2022/11/01 by Aziz Kharoof, Kharoof, Aziz, Cihan Okay +1 · 1 citation
Computer Science · Mathematics · #Advanced Topology and Set Theory #Algebraic Topology (math.AT) #Category Theory (math.CT) #FOS: Mathematics #FOS: Physical sciences #Homotopy and Cohomology in Algebraic Topology #Quantum Physics (quant-ph) #Topological and Geometric Data Analysis
paper · pdf · doi:10.48550/arxiv.2211.00571
openalex publication_date 2022/11/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
The data of a physical experiment can be represented as a presheaf of probability distributions. A striking feature of quantum theory is that those probability distributions obtained in quantum mechanical experiments do not always admit a joint probability distribution, a celebrated observation due to Bell. Such distributions are called contextual. Simplicial distributions are combinatorial models that extend presheaves of probability distributions by elevating sets of measurements and outcomes to spaces. Contextuality can be defined in this generalized setting. This paper introduces the notion of convex categories to study simplicial distributions from a categorical perspective. Simplicial distributions can be given the structure of a convex monoid, a convex category with a single object, when the outcome space has the structure of a group. We describe contextuality as a monoid-theoretic notion by introducing a weak version of invertibility for monoids. Our main result is that a simplicial distribution is noncontextual if and only if it is weakly invertible. Similarly, strong contextuality and contextual fraction can be characterized in terms of invertibility in monoids. Finally, we show that simplicial homotopy can be used to detect extremal simplicial distributions refining the earlier methods based on Cech cohomology and the cohomology of groups.