Source
Butterfield & Isham (2001; IJTP 41, 2002), LIT-382, read in full in NOTE-328.
What was actually shown
Setting. The base category V is the poset of commutative von Neumann subalgebras of B(H). The coarse-graining presheaf G assigns V its projection lattice L(V), and sends P̂ to the least projector of a smaller V₂ above it. For a projector, V(P̂) := {κ ∈ σ(V) | κ(P̂) = 1} is a clopen subset of the Gelfand spectrum.
G ≅ CloΣ (§2.4, eqs. 2.7–2.11). For V₂ ⊆ V₁ and P̂ ∈ L(V₁), {κ|_{V₂} | κ ∈ σ(V₁), κ(P̂) = 1} = {λ ∈ σ(V₂) | λ(G(P̂)) = 1}. One inclusion is easy. For the other, a character λ of V₂ with λ(G(P̂)) = 1 extends to a character κ of V₁ with κ(P̂) = 1, by Kadison–Ringrose Thm 4.3.13 (credited to Halvorson). The step from a state extension to a character extension is left implicit; NOTE-328 calls it standard and repairable. So P̂ ↦ V(P̂) is a natural isomorphism between G and the presheaf of clopen subsets with direct image as its arrow map.
Supports determine the semantics (Thm 3.1). Let supports s(α, V) = inf T^α(V) form a global element of G, and let i_{V₂V₁} ∈ α_{V₁}(P̂) iff s(α, V₂) ≤ G(P̂). Then α is sieve-valued, obeys FUNC, and α_{V₁}(P̂) = {V₂ | G(s(α, V₁)) ≤ G(P̂)} (3.3). The proof uses only monotonicity and functoriality of G. A state’s valuation ν^ρ meets the conditions: its support at V is the least projector of V that ρ makes certain. Theorem 3.2 gives the same with intervals of the spectrum in place of supports, when they form a “tight” subobject of Σ.
Choice of relation (§4). Defining a valuation by “a(V₂) R G(P̂)” for a global element a of G, FUNC holds for every R, and R = ≤ gives sievehood, null, monotonicity and exclusivity, provided a never assigns 0̂. NOTE-328 checked each item.
What could have come out otherwise. Restriction of a clopen region could have landed strictly inside the region of the coarse-grained projector, which would have made coarse-graining lossy on the spectral side. The extension theorem rules it out.
What this does not say
- That the theorems are deep. Condition (i) of Theorem 3.1 is essentially its conclusion once (ii) holds, and the paper calls the converses “trivially” true. The substance is G ≅ CloΣ, which sits in the “review” section and is not mentioned in the abstract.
- Anything about the Heyting operations on clopen sub-objects. No implication or negation is built, so this does not meet THEORY-037’s promotion condition. It does sharpen THEORY-037’s caveat: daseinised projectors are tight subobjects, global elements of CloΣ, and not every subobject of Σ is one.
- That probability-r valuations fail to form global elements. That is asserted without counterexample.
- Anything on the operator category beyond discrete spectra. The claim f(s(·, Â)) ⊆ s(·, f(Â)) “even if  has in part a continuous spectrum” (3.26) is undefined as written for continuous spectra, and NOTE-328 gives a counterexample under the closed-support reading.
- Anything about probabilities, dynamics or composites.
Connections
- Part III (Hamilton, Isham & Butterfield, LIT-380, Superseded) introduced the base category V and asserted G ≅ CloΣ; this paper proves it.
- THEORY-037 rests on daseinisation sending each projector to a clopen sub-object of Σ; Döring–Isham’s daseinisation is G here, and the isomorphism proved here is what makes the sub-object clopen.
- The support of a pure state ψ at V is the least projector of V above |ψ⟩⟨ψ|, the later “pseudo-state” of LIT-378 (NOTE-328’s gloss).
- Pairwise, between nested contexts, local two-valued valuations always extend; the Kochen–Specker obstruction appears only over the whole poset, which fits THEORY-020.
- The sieve semantics in general, and why it does not depend on global sections, is THEORY-516.