Information structures and their cohomology

Juan Pablo Vigneaux

Theory and applications of categories, Tome 35 (2020), pp. 1476-1529.

Résumé

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.

Détail

Publié le : 2020-08-24

Classification : 55N35, 94A15, 39B05, 60A99
Keywords: information cohomology, entropy, nonextensive statistics, information structures, sheaves, topos