Entropy as a Topological Operad Derivation

Tai-Danae Bradley

doi:10.3390/e23091195

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arXiv · 2107.09581

Entropy as a Topological Operad Derivation

Tai-Danae Bradley

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Abstract

We share a small connection between information theory, algebra, and topology - namely, a correspondence between Shannon entropy and derivations of the operad of topological simplices. We begin with a brief review of operads and their representations with topological simplices and the real line as the main example. We then give a general definition for a derivation of an operad in any category with values in an abelian bimodule over the operad. The main result is that Shannon entropy defines a derivation of the operad of topological simplices, and that for every derivation of this operad there exists a point at which it is given by a constant multiple of Shannon entropy. We show this is compatible with, and relies heavily on, a well-known characterization of entropy given by Faddeev in 1956 and a recent variation given by Leinster.

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Tai-Danae Bradley. 2021-07-20. Entropy as a Topological Operad Derivation. https://doi.org/10.3390/e23091195

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