Information structures and their cohomology

TL;DR

This paper develops a mathematical framework for modeling and analyzing contextuality in classical and quantum systems using information structures and their cohomology, revealing entropy measures as key cocycles.

Contribution

It introduces the category of information structures, extends cohomology theory to this setting, and identifies entropy functions as fundamental cocycles.

Findings

Information structures form a ringed site with a trivial topology.

The cohomology framework recovers known cochain complexes.

Entropy functions emerge as the only 1-cocycles for certain coefficients.

Abstract

We introduce the category of information structures, whose objects are suitable diagrams of measurable sets that encode the possible outputs of a given family of observables and their mutual relationships of refinement; they serve as mathematical models of contextuality in classical and quantum settings. Each information structure can be regarded as a ringed site with trivial topology; the structure ring is generated by the observables themselves and its multiplication corresponds to joint measurement. We extend Baudot and Bennequin's definition of information cohomology to this setting, as a derived functor in the category of modules over the structure ring, and show explicitly that the bar construction gives a projective resolution in that category, recovering in this way the cochain complexes previously considered in the literature. Finally, we study the particular case of a one-parameter family of coefficients made of functions of probability distributions. The only 1-cocycles are Shannon entropy or Tsallis $\alpha$-entropy, depending on the value of the parameter.