Comparing the homotopy calculi

拓扑学研究的是抽象的形状概念，即拓扑空间。简单的例子包括圆、球和环面（一个美国甜甜圈），实际上，现实生活中的任何物体都代表一个空间。拓扑学还包括对这些空间之间关系的研究，即地图。例如，圆可以被认为是球体的赤道。但是我们可以用许多其他的方法把圆送到球体上，例如在球体上画一个圈（它可以交叉）。代数拓扑学关注的是空间和映射的性质，这些性质在连续变形中保持不变，被称为同伦。直觉是，虽然你可以把一个圆柱体压成圆盘，但你可能不会把一个洞撕开成一个形状。例如，任何从圆到球的映射都是同伦的常数映射，但从圆到环面的映射则不是如此。几何和代数的结合以及空间的无所不在帮助代数拓扑成为一个迷人的数学领域，可以将其强大的技术应用于各种其他数学学科中的许多类型的问题。代数拓扑的基本结构之一是函子。这些机器以一种数学对象作为输入，并给出另一种数学对象作为输出。作为一个简单的例子，有一个函子，它接受一个拓扑空间作为输入，并作为输出返回输入空间的两个不相交的副本。作为代数拓扑的核心，研究某些函子如何工作的好方法是很重要的。Goodwillie和Weiss在20世纪90年代开发了两种这样的方法。这些方法被称为函子演算和正交演算，自发明以来在很短的时间内产生了许多令人兴奋的结果。这两种形式的同伦微积分都是通过取一个函子并把它分解成一系列的近似而起作用的。一个阶段和下一个阶段之间的区别在于每个阶段都有一个结构特别良好的空间，即谱。这种结构使得光谱比空间更容易研究。这样做的好处是，我们可以证明关于原始函子的陈述，并通过观察它的近似来研究它的性质，并利用我们的光谱知识从一个近似移动到下一个近似。函子演算和正交演算被认为是相关的，但是没有对这种关系进行正式的研究。因此，这种关系只是含糊地提到，并没有很好的描述。本课题的目的是明确这一关系，并利用它来构造同伦微积分的一般形式。这种一般形式将允许Goodwillie和Weiss的结果应用于其他背景和其他数学领域，例如代数。更好的基础会鼓励用户更深入、更有趣的结果。对这种关系的清晰描述也有许多用途。例如，它将有助于计算，因为现在可以使用和比较两种形式。总的来说，这个项目将改进代数拓扑中一些已经很有用的工具，将它们扩展到新的数学领域，并帮助进行重要的计算。

Topology is the study of an abstract notion of shape known as a topological space. Simple examples include the circle, the sphere and the torus (an American doughnut), indeed any object in real life represents a space. Topology also encompasses the study of relations between these spaces, known as maps. For example, the circle can be thought of as the equator of the sphere. But we can send the circle to the sphere in number other ways, for example by drawing a loop on the sphere (that may cross itself).Algebraic topology focuses on the properties of spaces and maps which are left unchanged by continuous deformations, known as homotopies. The intuition is that while you can crush a cylinder to a disc, you may not rip a hole into a shape. As an example, any map from a circle into a sphere is homotopic to a constant map, but the same is not true for maps from a circle to a torus. The combination of geometry and algebra and the ubiquity of spaces has helped algebraic topology to become a fascinating area of mathematics that can apply its powerful techniques to many kinds of problems in a wide variety of other mathematical disciplines.One of the fundamental constructions in algebraic topology are functors. These are machines which take one kind of mathematical object as an input and give another kind as an output. As a simple example, there is a functor which accepts a topological space as an input, and as output gives back two disjoint copies of the input space. Being so central to algebraic topology, a good method for studying how certain functors work is important. Goodwillie and Weiss developed two such methods in the 1990s. These methods, known as functor calculus and orthogonal calculus, have produced a number of exciting results in the short time since their invention. Both of these two forms of homotopy calculus work by taking a functor and splitting it into a series of approximations. The difference between one stage and the next is in each case a particularly well-structured space known as a spectrum. This structure makes spectra much easier to study than spaces. The advantage is therefore that we can prove statements about the original functor and study its properties by looking at its approximations and using our knowledge of spectra to move from one approximation to the next.The functor calculus and the orthogonal calculus are known to be related, but no formal study of this relation has ever been undertaken. Consequently the relation is only vaguely alluded to and no good description exists. The purpose of this project is to make this relation clear and use it to construct a general form of homotopy calculus. This general form will allow the results of Goodwillie and Weiss to be applied to other contexts and other areas of mathematics, such as algebra. The better foundations will encourage the users to work on deeper, more interesting results. A clear description of the relation also has a number of uses. For example, it will assist with calculations as now both forms can be used and compared. Overall this project will improve some already useful tools in algebraic topology, extend them to new areas of mathematics and help with important calculations.