Information topology through topos cohomology

An observer describes a system by collecting data from many local points of observation whose domains can be coherently glued together. A Grothendieck topos formalizes this process through a site of observation points and their relations. In this view, a topos is a space of phenomena where empirical data are relationally structured. The cohomology of these sheaves measures obstructions to forming coherent global descriptions or finer incompatibilities between different descriptions.

In their 2015 article The Homological Nature of Entropy, Pierre Baudot and Daniel Bennequin use cohomological tools from topos theory to reinterpret Shannon entropy. A system is modeled as a ringed site of experiments that partially or fully partition its space of possible states. On this site, a sheaf of modules encodes all conceivable measures of probabilistic information derived from these experiments, with each experiment acting on the sheaf to determine how information changes.
The cohomology of this sheaf captures how well these information measures satisfy desirable properties under observation, interpreting entropy as a generator of first-degree cohomology and extending information measures to higher dimensions.

In this talk, I will detail three cases of probabilistic information cohomology : classical, quantum and dynamical. You can find more materials here.