Calculus of Functors

DMS-0204969托马斯·G·古德威廉我们建议继续研究几种版本的“函数式演算”。这些都是一种利用多相对连通性估计在某些上下文中用特定值来描述连续函子的技术；例如，空间的函子可以从它在高度连通空间的值恢复，流形的子空间的函子可以从它在低维子空间的值恢复，或者真实内积空间的函子可以从它在高维空间的值恢复。该提议的一部分是提炼纯同伦理论版本的“微积分”。其他部分则涉及组合几种版本并将它们应用于高维和低维微分拓扑中的各种问题。上面提到的每一种“函数演算”之所以被称为函数演算，是因为它们与牛顿和莱布尼茨的普通微积分并不完全是想象中的相似之处。有时，把一个关于数字的事实放在一个上下文中来证明它是最好的，在这个上下文中，数字是一个巨大的数字家族的一部分--一个数字函数。然后，函数的性质通过微积分的一般定理，在第一次遇到时可能看起来有点神奇，从而导致数字的计算。这里就是这样：有时，将关于某个几何定义的对象的事实放在这样一个对象的一个庞大家族--函数器--的上下文中，并使用某种更现代的魔法来证明它是最好的。这种类比可能会显示出作品的某种味道；内容更难理解，因为大多数有问题的“几何”物体都是通过相当长的抽象概念链与日常现实联系在一起的。

DMS-0204969Thomas G. GoodwillieWe propose to continue our investigations in several versions of "functorial calculus". Each of these is a technique that exploits multirelative connectivity estimates to describe continuous functors in some context in terms of special values; for example a functor of spaces might be recovered from its values at highly connected spaces, or a functor of subspaces of a manifold from its values at low-dimensional subspaces, ora functor of real inner product spaces from its values at high-dimensional spaces. Part of the proposal is to refine the purely homotopy-theoretic version of "calculus". Other parts are concerned with combining several versions and applying them to various questions in both high- andlow-dimensional differential topology.Each "functorial calculus" mentioned above is so called because of a not-entirely-fanciful resemblance to the ordinary diferential calculus of Newton and Leibniz. Sometimes a fact about numbers is best proved by placing it in a context where the number is part of a huge family ofnumbers -- a numerical function. Properties of the function then lead, by general theorems of calculus that may seem a bit magical when encountered for the first time, to a computation of the number. So it is here: sometimes a fact about some geometrically defined object is best proved byplacing it in a context where the object is part of a huge family of such objects -- a functor -- and using some magic of a more modern kind. This analogy may show something of the flavor of the work; the content is harder to get at, because most of the "geometric" objects in question areconnected to everyday reality by rather long chains of abstract ideas.