The Topology of Causality

TL;DR

This paper develops a comprehensive, theory-independent framework for understanding causality, non-locality, and contextuality using sheaf theory, extending previous models to include arbitrary causal orders and revealing new phenomena like causally-induced contextuality.

Contribution

It introduces a unified operational framework for causality that encompasses arbitrary causal orders and extends sheaf-theoretic approaches to include dynamical and indefinite causal structures.

Findings

Causality is equivalent to continuity in the lowerset topology.

Causal functions can be organized into a presheaf, but may not always form a sheaf.

Causally-induced contextuality can arise when causal constraints are context-dependent.

Abstract

We provide a unified operational framework for the study of causality, non-locality and contextuality, in a fully device-independent and theory-independent setting. Our work has its roots in the sheaf-theoretic framework for contextuality by Abramsky and Brandenburger, which it extends to include arbitrary causal orders (be they definite, dynamical or indefinite). We define a notion of causal function for arbitrary spaces of input histories, and we show that the explicit imposition of causal constraints on joint outputs is equivalent to the free assignment of local outputs to the tip events of input histories. We prove factorisation results for causal functions over parallel, sequential, and conditional sequential compositions of the underlying spaces. We prove that causality is equivalent to continuity with respect to the lowerset topology on the underlying spaces, and we show that partial causal functions defined on open sub-spaces can be bundled into a presheaf. In a striking departure from the Abramsky-Brandenburger setting, however, we show that causal functions fail, under certain circumstances, to form a sheaf. We define empirical models as compatible families in the presheaf of probability distributions on causal functions, for arbitrary open covers of the underlying space of input histories. We show the existence of causally-induced contextuality, a phenomenon arising when the causal constraints themselves become context-dependent, and we prove a no-go result for non-locality on total orders, both static and dynamical.