Contribution
It extends the Abramsky–Brandenburger sheaf framework from no-signalling scenarios to arbitrary causal structure: definite, indefinite, and dynamical, where the order depends on inputs. The causal structure is a space of input histories (LIT-800). Causality is shown to be continuity, causal functions form a presheaf, and empirical models are compatible families of distributions on arbitrary open covers. Two new facts follow. Causal constraints can themselves make deterministic local data unglueable, so the presheaf of causal functions is not always a sheaf. And on causal switch spaces, total orders included, no standard empirical model is non-local.
Key insight
Make the space of input histories a topological space with the lowerset topology, and a causal function is just a continuous one. All of Abramsky–Brandenburger’s machinery (contexts as opens, empirical models as compatible families, contextuality as failure of a global section) then applies with “no-signalling” replaced by “this causal structure”. The new twist is that different contexts can now carry different causal structures. Then even deterministic assignments that are causal in every context can fail to be causal together.
Assumptions
- Finite events, inputs and outputs. A space of input histories as in LIT-800. Most results do not need causal completeness or tightness, but some do, and the statements say which.
- Distributions are finitely supported and valued in ℝ⁺. The definitions extend to commutative semirings (Remark 2.8), but only the probabilistic case is developed.
- A cover is a maximal antichain of opens covering the space (Definition 2.29). This is narrower than an arbitrary open cover.
- Theorem 2.34’s sheaf condition needs at least two outputs at every event.
Key results
- Proposition 2.1. On order-induced spaces, causal joint IO functions are exactly F(k)ω = G_ω(k|{ω↓}).
- Theorems 2.3 and 2.14, Propositions 2.4 and 2.16. Causal functions correspond one-to-one to free maps from histories to outputs at their tips, modulo tip-equivalence (∼_ω) in non-tight spaces. They also correspond to extended functions satisfying the consistency (equivalently gluing) condition, and to causal joint IO functions.
- Theorem 2.9. A causal function is inseparable (arises from no causally complete refinement) iff it has an inseparability witness. Example: on total(A, {B, C}) with binary inputs, 50,176 of 262,144 causal functions are separable.
- Theorems 2.17–2.19. CausFun factors over parallel composition (a product), sequential composition (a product with a family indexed by the first space’s maximal histories) and conditional sequential composition.
- Theorem 2.26. An extended function is causal iff it is continuous Ext(Θ) → PFun(O) for the lowerset topologies. Proposition 2.22: for partial orders, continuity in the lowerset topology is monotonicity.
- Theorem 2.30. CausFun(Λ(Θ), O) is a separated presheaf. A compatible family has a gluing iff its compatible join is causal.
- Theorem 2.34. With |O_ω| ≥ 2 for all ω, the presheaf is a sheaf iff Θ admits no solipsistic contextuality, and in particular whenever Θ is tight. Proposition 2.33 characterises solipsistic contextuality as a pair of tip-equivalent histories whose minimal common extensions are not all histories.
- Proposition 2.42. Covers form a lattice, with SolCov ⪯ StdCov ⪯ ClsCov.
- Theorem 2.48, Corollaries 2.47 and 2.50. A solipsistic contextuality witness yields a deterministic solipsistic empirical model that does not extend to a standard one. If the presheaf is a sheaf, every deterministic model is globally deterministic. Deterministic standard models are always globally deterministic.
- Theorem 2.51, Corollary 2.52. Every standard empirical model on a non-empty causal switch space is local.
- Proposition 2.54. On spaces built by composition from indiscrete ones, a local model that is a mixture of models each lifting to a causal completion is local in a causally separable way. The paper shows the converse fails.
- Examples. Leggett–Garg: local on total(A, B, C), with an explicit 12-function decomposition. The model also lifts to a causally complete refinement that encodes “no measurement on input 0”, but it is not an empirical model for two of the three indefinite orders that the paper reads macro-realism as imposing. BFW: a 50–50 mixture of two inseparable causal functions (circular identity and circular bit-flip) on the indiscrete space.
