Polynomial Functors and Shannon Entropy

Abstract

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 twostep 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 A √ B and can be thought of as the log aspect ratio of the rectangle.

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