Source
Döring & Isham (2007; J. Math. Phys. 49, 2008), “A topos foundation for theories of physics: IV. Categories of systems”, LIT-310, read in full in NOTE-285. The same construction, restated, is LIT-378 §§11–13, read in full in NOTE-326.
What was actually shown
The scheme. Systems form a category Sys whose arrows translate one system’s physical quantities into another’s, with composite ⋄ and disjoint sum ⊔ as monoidal structures. Each arrow j is to be realised by a triple (τ(j), φ(j), β(j)) that pulls a quantity arrow of S back to one of S₁ (Def 3.2, eq. 3.5). The target is the translation condition (3.18): the pull-back of A’s arrow is the arrow of A’s translation. The rules are “partly ‘experimental’”.
Classical physics (§2.4). Pull-back along the inclusion into a disjoint union, or the projection from a product, satisfies the condition exactly (2.33).
Quantum disjoint sums (§4.2). The functor V ↦ V ⊕ ℂ1̂_{H₂} induces the pull-back. Lemma 4.2, δ(Â₁⊕Â₂){V₁⊕V₂} = δ(Â₁){V₁} ⊕ δ(Â₂)_{V₂}, gives the pull-back of δ̆(⟨A₁, A₂⟩) as δ̆(A₁) (4.24). NOTE-285 checked it.
Quantum composites (§4.3). The functor W ↦ V_W, the largest V with V⊗ℂ1̂ ⊆ W, induces the pull-back, into an augmented context category that re-admits ℂ1̂. At each context W the pulled-back arrow is the Gel’fand transform of δ(Â₁)_{V_W} ⊗ 1̂ (4.44–4.45). It agrees with δ̆(A₁⋄1) at every W of the form V₁⊗ℂ1̂ (fn. 27). NOTE-285 checked the construction.
The asserted failure. “It is relatively easy to show” that δ(Â₁⊗1̂)W ≠ δ(Â₁){V_W}⊗1̂ in general (4.46), so (3.18) fails (4.47), and “there appears to be no operator B̂” whose arrow equals the pull-back. The authors attribute this to “operator entanglement”: contexts of V(H₁⊗H₂) not of the form V₁⊗V₂. Their summary is that quantum physics “almost” satisfies the scheme (§5).
What could have come out otherwise. The pull-back could have agreed with the composite’s quantity on every context, as it does for disjoint sums. On the contexts the factorisation picks out it does; elsewhere the authors say it does not, without proof.
What this does not say
- That the source proves the failure. Neither (4.46) nor the non-existence of B̂ is proved in the paper, and fn. 25 leaves open the case W = V₁⊗V₂ with V₂ non-trivial. Both are proved in the dated note below, for finite dimensions, and that case fails.
- That the failure follows the factorisation. It does not: a context whose projections all have full support on H₁, such as one spanned by maximally entangled projections, agrees for every quantity, while V₁⊗V₂ with both factors non-trivial does not.
- That it concerns entanglement of states, Bell non-locality or contextuality. The “entanglement” is in the context poset of H₁⊗H₂. The paper treats no states on composites and nothing like a partial trace.
- That the rules are settled. Several are conjectures (Rule 6), and the rule that the trivial system’s topos is Sets contradicts the series’ exclusion of ℂ1̂ from the contexts, under which the trivial system’s topos is degenerate (NOTE-285’s observation, repeated in NOTE-326).
- That there is a single topos for all systems. The “gros topos” is posed in §5, not built.
Connections
- Presupposes THEORY-553: the arrows being translated are the quantity representations into ℝ^⪰ of paper III.
- NOTE-285 relates the result to THEORY-017: constituent quantities translate faithfully only on the sub-poset of contexts a given tensor factorisation picks out.
- Mereology in the record does not engage physical composition in these terms; the tag is for the subject, what counts as a system and its parts.
Note of 2026-10-19: the composite’s failure, proved
The promotion audit (batch 044) proved what the source asserts, for finite-dimensional H₁ and H₂ with dim H₂ ≥ 2. δ(Â)_V is outer daseinisation, ∫λ d(δ^i(Ê_λ)_V), where δ^i(P)V is the largest projection of V below P. The pulled-back arrow at a context W is δ(Â₁){V_W}⊗1̂, with V_W = W ∩ (B(H₁)⊗1̂), which may be ℂ1̂ in the augmented category. Two arrows into ℝ^⪰ agree at W exactly when their operators agree at every sub-context of W, since restriction of states is onto and the Gel’fand transform is injective. Call the support of a projection R on H₁ the range of its partial trace, and call W good if every projection of W with proper support has the form e⊗1̂.
- Agreement exactly on good contexts. If W is good, so is every sub-context V′. For λ below the top of Â₁’s spectrum, a projection of V′ under Ê_λ⊗1̂ has support under Ê_λ, so it is e⊗1̂ with e ∈ V_{V′}. Hence δ^i(Ê_λ⊗1̂){V′} = δ^i(Ê_λ){V_{V′}}⊗1̂ for every λ, and the operators agree. If W is not good, take an offending R with support P and V′ = {0, R, 1̂ − R, 1̂}. Then V_{V′} = ℂ1̂, so for Â₁ = 1̂ − P the pull-back is 1̂, while δ((1̂ − P)⊗1̂)_{V′} has eigenvalue 0 on R ≤ P⊗1̂.
- Every non-scalar quantity fails. With u an eigenvector for Â₁’s least eigenvalue and q any unit vector of H₂, R = |u⟩⟨u|⊗|q⟩⟨q| has proper support and is not e⊗1̂. At V′ as above the pull-back is λ_max·1̂, while δ(Â₁⊗1̂)_{V′} takes the value λ_min on R. This is (4.46).
- No operator realises the pull-back. If δ(B̂)W = δ(Â₁){V_W}⊗1̂ for every W, then at a maximal W containing B̂, B̂ = δ(B̂)_W = b⊗1̂. At W = V⊗V₂ with V maximal containing b, V_W = V gives b = δ(Â₁)_V, so b dominates Â₁ in spectral order. At W = V⊗V₂ with V maximal containing Â₁, Â₁⊗1̂ = δ(b⊗1̂)_W dominates b⊗1̂. So b = Â₁, and B̂ = Â₁⊗1̂, which the previous point rules out.
- Disjoint sums. Projections of V₁⊕V₂ are e₁⊕e₂, so the largest one under E₁⊕E₂ is the pair of largest ones: Lemma 4.2, and the exact composition for disjoint sums.
V₁⊗V₂ with both factors non-trivial is not good, since p⊗q has proper support; ℂ1̂⊗V₂ and a context of maximally entangled projections are. Infinite dimensions are not treated.