Entropy, cocycles, and their diagrammatics

TL;DR

This paper develops a diagrammatic calculus for encoding cocycles on groups using planar graphs, connecting entropy, dilogarithms, and monoidal categories to provide a new categorical perspective.

Contribution

It introduces a novel diagrammatic framework for cocycles and entropy, linking group cohomology with monoidal categories and graphical calculus.

Findings

Diagrammatic encoding of cocycles on groups using planar graphs.

Interpretation of entropy and dilogarithms via network equalities.

Connection of four-term relations to associativity in monoidal categories.

Abstract

The first part of the paper explains how to encode a one-cocycle and a two-cocycle on a group $G$ with values in its representation by networks of planar trivalent graphs with edges labelled by elements of $G$, elements of the representation floating in the regions, and suitable rules for manipulation of these diagrams. When the group is a semidirect product, there is a similar presentation via overlapping networks for the two subgroups involved. M. Kontsevich and J.-L. Cathelineau have shown how to interpret the entropy of a finite random variable and infinitesimal dilogarithms, including their four-term functional relations, via 2-cocycles on the group of affine symmetries of a line. We convert their construction into a diagrammatical calculus evaluating planar networks that describe morphisms in suitable monoidal categories. In particular, the four-term relations become equalities of networks analogous to associativity equations. The resulting monoidal categories complement existing categorical and operadic approaches to entropy.