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.
Highlights
We describe a simple connection between information theory, algebra, and topology
Our work is based on a particular characterization of Shannon entropy that is compatible with an operadic viewpoint
Baez explored an algebraic interpretation of Equation (2) in the informal article [6], where the reader is reminded that Shannon entropy is a derivative of the partition function of a probability distribution with respect to Boltzmann’s constant, considered as a formal parameter
Summary
Consider the function d : [0, 1] → R defined by This map satisfies an equation reminiscent of the Leibniz rule from Calculus, d( xy) =. D is a nonlinear derivation [1], (Lemma 2.2.6). This derivation may bring to mind the Shannon entropy of a probability distribution. Pn ) satisfying ∑in=1 pi = 1, and the Shannon entropy of p is defined to be n. We describe one such setting below by showing a correspondence between Shannon entropy and derivations of the operad of topological simplices
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