Contribution
It is the first attempt to put definite causal order into the sheaf-theoretic framework of LIT-016. In LIT-016 every event is causally unrelated to every other. Here an output may depend on inputs in its causal past. Locality becomes the existence of a classical model built from deterministic causal functions, and the no-signalling condition becomes a family of causality equations, one per lower set. It reduces exactly to LIT-016’s construction on the discrete order. It also sketches pre-orders for indefinite causal order and a causal-fraction analysis of the Baumeler–Feix–Wolf model.
Key insight
Replace “measurements that can be performed together” by “inputs at events closed under the past”. On a lower set, a function from joint inputs to joint outputs that respects the causal order can be restricted to a smaller lower set, because nothing in the smaller set depends on what was cut away. So causal functions form a sheaf, and the whole Abramsky–Brandenburger apparatus (distributions, compatible families, global sections) can be reused with causality in the sections rather than in a no-signalling condition.
Assumptions
- Ω a finite poset; finite, nonempty input and output sets at each event.
- Non-determinism valued in a commutative semiring R: R₊ for probabilities, ℝ for signed, 𝔹 for possibilistic.
- An empirical model is a compatible family over the cover induced by all joint inputs I_Σ (Definition 11). Every event’s input is chosen in every run. There is no notion of measuring only some events.
Key results
- Definition 3, Proposition 5 (the locale of inputs). L_Σ is the disjoint union over lower sets λ of the products over ω ∈ λ of nonempty subsets of I_ω, ordered componentwise. Its meet is given as (U_ω ∩ V_ω) on λ_{U∩V} = {ω ∈ λ_U ∩ λ_V : U_ω ∩ V_ω ≠ ∅}. This fails. Take Ω = {C < A} and I_C = I_A = {0, 1}. Let U = ({0}, {0}) and V = ({1}, {0}), both on {C, A}. Then λ_{U∩V} = {A}, which is not a lower set, so U ∩ V is not in L_Σ. The true greatest lower bound is the bottom ∅. With a = U, b = ({0}) on {C} and c = V: b ∨ c = ({0, 1}, {0}), so a ∧ (b ∨ c) = a, while (a ∧ b) ∨ (a ∧ c) = ({0}) on {C} ≠ a. Distributivity fails, so L_Σ is not a locale. This is the reader’s check, and it agrees with the authors’ withdrawal of Definition 3.
- Definition 6. f on a lower set is causal if f(i)_ω depends only on i restricted to ω’s down-set.
- Proposition 8. Restriction of causal sections is well defined, and E_Σ is a sheaf. The gluing argument assumes U ∩ V as in Proposition 5, so it inherits that defect.
- Proposition 9. For a discrete Ω, L_Σ ≅ P(X) and E_Σ is the original sheaf of sections. Here every subset of Ω is a lower set, and the defect above cannot arise.
- Proposition 15. Empirical models over the joint-input cover are exactly causal conditional distributions e(o | i). Causality means that the marginal on each lower set λ is independent of inputs outside λ.
- Proposition 16. Probabilistic empirical models form a polytope: the product of |I_Σ| simplices intersected with the linear causality equations (35). Remark 17 notes that they are redundant.
- Proposition 21. e is local (has a global section) if and only if ê = Σ_ξ p(ξ) δ_{f(ξ)} for causal functions f(ξ) and p ∈ D_R. This is the causal analogue of LIT-016’s Theorem 8.1. The proof here is short because local sections are already functions on all joint inputs.
- Proposition 22. Diagrams over a framed multigraph in an R-probabilistic theory (the authors’ earlier formalism) with normalised processes yield R-valued empirical models. The proof discards maximal events one at a time.
- Section 9 (stated, proofs deferred). For the Baumeler–Feix–Wolf model, the Ω-causal fraction (the largest λ with λ·e ≤ e_BFW for an Ω-causal e) is 0 for every partial order on three events and 1 for four pre-orders. Fixing the first party’s input leaves a no-signalling model for the other two.
Claims
| id | claim | strength | support |
|---|---|---|---|
| C1 | On a discrete order the construction is exactly Abramsky–Brandenburger’s | strong (proof; unaffected by the defect) | Proposition 9 |
| C2 | Empirical models are exactly causal conditional distributions, and they form a polytope | strong (proof) | Propositions 15–16 |
| C3 | Locality is the same as a mixture of deterministic causal functions | strong (proof, given the sheaf of sections) | Proposition 21 |
| C4 | L_Σ is a locale and E_Σ a sheaf over it | fails as stated | Proposition 5; counterexample above; authors’ withdrawal |
| C5 | Physical diagrams respecting the causal order yield empirical models | moderate (proof sketch in another formalism) | Proposition 22 |
| C6 | The BFW model has causal fraction 0 for every definite order and 1 for exactly four pre-orders | not supported here (proof deferred) | Section 9 |
Concepts
- definite causal scenario. (Ω, I, O) with Ω a finite poset.
- lower set. A set of events closed under the past. Λ(Ω) is the set of all of them.
- locale of inputs L_Σ. Families of nonempty input sets indexed by lower sets (Definition 3). Defective, as above.
- causal function. Its output at ω depends only on inputs at events ≤ ω.
- local / non-local. A global section of the presheaf of R-distributions exists or does not. Here this means “explained by classical causal functions”, which generalizes Bell locality.
- Ω-causal fraction. The largest weight of an Ω-causal sub-model, by analogy with LIT-265’s non-contextual fraction.
Connections
It builds on LIT-016 and cites LIT-277 (cohomology) and LIT-265 (contextual fraction) in its literature review. It also builds on the authors’ process-theoretic causal frameworks (Pinzani & Gogioso 2020; Gogioso & Scandolo 2018). It contrasts itself with Mansfield’s 2017 presentations, which impose a causal order on the powerset of measurements and, the authors argue, cannot handle Bell scenarios or classify causal classical functions as local. Fritz’s correlation scenarios and Henson, Lal and Pusey’s generalized Bayesian networks are named as the other traditions. Contextuality-by-Default (Dzhafarov, Kujala and Cervantes) is cited once, for the idea that context changes “the very conditions” of prediction.
Bearing on the record
- LIT-016 and THEORY-012. Proposition 21 extends THEORY-012’s criterion: noncontextual (local) exactly when there is a global section, equivalently a classical model, with “classical” now meaning causal functions on a causal order. Within this paper the extension rests on a construction the authors withdrew. On the discrete order it is LIT-016’s result unchanged. No edit to THEORY-012 follows.
- THEORY-014. The R-valued generality (ℝ for signed distributions) is set up but not used. The paper proves no “signed global sections always exist” result for causal scenarios, so it adds no instance to THEORY-014.
- LIT-265 and THEORY-165. The Ω-causal fraction of Section 9 is the contextual fraction’s construction with causality in place of locality. Its properties are not proved here.
- The manuscript. The reading supplies no premise for the claims
around
what-survives-translation. Its relevance would be the idea of restricting the admissible covers by an order, and that is the withdrawn part. - No new THEORY. No instruction for machine-learning practice.
Limitations
- The central construction (Definition 3, Proposition 5) is defective, as the authors acknowledge. Propositions 8 and 21 inherit it outside the discrete case.
- Empirical models are defined only over the cover of all joint inputs. Scenarios in which some events are not run are not represented.
- The indefinite-causality section states results without proof.
- No example is analysed for locality. The quantum diamond model is constructed but not tested.
Open questions
- Whether the successor’s “spaces of input histories” (arXiv 2206.08911) repair Propositions 5 and 8 while keeping Propositions 15–21. Settling that means filing and reading the successor.
- The status of the BFW causal-fraction results, proved only in later work.