论文标题
Ramsey学位和组合结构的熵
Ramsey degrees and entropy of combinatorial structures
论文作者
论文摘要
熵的各种概念与类别理论的设备之间的密切联系已经在1980年代观察到，并且在过去十年中更具剧烈的发展。本文的起点是对结构性拉姆西学位的最新分类理解，然后导致一种方法来计算小型类别中对象的熵，而不是统计的量度，而是对其组合复杂性的量度。我们提出的新熵函数是拉姆西熵，是任意小型类别中对象的实用值不变的。我们不需要其他分类机器来引入和证明此熵的特性。由组合现象（结构性拉姆西学位）的动机，我们构建了必要的基础设施，并仅使用对Homsets的特殊分区证明了基本属性。我们以对我们称为Ramsey-Boltzmann熵的类别的最大拉姆西熵的讨论来结束论文。
Close connections between various notions of entropy and the apparatus of category theory have been observed already in the 1980s and more vigorously developed in the past ten years. The starting point of the paper is the recent categorical understanding of structural Ramsey degrees, which then leads to a way to compute entropy of an object in a small category not as a measure of statistical, but as a measure of its combinatorial complexity. The new entropy function we propose, the Ramsey entropy, is a real-valued invariant of an object in an arbitrary small category. We require no additional categorical machinery to introduce and prove the properties of this entropy. Motivated by combinatorial phenomena (structural Ramsey degrees) we build the necessary infrastructure and prove the fundamental properties using only special partitions imposed on homsets. We conclude the paper with the discussion of the maximal Ramsey entropy on a category that we refer to as the Ramsey-Boltzmann entropy.
