Theory and applications of tangent categories

微积分是数学中最有用、最实用的领域。它研究如何计算变化量，因此是科学、经济学和工程学中的基本工具。然而，尽管它在应用环境中极其重要，但它的研究只是更广泛的数学世界的一部分，数学世界包括许多其他主题，如组合学(计数)、拓扑学(研究抽象形状)和计算机科学。 另一方面，在数学的其他一些领域，研究人员已经开始开发与微积分有许多形式上相似之处的想法。它们有不同的名称：多项式函子(在组合学中)，函子演算(在拓扑学中)和微分线性逻辑(在计算机科学中)。随着这些想法的发展，还不清楚它们之间是如何联系的，或者它们是否真的与普通微积分有直接的相似之处。 为这笔拨款提出的研究提供了一种解决这个问题的方法，允许人们将所有这些不同的概念视为一个共同想法的所有方面，即切线范畴的概念。通过为这些不同的思想提供一个公共框架，切线范畴提供了一种在数学的不同领域之间转换思想的基本语言。此外，人们可以在正切范畴的抽象设置中发展许多微分几何，允许将微分几何的复杂思想转移到多项式函子、函子演算和微分线性逻辑的研究中。因此，这项工作使我们能够以一种全新的方式来促进我们对这些重要思想的理解。

Calculus is the most useful and practical area of mathematics. It deals with how to calculate with changing quantities, and is thus a fundamental tool in sciences, economics, and engineering. However, while it is extremely important in applied contexts, its study is but one part of a much wider world of mathematics, which includes many other topics, such as combinatorics (counting), topology (the study of abstract shapes) and computer science. On the other hand, in some of these other areas of mathematics, researchers have started to develop ideas that share many formal similarities with calculus. These have gone by various names: polynomial functors (in combinatorics), the functor calculus (in topology) and differential linear logic (in computer science). As these ideas have been developed, it has not been clear how they are related to each other, or if they truly share a direct similarity with ordinary calculus. The research proposed for this grant offers a way to resolve this problem, by allowing one to view all these various notions as all aspects of one common idea, the notion of a tangent category. By providing a common framework for these different ideas, tangent categories provide an essential language to translate ideas between different areas of mathematics. Moreoever, one can develop much of differential geometry within the abstract setting of a tangent category, allowing the sophisticated ideas of differential geometry to be transferred into the study of polynomial functors, functor calculus, and differential linear logic. This work thus allows us an entirely new way to advance our understanding of these important ideas.