Pierre Baudot, Daniel Bennequin (2015), Entropy 17(5):3253–3318 (MDPI) — DOI-10.3390/e17053253
Key takeaways
- Shannon entropy is the (first) universal cohomology class of an “information structure” built from a family of observables and a family of probability laws. The same holds for quantum probabilities and for dynamic observation trees, which gives a topology of information that recovers multivariate mutual informations and KL divergence.
Seeded from the abstract alone, before the reading in NOTE-088. What follows is what the work says about itself.
The authors propose that entropy is a universal cohomological class in a theory attached to a family of observable quantities and a family of probability distributions. They develop three cases: classical probabilities with random variables, quantum probabilities with observable operators, and dynamic probabilities with observation trees. The result is a new kind of topology for information processes. It accounts for entropy, mutual informations of all orders and Kullback–Leibler divergence, and generalizes them. The article has two independently readable parts. The first is an overview of results, open questions, future directions and application to complex data. The second gives complete definitions and proofs of the main theorems (A, C, E), showing that entropy is the first homological invariant of an information structure in four settings: static classical or quantum probability, and dynamics of classical or quantum observation strategies.
Standing in the record
Filed from the survey of 2026-09-25 of work the anthology set aside as out of scope (tier C): 455 seconds of active reading over 1 session in the papers-feed tracker. It was filed Deferred, unread. NOTE-088 is the close reading of 2026-09-25, and it placed the work: Active — the founding paper of information cohomology, with complete proofs of the degree-one results (Thms 1, 3, 4). Read it for those. Its broader programme (higher mutual informations as cocycles, super/sub-harmonicity, max-entropy discrimination) is sketched or deferred here to unpublished companion papers.
When seeded, before the reading in NOTE-088, the entry gave a priority for a deeper reading: medium — Owner time is one long session (t=455, n=1). The abstract states the headline claim, but none of it is verified from the text, and the full paper was unreachable here.
What the seeding asked a deeper reading to check (NOTE-088 is that reading):
- It is a foundational paper for “information cohomology” / topological data analysis of multivariate dependence. The claim that higher mutual informations arise as cocycles is the part most likely to matter for analysing feature interactions.
- A deeper reading must reach the text (MDPI PDF from a normal browser) and check exactly which “information structure” category is used, what “universal” means in Theorem A (uniqueness up to a multiplicative constant?), and how the quantum case (§ on observable operators) differs.
- It connects to later work by Vigneaux (information cohomology of generalized entropies) and to Baudot’s applications to gene-expression data. It is worth deciding whether it belongs under information-theory, primarily, rather than mathematics.
Access when seeded: I had Crossref metadata and the full abstract (Entropy 17(5):3253–3318, published 2015-05-13, which Semantic Scholar also gives). OpenAlex and Semantic Scholar list it as gold open access (CC BY) at MDPI, but every MDPI URL (PDF and HTML) returned “Access Denied” through the proxy. The MPG edoc repository copy sits behind an Anubis anti-bot proof-of-work page, which I did not try to get past. The CiteSeerX connection was reset. An arXiv title search found no preprint under this title, and Europe PMC and HAL had no record. I did not read the full text. The keywords above are mine, not the journal’s.