Polynomial Functors and Shannon Entropy

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