Calculus of Homotopy Functors, Algebraic K-theory and Universal Constructions of Finite Degree

同伦函子微积分、代数K理论和有限度泛结构

• 批准号: 9703655
• 负责人: Randy McCarthy
• 金额: $ 7.5万
• 依托单位: University of Illinois at Urbana-Champaign
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 1997
• 资助国家: 美国
• 起止时间: 1997-08-15 至 2000-07-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=9703655&HistoricalAwards=false
• 关键词: Calculus; Homotopy; Functors; Algebraic; theory

9703655 McCarthy Tom Goodwillie defined a Taylor tower for homotopy functors. These are functors from spaces (or spectra) to spaces (or spectra) and they must preserve homotopy equivalences. The tower is an inverse limit of functors with the fibers between stages being equivariant homology theories. The tower does not necessarily recover the original functor for all spaces or spectra but does for suitably connected objects. One way in which the investigator will study this tower is by studying another tower that agrees with the Taylor tower within the radius of convergence for a given functor but whose defining characteristic at the n-th level is the universal degree n construction as compared to the universal n-excisive construction given by the Taylor tower. He will also continue to study algebraic K-theory by use of linearizing functors of exact categories to spectra. A functor F from exact categories to spaces or spectra is said to satisfy ``additivity'' if F applied to the exact category of short exact sequences of a category is naturally equivalent to the two-fold product of F applied to the category given by the exact functors that take kernel and cokernel of a short exact sequence. There is a universal construction, called ``linearizing,'' which takes an arbitrary functor from exact categories to spectra and produces a new functor of exact categories that satisfies additivity. This is completely analogous to taking the ``derivative'' in the sense of Goodwillie when reinterpreted with these new towers. Algebraic K-theory itself is the linearization of a free functor from exact categories to spectra. Topological Hochschild homology and topological cyclic homology are also examples of linearizing a functor, and the trace maps that connect these theories may be studied from this point of view. The investigator will be examining different constructions for the study of algebraic K-theory obtained by linearizing various types of functors. The hope is to obtain new theories that one can use to study algebraic K-theory effectively. An important tool for studying functions from the complex numbers to itself is the Taylor series expansion of the function about a point. For a functor -- a kind of generalized function -- from spaces to spaces, Tom Goodwillie has similarly defined a Taylor series expansion of the functor about a space. In standard analysis one must assume a function has all its derivatives about a point to ensure that the Taylor tower exists, and then one can be sure that this approximation to the original function is accurate only within a radius of convergence about the point. Similarly, for functors of spaces, one must make assumptions about the functor to ensure its Taylor tower exists, and even when these are satisfied, one obtains accurate estimates of the original functor only for spaces sufficiently close to the space of expansion. The investigator will be examining a modification of Goodwillie's original definition for the Taylor tower of functors from spaces to spaces that tends to agree with his definition for spaces within the radius of convergence of a point but differs in general. Two advantages of this new tower are that it is easier to define and that it can be applied to an even greater variety of interesting situations. One area to be explored with this new technology is algebraic K-theory, which from this point of view is simply the derivative of a particularly easy functor. ***

小行星9703655 汤姆·古德威利为同伦函子定义了泰勒塔。 这些是从空间（或谱）到空间（或谱）的函子，它们必须保持同伦等价。 塔是函子的逆极限，阶段之间的纤维是等变同调理论。 塔并不一定恢复所有空间或谱的原始函子，但对适当连接的对象。 研究者研究这个塔的一种方法是研究另一个塔，这个塔在给定函子的收敛半径内与泰勒塔一致，但与泰勒塔给出的泛n-排除结构相比，它在n层的定义特征是泛n次结构。 他还将继续研究代数K-理论的使用线性函子的确切类别的频谱。 从正合范畴到空间或谱的函子F被称为满足“可加性”，如果F应用于范畴的短正合序列的正合范畴自然等价于F应用于由取短正合序列的核和上核的正合函子给出的范畴的二重积。 有一个通用的构造，称为“线性化”，它将一个任意函子从精确范畴带到谱，并产生一个满足可加性的精确范畴的新函子。 这完全类似于在古德威利的意义上，当重新解释这些新的塔楼时，采取“派生”。 代数K-理论本身就是一个自由函子从正合范畴到谱的线性化。 拓扑Hochschild同调和拓扑循环同调也是线性化函子的例子，连接这些理论的迹映射可以从这个角度来研究。 研究人员将检查不同的结构，通过线性化各种类型的函子获得的代数K理论的研究。 希望是获得新的理论，可以用来有效地研究代数K-理论。 研究函数从复数到自身的一个重要工具是函数关于一点的泰勒级数展开。 对于一个函子--一种广义函数--从空间到空间，Tom Goodwillie同样定义了函子关于空间的泰勒级数展开。 在标准分析中，必须假设一个函数的所有导数都是关于一点的，以确保泰勒塔存在，然后可以确定对原始函数的这种近似仅在关于该点的收敛半径内是准确的。 类似地，对于空间的函子，必须对函子做出假设以确保其泰勒塔存在，即使满足这些假设，也只能对足够接近展开空间的空间获得原始函子的准确估计。 调查将审查修改古德威利的原始定义泰勒塔的函子从空间到空间，往往同意他的定义为空间内的半径收敛的一个点，但不同的一般。 这种新塔的两个优点是，它更容易定义，它可以应用于各种有趣的情况。 用这种新技术探索的一个领域是代数K理论，从这个角度来看，它只是一个特别容易的函子的导数。 ***