nucleation

LIT-310: A topos foundation for theories of physics: IV. Categories of systems

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Properties5
StatusProposed — read in full 2026-09-29 (NOTE-285); promising and unproven. The rules are explicitly "experimental", the quantum composite fails the commutativity condition the rules state, and the key negative claims ((4.46), and that no operator B̂ gives the pulled-back arrow) are asserted, not shown. What would settle it is a proof of (4.46) and of the non-existence of B̂, plus a variant of the rules (or of the context category) under which tensor-product composites satisfy (3.18).
Version2
Tagsquantum-foundations, mereology, mathematics, logic, philosophy-of-science
Cited as “paper” byNOTE-285 — A topos foundation for theories of physics: IV. Categories of systems
Cited as “source” byTHEORY-585 — In the Döring–Isham scheme for realising collections of systems in topoi, disjoint sums of quantum systems compose exactly; along a tensor product of finite-dimensional systems, the pull-back of a constituent's quantity agrees with the composite's own quantity at a context exactly when every projection in it with proper support on the first factor has the form P⊗1, so it fails for every non-scalar quantity and no operator on the composite realises the pull-back

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Andreas Döring, Chris Isham (2008), Journal of Mathematical Physics 49(5):053518 — DOI-10.1063/1.2883826

Standing in the record

Registered on 2026-09-29 at the owner’s request, because the owner’s prior-art novelty map (a private working document on concepts as projectors, the linear representation hypothesis and their neighbours) cites it: §0 and row 8: the Döring–Isham topos program (spectral presheaf) that the map says owns §2.1 and should be instantiated as the “Topos of Bricks”. It is seeded from metadata and published descriptions and has not been read here.

Identification notes: Part of the four-paper 2008 series the map cites as one reference; all four registered. arXiv v1 2007-03-07; journal online 2008-05-13.

It was filed Deferred, unread. NOTE-285 is the close reading of 2026-09-29, and it placed the work: Proposed — promising and unproven. The rules are explicitly “experimental”, the quantum composite fails the commutativity condition the rules state, and the key negative claims ((4.46), and that no operator B̂ gives the pulled-back arrow) are asserted, not shown. What would settle it is a proof of (4.46) and of the non-existence of B̂, plus a variant of the rules (or of the context category) under which tensor-product composites satisfy (3.18).

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