Source
Baudot & Bennequin (2015), “The Homological Nature of Entropy”, Entropy 17, LIT-053, read in NOTE-088.
What was actually shown
The complex. On a finite Ω, an information structure S is a set of partitions containing the trivial one and closed under joins of partitions with a common refinement in S. A partition Y acts on functions of P by averaged conditioning, Y.F(P) = Σ_j P(Y=y_j) F(P|Y=y_j). The bar complex of this action has the 1-cocycle equation f((Y,Y′);P) = f(Y;P) + Y.f(Y′;P), which is the chain rule.
Theorem 1. If S is connected and every element lies on a rich edge (X, Y with at least two values each, XY cutting them into at least four pieces), Q is complete, and cochains are measurable, then H¹(S;Q) is one dimensional and spanned by Shannon entropy. Because H⁰ consists of constants, there are no non-trivial 1-coboundaries, so H¹ = Z¹: every measurable 1-cocycle equals C·H, one constant per connected component. The proof reduces to Tverberg’s functional equation on Ω = {00, 01, 10, 11} (Lemma 2).
Theorem 3 (quantum). For unitary-invariant, connected, sufficiently rich S with the canonical density functor, H¹_U = 0. Shannon entropy of a decomposition is still a cocycle, but δ̂(S_X)(Y;ρ) = −H_X(Y;ρ): it is the coboundary of minus the von Neumann entropy, which is the unique invariant 0-cochain with that property up to an additive constant. The proof’s step IV rests on a two-sentence lemma (Lemma 7).
Also proved. KL divergence of image laws is a 1-cocycle for pairs of laws (Prop. 4), and entropy satisfies the finer, operadic chain rule of Faddeev and Baez–Fritz–Leinster (Prop. 8).
What could have come out otherwise: other measurable functions could have satisfied the cocycle equation. On rich structures none does.
What this does not say
- That the quantum case is “the same”. In degree one it is the opposite: classical entropy is a non-trivial class, and in the invariant quantum theory it becomes trivial (NOTE-088, Corrections).
- That higher mutual informations are cohomology classes. The paper’s formulas present them as images of δ or δ_t, coboundaries, and §3 is titled “A Sketch”. Higher-degree cohomology is not computed.
- That I₃ < 0 is synergy, or a Borromean linking invariant. Asserted with citations (C6, weak).
- That KL divergence is unique. It is shown to be a cocycle, nothing more.
- Anything infinite. Ω is finite; the infinite case is deferred.
- That cohomology has been applied to data. The §1.7 applications are others’ uses of three-way mutual information, not of the cohomology.
Connections
- Shannon (LIT-764, THEORY-602) says his axiomatic uniqueness theorem is “in no way necessary” and that the coding theorems justify entropy. This is a second, axiom-light uniqueness result. It does not replace the operational justification.
- The interaction information it reads as synergy is, for two sources, exactly redundancy minus synergy under any PID (THEORY-568).
- Like Abramsky & Brandenburger’s sheaf formulation (LIT-016, THEORY-012), it attaches probability data to a site of commuting observables; there the question is gluing, here it is the derived cohomology of the conditioning module.
- The anthology’s ANTH-LIT-698 uses this reading by analogy.