论文标题
多项式函子和香农熵
Polynomial Functors and Shannon Entropy
论文作者
论文摘要
过去的工作表明，一个人可以将香农熵的概念与被视为经验分布的多项式多项式相关联。实际上，可以通过两步过程从任何D：DIR中提取熵，其中第一步是DIR的钻机同构，即Dirichlet多项式的 *集合 *，其钻机结构由标准添加和乘法给出。在此简短说明中，我们表明，当我们用普通（笛卡尔）多项式的 *类别 *替换Dirichlet多项式 *时，可以将这种钻机同态升级到钻机 *函子 *。 在笛卡尔案中，该过程有三个步骤。第一步是钻机函数polycart-> poly将多项式p发送到（dp）y，其中dp是p的导数。第二个是钻机函数poly-> set x set^op，将多项式q发送到对（q（q（1），伽马（q）），其中gamma（q）= poly（q）= poly（q，y）可以解释为q的全局段，将其视为bundledle，q（1）和q（1）。为了确切地说，我们定义了集合X集合上的新分布单体结构，可以从矩形上以几何理解。对于Dirichlet多项式，最后一步只是将熵从一对集（a，b）中提取为真实数字；它由log a -log b^（1/a）给出，可以将其视为矩形的日志纵横比。
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.
