Entropy as a Topological Operad Derivation

TL;DR

This paper establishes a novel mathematical connection between Shannon entropy and derivations of the operad of topological simplices, linking information theory with algebraic topology.

Contribution

It introduces a new perspective by showing Shannon entropy as a derivation of a topological operad, unifying concepts from information theory, algebra, and topology.

Findings

Shannon entropy defines a derivation of the operad of topological simplices.

Every derivation of this operad corresponds to a constant multiple of Shannon entropy at some point.

The work relies on classical characterizations of entropy by Faddeev and Leinster.

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.