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.

中文翻译：

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线性化组合器

2017 年，Bauer、Johnson、Osborne、Riehl 和 Tebbe (BJORT) 表明，阿贝尔函子微积分提供了笛卡尔微分范畴的一个例子。笛卡尔微分范畴的定义基于微分组合器，该组合器直接将多变量微积分的总导数形式化。然而，在上述工作中，作者使用 Goodwillie 的函子演算中的技术建立了一个线性化过程，然后他们从中推导出了一个微分组合器。这提出了线性化和具有差分组合器之间的精确关系可能是什么的问题。在本文中，我们介绍了线性化组合器的概念，它抽象了阿贝尔函子演算中的线性化。然后我们使用它来提供笛卡尔微分类别的替代公理化。每个笛卡尔微分类别都配备了通过零微分获得的规范线性化组合器。相反，当一个人在每个上下文中都有一个部分线性化组合器的系统时，可以构建一个微分组合器\`a la BJORT。因此，虽然线性化组合器确实提供了笛卡尔微分类别的替代公理化，但需要明确的部分线性化概念。这与微分组合器的情况相反，在全微分存在的情况下，偏微分是自动的。从总线性化组合器形成部分线性化组合器系统的能力，虽然通常不可能，