A bundle perspective on contextuality: Empirical models and simplicial distributions on bundle scenarios

TL;DR

This paper introduces a new bundle perspective on contextuality using simplicial complexes and sets, generalizing existing frameworks and establishing functors and isomorphisms to compare different formulations.

Contribution

It develops a formalism connecting bundle categories with sheaf-theoretic and simplicial set approaches to contextuality, facilitating future resource theory applications.

Findings

Constructed bundle categories for contextuality scenarios

Established functors and natural isomorphisms between frameworks

Lays groundwork for resource theory of contextuality

Abstract

This paper provides a bundle perspective to contextuality by introducing new categories of contextuality scenarios based on bundles of simplicial complexes and simplicial sets. The former approach generalizes earlier work on the sheaf-theoretic perspective on contextuality, and the latter extends simplicial distributions, a more recent approach to contextuality formulated in the language of simplicial sets. After constructing our bundle categories, we also construct functors that relate them and natural isomorphisms that allow us to compare the notions of contextuality formulated in two languages. We are motivated by applications to the resource theory of contextuality, captured by the morphisms in these categories. In this paper, we develop the main formalism and leave applications to future work.