On the Cohomology of Contextuality

TL;DR

This paper explores the cohomological approach to contextuality, revealing limitations of existing invariants, extending the theory to higher cohomology, and offering new interpretations of the obstructions.

Contribution

It demonstrates that cohomological obstructions are not complete invariants, extends the framework to higher cohomology groups, and provides a torsor-based interpretation.

Findings

Cohomological obstruction is not a complete invariant for strong contextuality.

Higher cohomology groups refine the hierarchy of contextuality levels.

First cohomology group can be described using torsors.

Abstract

Recent work by Abramsky and Brandenburger used sheaf theory to give a mathematical formulation of non-locality and contextuality. By adopting this viewpoint, it has been possible to define cohomological obstructions to the existence of global sections. In the present work, we illustrate new insights into different aspects of this theory. We shed light on the power of detection of the cohomological obstruction by showing that it is not a complete invariant for strong contextuality even under symmetry and connectedness restrictions on the measurement cover, disproving a previous conjecture. We generalise obstructions to higher cohomology groups and show that they give rise to a refinement of the notion of cohomological contextuality: different "levels" of contextuality are organised in a hierarchy of logical implications. Finally, we present an alternative description of the first cohomology group in terms of torsors, resulting in a new interpretation of the cohomological obstructions.