Entropy, cocycles, and their diagrammatics
<|reference_start|>Entropy, cocycles, and their diagrammatics: 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.<|reference_end|>
@article{im2024entropy,,
    title={Entropy, cocycles, and their diagrammatics},
    author={Mee Seong Im, Mikhail Khovanov},
    journal={arXiv preprint arXiv:2409.08462},
    year={2024},
    archivePrefix={arXiv},
    eprint={2409.08462},
    primaryClass={math.KT cs.IT math-ph math.CT math.IT math.MP}
}