Polynomial functors and Shannon entropy

01/30/2022

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by   David I. Spivak, et al.

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Past work shows that one can associate a notion of Shannon entropy to a Dirichlet polynomial, regarded as an empirical distribution. Indeed, entropy can be extracted from any d∈𝖣𝗂𝗋 by a two-step process, where the first step is a rig homomorphism out of 𝖣𝗂𝗋, the set of Dirichlet polynomials, with rig structure given by standard addition and multiplication. In this short note, we show that this rig homomorphism can be upgraded to a rig functor, when we replace the set of Dirichlet polynomials by the category of ordinary (Cartesian) polynomials. In the Cartesian case, the process has three steps. The first step is a rig functor 𝐏𝐨𝐥𝐲^𝐂𝐚𝐫𝐭→𝐏𝐨𝐥𝐲 sending a polynomial p to ṗ𝓎, where ṗ is the derivative of p. The second is a rig functor 𝐏𝐨𝐥𝐲→𝐒𝐞𝐭×𝐒𝐞𝐭^op, sending a polynomial q to the pair (q(1),Γ(q)), where Γ(q)=𝐏𝐨𝐥𝐲(q,𝓎) can be interpreted as the global sections of q viewed as a bundle, and q(1) as its base. To make this precise we define what appears to be a new distributive monoidal structure on 𝐒𝐞𝐭×𝐒𝐞𝐭^op, which can be understood geometrically in terms of rectangles. The last step, as for Dirichlet polynomials, is simply to extract the entropy as a real number from a pair of sets (A,B); it is given by log A-log√(B) and can be thought of as the log aspect ratio of the rectangle.