Claims
| id | claim | strength | support |
|---|---|---|---|
| C1 | Causality for extended functions is continuity in the lowerset topology | strong (proof, not checked) | Theorem 2.26 |
| C2 | Causal functions form a separated presheaf that is a sheaf iff the space has no solipsistic contextuality; tight spaces always give a sheaf | strong (proof, not checked) | Theorems 2.30, 2.34; worked non-gluing example on Θ₃ followed |
| C3 | Most causally complete 3-event binary spaces (1767 of 2644) admit solipsistic contextuality | moderate (computation, stated in the introduction) | not reproduced; no table of the 1767 in the main text |
| C4 | Standard empirical models on causal switch spaces, total orders included, are always local | strong (proof, not checked) | Theorem 2.51 |
| C5 | The Leggett–Garg violation is a failure of no-signalling constraints, not contextuality | moderate | Theorem 2.51 plus the paper’s reading of macro-realism as causal constraints (2.6.4) |
| C6 | Local and causally separable does not imply local in a causally separable way | conjecture, here only in one direction | Proposition 2.54; Conjecture 2.30 of LIT-788 |
Method
Order theory and sheaf theory. Spaces are given the lowerset topology, causal data are sections of presheaves over it, and probabilistic data come from the distribution monad applied to those presheaves. Contextuality is then the failure of a compatible family to extend to a coarser cover. Factorisation results are proved by composing spaces, and the examples compute decompositions into causal functions by hand or linear programming.
Concepts
- causal function: a map from input histories to outputs at their tips, with tip-equivalent histories forced to agree.
- extended causal function: its gluing over extended histories, which assigns outputs to every event in an extended history’s domain.
- lowerset topology: the opens of a poset are its lowersets. It is the dual of the Alexandrov topology.
- solipsistic contextuality: the existence of contexts whose causal structures are incompatible, so that locally causal deterministic data cannot be glued. Witnessed by a quadruple (k, ω, h, h′).
- covers: maximal antichains of opens. Standard: the downsets of maximal extended histories. Fully solipsistic: the downsets of maximal histories. Classical: the whole space.
- empirical model: a compatible family of distributions over extended causal functions on a cover.
- local / non-contextual: the restriction of a classical empirical model.
- separable causal function: one arising from a causal function on a causally complete refinement.
Connections
It extends Abramsky and Brandenburger (LIT-016), to which it reduces on the discrete space, where causal functions are the “sheaf of events”. It adopts the contextual fraction and related machinery from that line. It recovers, in this framework, the known result that processes not violating causal inequalities are local when mixtures of fixed or dynamical total orders are allowed. It reinterprets Leggett and Garg’s macro-realism. It sets its framework apart from Spekkens’s operational contextuality and from ontological treatments of contextuality in time and indefinite causality. It redoes, over a new base, the construction of the authors’ withdrawn LIT-813, which it does not cite.
Bearing on the record
- LIT-813. This paper carries the withdrawn paper’s content. Its Proposition 21 (local iff a mixture of deterministic causal functions) reappears as the definition of noncontextual via the classical cover. Its sheaf of causal functions over lower sets becomes a presheaf over the lowerset topology of a space of input histories. That base is a genuine topology and so a locale, which repairs the defect the earlier NOTE found. One claim changes: on definite orders the earlier paper’s sheaf was asserted, while here sheafhood is a theorem for tight spaces only. Order-induced spaces are tight (LIT-800, Proposition 3.32), so the definite case survives.
- THEORY-012 (noncontextual iff the context distributions glue) holds here by definition, generalised to causal covers. This paper adds a second, purely causal obstruction to gluing that is present even for deterministic data. That is the content of THEORY-171, filed from this reading.
- THEORY-165 (the contextual fraction is Lipschitz within a scenario) is stated for fixed measurement scenarios. Here contextual fractions are defined relative to a cover of a space of input histories. The reading neither tests nor extends that THEORY.
- No instruction for machine-learning practice; nothing for the anthology.
Limitations
- Proofs were not read here, and the paper is an unrefereed preprint.
- The census figure for solipsistic contextuality (1767 of 2644) is stated in the introduction without a table in the main text.
- Covers are restricted to maximal antichains.
- The interpretation of the fully solipsistic cover is operationally thin: the paper describes it as settings where distributions exist only over an event’s past, “witnessed” by that event.
- Internal cross-references to the companion papers point to section numbers that do not exist in the current versions (e.g. “Subsubsection 4.5.5 of [6]” in LIT-788). They appear to come from the combined v1 of 2206.08911.
Open questions
- An operational experiment, or a quantum example, that displays solipsistic contextuality in data rather than in the presheaf.
- Whether “local and causally separable” implies “local in a causally separable way” (Conjecture 2.30 of LIT-788).
- The semiring generalisation (possibilistic, signed), which is stated but not developed.