Calculus of Functors

DMS-0204969Thomas G.我们建议继续我们的调查在几个版本的“函演算”。这些都是一种技术，利用多相对连通性估计来描述连续函子在某些情况下的特殊值;例如，空间的函子可以从其在高连通空间的值中恢复，或者流形的子空间的函子可以从其在低维子空间的值中恢复，或者真实的内积空间的函子可以从其在高维空间的值中恢复。部分建议是完善纯粹同伦理论版本的“微积分”。其他部分涉及到结合几个版本，并将其应用于各种问题，在高-和低-维微分拓扑。每个“函演算”上面提到的是所谓的，因为一个不完全是幻想的相似之处，普通微分演算牛顿和莱布尼茨。有时候，一个关于数字的事实最好的证明是把它放在一个上下文中，这个数字是一个巨大的数字家族的一部分-一个数字函数。函数的性质，然后导致，一般定理的微积分，可能似乎有点神奇时，遇到的第一次，以计算的数量。所以它是这样的：有时候，一个关于几何定义的对象的事实，最好的证明方法是把它放在一个上下文中，这个对象是一个庞大的此类对象家族的一部分--一个函子--并使用一些更现代的魔法。这种类比可能显示出作品的某种风格;内容更难理解，因为大多数被讨论的“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.