Towards a unified approach to functor calculus

拓扑学是数学的一部分，它提供了对形状或拓扑空间的仔细分析。 其中一些空间可以直观地观察，因为我们可以绘制它们或对其建模。 但其他人有这么多的维度，他们几乎无法想象。 代数拓扑学家的工作是通过给拓扑空间分配数学值来找到想象不可想象的空间的方法。 这个分配是使用同伦函子来完成的，这是一种用更容易想象甚至计算的东西来替换难以可视化，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--本身就很难计算。 为了使这个任务更容易，有一种同伦函子的演算，最初是由古德威利开创的，被称为古德威利演算。 古德威利演算的思想是用更容易计算的简单函子来近似有价值但复杂的同伦函子。 古德威利演算的一些最早的应用是著名的函子，如K理论，这是重要的许多数学分支，包括数论，代数，代数几何，拓扑和分析。 我的主要研究目标是使它更容易使用古德威利演算。 特别是，我的研究目标之一是形式化古德威利微积分与我们教给一年级大学生的牛顿和莱布尼茨微积分之间的关系。 它们并不相同，但它们有许多相同的属性。 因为数学家已经很好地理解了函数的微积分几百年了，如果我能使这个关系精确，那么这将使它更容易使用古德威利的微积分，太。 我们与我的同事约翰逊、奥斯本、里尔和特伯一起，已经用微分范畴理论找到了我们在一个特殊情况下所需要的精确关系。 拓扑学被用来对各种问题建模。 机器人在自动化仓库中的位置由拓扑空间建模，时空也是如此。 在一个数据收集的世界中，参数数量不断增加，拓扑学为我们提供了一种整体理解数据的方法，对数据进行建模，并利用这些数据从我们获得的信息中得出结论并进行预测。 对于拓扑在真实的世界中的每一个应用，都潜伏着同伦函子，它可以简化这些信息，使其更容易理解。 事实上，这已经在做了：通过使用一个简单的同伦函子来分析肝脏病变的扫描，Carlsson等人能够将这些病变分类为少数疾病类型。 随着拓扑学的应用变得越来越普遍，我们使用的工具将变得更加复杂。 我的工作将简化复杂的同伦函子时，这些工具是必要的。

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.