论文标题
无COF的笛卡尔差异类别的属性和特征
Properties and Characterisations of Cofree Cartesian Differential Categories
论文作者
论文摘要
笛卡尔差异类别配备了一个差分运算符，该操作员从多变量演算中形式化了总导数。无COF的笛卡尔差异类别始终存在于指定的基础类别上，其中一般结构基于FaàdiBruno的公式。一个自然的问题是，当给出任意的笛卡尔差异类别时，如何在不知道基本类别的情况下检查它是否是无用的？在本文中，我们提供了无COF的笛卡尔差异类别的特征，而无需指定基本类别。令人惊讶的是，这些特征的关键是其衍生物为零的地图，我们称之为差异常数。一个表征是从整体的超量学空间来看，其中超量化是由差分常数诱导的，这与功率序列的度量标准相似。另一个表征是单子的代数。在任何一个表征中，基本类别都是差分常数的类别。我们还讨论了无COF的笛卡尔差异类别（例如线性图）的其他基本特性，并解释了多少众所周知的笛卡尔差异类别（例如多项式或光滑函数）不是无关的。
Cartesian differential categories come equipped with a differential operator which formalises the total derivative from multivariable calculus. Cofree Cartesian differential categories always exist over a specified base category, where the general construction is based on Faà di Bruno's formula. A natural question to ask is, when given an arbitrary Cartesian differential category, how can one check if it is cofree without knowing the base category? In this paper, we provide characterisations of cofree Cartesian differential categories without specifying a base category. The key to these characterisations is, surprisingly, maps whose derivatives are zero, which we call differential constants. One characterisation is in terms of the homsets being complete ultrametric spaces, where the ultrametric is induced by differential constants, which is similar to the metric for power series. Another characterisation is as algebras of a monad. In either characterisation, the base category is the category of differential constants. We also discuss other basic properties of cofree Cartesian differential categories, such as the linear maps, and explain how many well-known Cartesian differential categories (such as polynomial or smooth functions) are not cofree.
