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 $d\in\mathsf{Dir}$ by a two-step process, where the first step is a rig homomorphism out of $\mathsf{Dir}$, the \emph{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 \emph{functor}, when we replace the set of Dirichlet polynomials by the \emph{category} of ordinary (Cartesian) polynomials. In the Cartesian case, the process has three steps. The first step is a rig functor $\mathbf{Poly}^{\mathbf{Cart}}\to\mathbf{Poly}$ sending a polynomial $p$ to $\dot{p}\mathcal{y}$, where $\dot{p}$ is the derivative of $p$. The second is a rig functor $\mathbf{Poly}\to\mathbf{Set}\times\mathbf{Set}^{\text{op}}$, sending a polynomial $q$ to the pair $(q(1),\Gamma(q))$, where $\Gamma(q)=\mathbf{Poly}(q,\mathcal{y})$ 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 $\mathbf{Set}\times\mathbf{Set}^{\text{op}}$, 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 $\log A-\log \sqrt[A]{B}$ and can be thought of as the log aspect ratio of the rectangle.
翻译:以往的工作显示, 您可以将香农( 香农) entropy 的概念与 Dirichlet 多边协议挂钩, 被视为实证分布 。 事实上, 当我们用普通( 卡泰什) 多边协议 的 $\ mathsf{Dir} 程序从任何 $\ math{ dir} 中提取 。 在Carteian 案中, 程序有三个步骤 。 第一个步骤是 richctor $\ mathb{ polopblation{ 和乘法 。 在这个简短的注解中, 我们显示, 当我们用普通( 卡提亚) 多边协议的 $\ may\ 美元来替换 美元 美元 美元 美元 美元 美元 美元 美元 美元 美元 美元 美元 美元 美元 和 美元 美元 美元 美元 美元 美元 美元 。