多项式函子与香农熵
Polynomial Functors and Shannon Entropy
AI总结：
本文证明可将提取 Dirichlet 多项式香农熵的 rig 同态提升为基于笛卡尔多项式范畴的 rig 函子，通过三步映射过程并引入新的分配幺半结构，将熵几何化解释为矩形的对数长宽比。
摘要：
先前的工作表明，可以将香农熵的概念与被视为经验分布的 Dirichlet 多项式相关联。事实上，可以通过两步过程从任何 d:Dir 中提取熵，其中第一步是从 Dir（即 Dirichlet 多项式的*集合*，其 rig 结构由标准的加法和乘法给出）出发的 rig 同态。在这篇短文中，我们证明了当用普通（笛卡尔）多项式的*范畴*代替 Dirichlet 多项式的集合时，该 rig 同态可以提升为 rig *函子*。在笛卡尔情形下，该过程包含三个步骤。第一步是一个 rig 函子 PolyCart -> Poly，将多项式 p 映射为 (dp)y，其中 dp 是 p 的导数。第二步是一个 rig 函子 Poly -> Set x Set^op，将多项式 q 映射为序对 (q(1), Gamma(q))，其中 Gamma(q)=Poly(q,y) 可以解释为将 q 视为丛时的全局截面，而 q(1) 视为其底空间。为了精确表述这一点，我们在 Set x Set^op 上定义了一种看似全新的分配幺半结构，该结构可以在几何上用矩形来理解。最后一步，如同 Dirichlet 多项式那样，仅仅是作为实数从集合对 (A,B) 中提取熵；其公式为 log A - log B^(1/A)，可以将其视为矩形的对数长宽比。
英文摘要：
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 PolyCart -> Poly sending a polynomial p to (dp)y, where dp is the derivative of p. The second is a rig functor Poly -> Set x Set^op, sending a polynomial q to the pair (q(1),Gamma(q)), where Gamma(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 x 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^(1/A) and can be thought of as the log aspect ratio of the rectangle.
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