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LIT-846: Categories of Empirical Models

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Properties7
StatusActive — read in full 2026-10-09 (NOTE-651); worth reading as the paper that first gives empirical models on different measurement scenarios a notion of morphism: a simulation d → e answers each measurement of e by a set of jointly measurable measurements of d (a simplicial relation, so the cover may change) and a stochastic map of outcomes natural in the context. With it, non-contextual models are exactly those simulable from the terminal (empty) model, the non-contextual fraction can only grow along a simulation, strong contextuality is reflected backwards, Graham reductions are simulations (Vorob'ev's theorem), and no contextual model can be cloned. It is restricted to no-signalling models and has no preprocessing.
Tagscontextuality, mathematics, quantum-foundations
ExtendsLIT-016 — The sheaf-theoretic structure of non-locality and contextuality, LIT-265 — The contextual fraction as a measure of contextuality
Extended byLIT-845 — Closing Bell: Boxing black box simulations in the resource theory of contextuality, LIT-847 — A comonadic view of simulation and quantum resources
SupportsCLAIM-070 — Transport induced by one global stochastic kernel carries a global extension of the source model to a global extension of the target model, CLAIM-121 — A transport whose local kernels commute with restriction maps overlap-consistent source models to overlap-consistent target models, CLAIM-125 — Directed transport between empirical models whose covers change has two open problems: when it preserves decision-relevant information, and how it extends to signalling data
Cited as “paper” byNOTE-651 — Categories of Empirical Models
Cited as “source” byTHEORY-174 — Classical simulations between no-signalling empirical models on different scenarios never create contextuality, and a convex map between two scenarios' model sets is induced by a classical non-adaptive procedure exactly when some non-contextual model of the hom scenario that satisfies its predicate induces it, THEORY-720 — Classical simulations between no-signalling empirical models on finite scenarios, which may change the cover, never create contextuality: a model is non-contextual exactly when the empty model simulates it, the non-contextual fraction cannot fall along a simulation, strong contextuality is reflected backwards, and only non-contextual models can be cloned

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Martti Karvonen (2018), in Proceedings of the 15th International Conference on Quantum Physics and Logic (QPL 2018), EPTCS 287 (2019), pp. 239–252, DOI 10.4204/EPTCS.287.14 — ARXIV-1804.01514

Key takeaways

  • Morphisms that change the cover. A morphism from scenario ⟨Y, N, P⟩ to ⟨X, M, O⟩ is a simplicial relation π : X → Y (each measurement x is answered by a set π(x) of source measurements, and each target context goes to a jointly measurable source set) with a natural transformation σ : E_Y(π(−)) → D_R ∘ E_X(−) (Definition 3.9). A simulation d → e is such a morphism whose pushforward of d is e. The covers of the two scenarios need not match; the relation, not a function, is what lets a target measurement depend on several source measurements (Example 3.7, parity).
  • Contextuality as non-simulability. Distributions on global sections explaining e correspond one to one with simulations from the terminal model 1 → e (Theorem 4.1), so e is non-contextual iff such a simulation exists. A semifield homomorphism gives a functor, so logical contextuality implies probabilistic (Theorem 4.2).
  • What simulation preserves. If d → e, then NCF(d) ≤ NCF(e) (Theorem 4.5: the non-contextual fraction is a functor to [0, 1]), and if e is strongly contextual so is d (Theorem 4.3). A simulation can remove contextuality but never create it.
  • Two further results. A Graham reduction of the cover induces a simulation e|X{x} → e (Theorem 4.7), so models on acyclic covers are non-contextual, which recasts Vorob’ev’s theorem. With the parallel product as a monoidal structure, e → e ⊗ e exists iff e is non-contextual (Theorem 4.8, no-cloning).
  • Limits stated by the author. The definition needs no-signalling for the pushforward to be well defined (Remark 3.2). There is no preprocessing: which source measurements are called cannot depend on randomness or on earlier outcomes (§5). Stochastic morphisms do not glue along arbitrary covers, only along partitions (Lemma 3.12), because D ∘ E is not a sheaf.

Standing in the record

Filed on 2026-10-09 at the owner’s direct request, as the first of the works that set up simulations between empirical models. It is not from the manuscript bibliography: the owner asked for it directly on 2026-10-09. It is read with its two successors, the comonadic generalisation (LIT-847) and the characterisation of free transformations (LIT-845). Together they are the source of THEORY-174. NOTE-651 says how the reading bears on the manuscript’s transport problem on line pragmatic-transport.

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