Towards a unified approach to functor calculus

拓扑学是数学的一部分，它提供对形状或拓扑空间的仔细分析。其中一些空间可以进行视觉检查，因为我们可以绘制它们或对它们进行建模。但其他的维度如此之多，以至于很难想象。代数拓扑学家的工作是通过将数学值赋给适当表示空间的拓扑空间，来找到想象不可想象的事情的方法。这项任务是使用同伦函子来完成的，同伦函子是一种用更容易想象甚至计算的东西来取代难以可视化、分析或分类的东西的方法。不幸的是，许多最有价值的同伦函子--那些最具描述性的函子--本身就很难计算。为了使这项任务变得更容易，有一种同伦函子的演算，它最初是由古德威利首创的，被称为古德威利演算。Goodwillie演算的思想是用更简单、更容易计算的函子来逼近有价值但复杂的同伦函子。古德威利演算最初的一些应用是应用于著名的函子，如K理论，它在数学的许多分支中都很重要，包括数论、代数、代数几何、拓扑学和分析。 我的主要研究目标是让古德威利微积分的使用变得更容易。特别是，我的研究目标之一是将古德威利微积分与我们教给一年级大学生的牛顿微积分和莱布尼茨微积分之间的关系正式化。这些是不同的，但它们有许多相同的属性。因为数学家对函数微积分的理解已经有几百年的历史了，如果我能把这个关系弄得很精确，那么这也会使古德威利的微积分更容易使用。与我的同事约翰逊、奥斯本、里尔和特贝一起，我们已经使用微分范畴理论找到了我们在特殊情况下想要的精确关系。 拓扑学用于对所有类型的问题进行建模。机器人在自动化仓库中的位置是由拓扑空间建模的，时空也是如此。在一个由越来越多的参数收集的数据世界中，拓扑学为我们提供了一种将数据作为一个整体来理解、对数据进行建模、并使用它来根据我们获得的信息得出结论和做出预测的一种方式。对于现实世界中拓扑学的每一种应用，都潜伏着同伦函子，这可以简化这些信息，使其更容易理解。事实上，这已经在做了：通过使用简单的同伦函数器来分析肝脏病变的扫描，Carlsson等人能够将这些病变归类为少数几种疾病类型。随着拓扑学的应用变得越来越普遍，我们使用的工具也会变得更加复杂。当需要这些工具时，我的工作将简化复杂的同伦函子。

Topology is the part of mathematics which provides a careful analysis of shapes, or topological spaces. Some of these spaces can be visually inspected because we can draw them or model them. But others have so many dimensions that they can hardly be imagined. The job of an algebraic topologist is to find ways to imagine the unimaginable by assigning mathematical values to topological spaces which appropriately represent the space. This assignment is accomplished using homotopy functors, a method of replacing something which is difficult to visualize, analyze or classify by something which is simpler to imagine or even compute. Unfortunately, many of the most valuable homotopy functors - those that are the most descriptive - are themselves very difficult to compute. In order to make this task easier, there is a kind of calculus for homotopy functors, which was originally pioneered by Goodwillie and which is known as Goodwillie calculus. The idea of Goodwillie calculus is to approximate valuable but complicated homotopy functors by simpler functors which are easier to compute. Some of the first applications of Goodwillie calculus were to famous functors like K-theory, which is important in many branches of mathematics including number theory, algebra, algebraic geometry, topology and analysis. My main research goal is to make it easier to use Goodwillie calculus. In particular, one of my research goals is to formalize the relationship between Goodwillie's calculus and the calculus of Newton and Leibniz that we teach to first-year university students. These are not the same, but they have many of the same properties. Since mathematician have understood calculus of functions very well for several hundred years, if I can make this relationship precise then this will make it easier to use Goodwillie's calculus, too. Together with my colleagues Johnson, Osborne, Riehl and Tebbe, we have already found the precise relationship we want in a special case using the theory of differential categories. Topology is used to model problems of all kinds. The positions of robots in an automated warehouse is modelled by a topological space, as is spacetime. In a world of data collected with an ever-increasing number of parameters, topology gives us one way of understanding the data as a whole, modelling the data, and using this to draw conclusions and make predictions from the information we have gained. For each application of topology in the real world, there are homotopy functors lurking which can simplify this information and make it easier to understand. Indeed, this is already being done: by using a simple homotopy functor to analyze scans of hepatic lesions, Carlsson et al were able to classify these lesions into a small number of disease types. As applications of topology become more prevalent, the tools we use will become more sophisticated. My work will simplify complicated homotopy functors when these tools are needed.