378 prefers ℝ^↔ “from a physical perspective” and then reverts to ℝ^⪰ (NOTE-326).
- That the spread brackets ⟨ψ|Â|ψ⟩ as least and greatest measurement results. That reading is cited from Döring’s antonymous-functions paper, not proved, and the authors disown its instrumentalism (fn. 14; NOTE-290, C6).
- That quantities can be added or squared as in quantum theory. Values form a monoid; differences need completion; general squares fail (fn. 23).
- Anything about unbounded operators. Position and momentum are outside the construction as given.
- That every step is proved. Thm 3.2, Thm 5.1 and the group-object structure of the completion are asserted.
Connections
- The points of Σ_V are exactly the ultrafilters of V’s Boolean projection lattice (3.28), the operator form of the Stone fact in THEORY-020; Σ has no global points by Kochen–Specker, for dim H > 2, a condition paper III drops.
- Propositions in this programme, and their Heyting logic, are THEORY-037. Composing these representations across systems is THEORY-585.
- Isham & Butterfield’s doubt that quantities should take real values (LIT-389, §1.3) is the motivation this construction makes concrete.
- Unitaries act covariantly by permuting contexts (§5), which fits THEORY-017’s claim that a context or basis is extra data.