In 2017, Bauer, Johnson, Osborne, Riehl, and Tebbe (BJORT) showed that the Abelian functor calculus provides an example of a Cartesian differential category. The definition of a Cartesian differential category is based on a differential combinator which directly formalizes the total derivative from multivariable calculus. However, in the aforementioned work the authors used techniques from Goodwillie's functor calculus to establish a linearization process from which they then derived a differential combinator. This raised the question of what the precise relationship between linearization and having a differential combinator might be. In this paper, we introduce the notion of a linearizing combinator which abstracts linearization in the Abelian functor calculus. We then use it to provide an alternative axiomatization of a Cartesian differential category. Every Cartesian differential category comes equipped with a canonical linearizing combinator obtained by differentiation at zero. Conversely, a differential combinator can be constructed \`a la BJORT when one has a system of partial linearizing combinators in each context. Thus, while linearizing combinators do provide an alternative axiomatization of Cartesian differential categories, an explicit notion of partial linearization is required. This is in contrast to the situation for differential combinators where partial differentiation is automatic in the presence of total differentiation. The ability to form a system of partial linearizing combinators from a total linearizing combinator, while not being possible in general, is possible when the setting is Cartesian closed.
翻译:2017年,Bauer, Johnson, Johnson, Osborne, Riehl, 和 Tebbe (BJORT) 显示, Abelian 的真菌计算器提供了一种Cartesian 差异分类的范例。 卡尔泰西亚差异类别的定义是基于一个差异组合器, 直接将多变量计算器产生的全部衍生衍生物正式化。 但是, 在上述工作中, 作者们使用了亲善派的真菌计算器的技术来建立一个线性化过程, 然后他们从中得出一个差分数组合。 这引起了一个问题: 直线化和差异组合之间的确切关系可能是什么? 在本文件中, 我们引入一个线性组合计算器的概念, 它将本部分线性组合器的直线性分类法, 而在每种情况下, 部分直线性分类法化的系统是直线性分类法化的直线性分析器, 而在每种情况下, 一种直线性分类法化的系统是直线性分析器, 在每种情况下, 一种直线性分析器是直线化的直线化的分类, 。