Cartesian differential categories are categories equipped with a differential combinator which axiomatizes the directional derivative. Important models of Cartesian differential categories include classical differential calculus of smooth functions and categorical models of the differential $\lambda$-calculus. However, Cartesian differential categories cannot account for other interesting notions of differentiation of a more discrete nature such as the calculus of finite differences. On the other hand, change action models have been shown to capture these examples as well as more "exotic" examples of differentiation. But change action models are very general and do not share the nice properties of Cartesian differential categories. In this paper, we introduce Cartesian difference categories as a bridge between Cartesian differential categories and change action models. We show that every Cartesian differential category is a Cartesian difference category, and how certain well-behaved change action models are Cartesian difference categories. In particular, Cartesian difference categories model both the differential calculus of smooth functions and the calculus of finite differences. Furthermore, every Cartesian difference category comes equipped with a tangent bundle monad whose Kleisli category is again a Cartesian difference category.
翻译：Cartesian 差分类别是配有不同组合的类别, 使方向衍生物具有分解性。 重要的Cartesian 差分类别模式包括光滑功能的经典差异计算法和差价 $\lambda$- calculus 的绝对模型。 但是, Cartesian 差分类别无法解释其他更离散性质差异的有趣概念, 如有限差异的微分。 另一方面, 已经展示了变化行动模式, 以捕捉这些例子以及更多的“ examinal” 差异示例。 但变化行动模式非常笼统, 不分享Cartesian 差分类别的良好特性。 在本文件中, 我们引入Cartesian 差分类别作为Cartesian 差分类别和差价行动模型之间的桥梁。 我们显示, 每个Cartesian 差分类别都属于Cartesian 差分类别, 以及某些稳妥的改变行动模式是Cartesian 差类别。 特别是, 差分分分分函数和差分差分差分差分的分类和差分差分。 此外, 每个Cartesian 差类别都配有卡列有卡列。
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