论文标题
笛卡尔差异类别
Cartesian Difference Categories
论文作者
论文摘要
笛卡尔差异类别是配备差分组合器的类别，该类别将方向衍生物公理。笛卡尔差异类别的重要模型包括平滑函数的经典差分计算以及差异$λ$ -calculus的分类模型。但是，笛卡尔差异类别不能说明更分散性质的区分的其他有趣概念，例如有限差异的演算。另一方面，已证明变更动作模型可以捕获这些示例以及分化的更多“异国”例子。但是，变更动作模型非常笼统，并不具有笛卡尔差异类别的良好属性。在本文中，我们将笛卡尔差异类别作为笛卡尔差异类别和变更行动模型之间的桥梁。我们表明，每个笛卡尔差异类别都是笛卡尔差异类别，以及某些行为良好的变更动作模型是笛卡尔差异类别。尤其是，笛卡尔差异类别模拟平滑函数的差分计算和有限差分的演算。此外，每个笛卡尔差异类别都配备了一个切线捆绑单，其Kleisli类别再次是笛卡尔差异类别。
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 $λ$-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.
