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LIT-808: The Topology of Causality

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Properties5
StatusActive — read 2026-10-09 (NOTE-612); worth reading as the direct successor of the withdrawn LIT-813's sheaf construction, now over spaces of input histories (LIT-800) rather than a "locale of inputs": a joint input–output function is causal when each event's output depends only on the histories with that event as a tip, causal functions are exactly free assignments of outputs to tips (up to tip-equivalence in non-tight spaces), and an extended function is causal iff it is continuous for the lowerset topology. Causal functions form a separated presheaf that is a sheaf exactly when the space admits no "solipsistic contextuality", so tight spaces give a sheaf and some dynamical causal structures do not, letting deterministic local data fail to glue. Empirical models are compatible families of distributions over causal functions on any open cover, from the fully solipsistic to the classical; standard empirical models on causal switch spaces, total orders included, are always local. Source of THEORY-171.
Tagscausality, contextuality, quantum-foundations, mathematics
SupersedesLIT-813 — The Sheaf-Theoretic Structure of Definite Causality
Cited as “paper” byNOTE-612 — The Topology of Causality
Cited as “source” byTHEORY-171 — When the causal constraints on events depend on context, deterministic assignments that are causal in each context can fail to glue into one causal assignment, so causal structure is itself a source of contextuality

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Stefano Gogioso and Nicola Pinzani (2023), arXiv 2303.07148, Part 2 of the trilogy after LIT-800 and before LIT-788

Key takeaways

  • Causal functions. A joint IO function is causal for a space Θ if, for every history h ∈ Θ, the outputs at h’s tips are the same for all joint inputs extending h (Definition 2.2). For order-induced spaces this is the familiar F(k)ω = G_ω(k|{ω↓}) (Proposition 2.1). For a switch space it is not a fixed dependency pattern. Causal functions are equivalently free maps from histories to outputs at their tips, with outputs at an event identified across tip-equivalent histories when the space is not tight (Definitions 2.3, 2.7, 2.13; Theorems 2.3, 2.14; Propositions 2.4, 2.16).
  • Inseparable functions. On causally incomplete spaces some causal functions arise from no causal completion: of 262,144 on total(A, {B, C}) with binary inputs, 211,968 are inseparable, such as a controlled swap. These are characterised by an inseparability witness (Theorem 2.9).
  • Factorisation. Causal functions on parallel, sequential and conditional sequential compositions factor as products (Theorems 2.17–2.19).
  • Topology. With the lowerset topology on Ext(Θ) and on partial output functions, an extended function is causal iff it is continuous (Theorem 2.26). The opens of a space are themselves spaces, and causal functions restrict to them (Propositions 2.27–2.28, Corollary 2.29).
  • Sheaf or not. Causal functions form a separated presheaf (Theorem 2.30). It is a sheaf iff Θ admits no solipsistic contextuality, so in particular when Θ is tight (Theorem 2.34). On the non-tight space Θ₃ there are two contexts with incompatible causal structure (A before B, and C before B), whose deterministic causal sections cannot be glued. The introduction says 1767 of the 2644 complete 3-event spaces exhibit this.
  • Empirical models and covers. Distributions over extended causal functions form a presheaf (Definition 2.27). An empirical model is a compatible family on an open cover (Definition 2.31). Covers form a lattice from the fully solipsistic cover through the standard cover to the classical one (Proposition 2.42). Contextual means not the restriction of a classical model. Contextual and local fractions are defined as in Abramsky–Brandenburger (Definitions 2.32–2.34).
  • No non-locality on switch spaces. Every standard empirical model on a causal switch space is local (Theorem 2.51). This includes every total order, and spaces built by conditional sequential composition of indiscrete spaces (Corollary 2.52). Other covers can still be contextual.
  • Examples. The Leggett–Garg model is local for total(A, B, C), a mixture of 12 causal functions. The paper reads Leggett and Garg’s macro-realist conditions as causal constraints that the model violates, so it ascribes the violation to signalling, not to contextuality. The BFW model is a 50–50 mixture of two inseparable functions on the indiscrete space.

Standing in the record

Filed on 2026-10-09 from the manuscript bibliography of 2026-10-09 (work what-survives-translation) at the owner’s request, as a successor of Gogioso and Pinzani 2021 (LIT-813). The authors withdrew that paper and named this one among those superseding it. Of the three successors, this one carries the withdrawn paper’s subject: its title, its sheaf of causal functions, locality as a global section, and empirical models as compatible families. LIT-813 is now Superseded by the trilogy. See the curation entry of that day.

Read on 2026-10-09 (NOTE-612). The reading is the source of THEORY-171: when causal constraints depend on context, deterministic locally causal data need not glue. It extends LIT-016’s framework, to which it reduces on the discrete space, and THEORY-012’s equation of noncontextuality with gluing applies within it unchanged.

The vocabulary has no topic for causal structure. The paper is tagged contextuality first because its results are about the sheaf-theoretic contextuality framework.

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