Comparing the homotopy calculi

拓扑学是对抽象的形状概念的研究，称为拓扑空间。简单的例子包括圆、球体和环面(美国甜甜圈)，事实上，现实生活中的任何物体都代表着一个空间。拓扑学还包括研究这些空间之间的关系，即所谓的映射。例如，圆可以被认为是球体的赤道。但我们可以用许多其他方法将圆发送到球体上，例如通过在球体上画一个环(可能会与自身相交)。代数拓扑学关注的是空间和映射的性质，这些空间和映射因连续变形而保持不变，称为同伦。人们的直觉是，虽然你可以把一个圆柱体压成一个圆盘，但你不能把一个洞撕成一个形状。例如，任何从圆到球面的映射都是常量映射的同伦，但从圆到环面的映射则不是这样。几何和代数的结合以及空间的无处不在使代数拓扑学成为一个令人着迷的数学领域，它可以将其强大的技巧应用于各种其他数学学科的许多类型的问题。这些机器将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，有一个函数器，它接受一个拓扑空间作为输入，作为输出返回输入空间的两个不相交的副本。作为代数拓扑学的核心，研究某些函子如何工作的好方法是很重要的。古德威利和韦斯在20世纪90年代开发了两种这样的方法。这些方法被称为函数式微积分和正交式微积分，自发明以来，在很短的时间内就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函子并将其分成一系列近似来工作的。一个阶段和下一个阶段之间的区别是，在每种情况下，都有一个特别结构良好的空间，称为频谱。这种结构使光谱比空间更容易研究。因此，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并利用我们的谱知识从一个近似转移到下一个近似来研究它的性质。函子演算和正交化演算已知是相关的，但这种关系还没有正式的研究。因此，这种关系只是含糊其辞地提及，没有好的描述。本课题的目的是弄清楚这一关系，并利用它来构造同伦演算的一般形式。这种一般形式将允许古德威利和韦斯的结果应用于其他上下文和其他数学领域，如代数。更好的基础将鼓励用户致力于更深层次、更有趣的结果。对这种关系的清晰描述也有很多用处。例如，它将帮助计算，因为现在这两种形式都可以使用和比较。总体而言，这个项目将改进代数拓扑学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.