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. ***

9703655 McCarthy Tom Goodwillie定义了一个用于同型函子的泰勒塔。 这些是从空间（或光谱）到空间（或光谱）的函子，它们必须保留同质等效。 该塔是函子的反向极限，阶段之间的纤维是模棱两可的同源性理论。 该塔并不一定会为所有空间或光谱恢复原始函子，而是用于适当连接的对象。 研究人员研究该塔的一种方法是研究另一个与给定函子相同的泰勒塔的塔，但与泰勒塔（Taylor Tower）给出的通用的N-效能结构相比，其在n级的定义特征是通用度n结构。 他还将通过使用与光谱的精确类别的线性化函数来研究代数K理论。 如果将f应用于类别的短序列的精确类别，则据说从精确类别到空间或光谱的函数f可以满足``添加性''，这自然等于f应用于f的两倍乘积，该类别是由将kernel和Short Short精确序列的kernel和Cokernel提供的确切函数给出的类别。 有一个通用的结构，称为``线性化''，它将任意函数从精确的类别中带到光谱，并产生一个满足添加性的确切类别的新函数。 这完全类似于在重新解释这些新塔时，从善意的意义上讲``衍生物''。 代数K理论本身是从精确类别到光谱的免费函子的线性化。 拓扑Hochschild同源性和拓扑循环同源性也是线性化函子的示例，并且可以从这个角度研究连接这些理论的痕量图。 研究者将检查通过线性化各种函子来获得的代数K理论研究的不同结构。 希望是获得可以有效研究代数K理论的新理论。 研究从复数到本身的功能的重要工具是该功能的泰勒级数扩展大约一个点。 对于从空间到空间的函子（一种广义函数），汤姆·古德里莉（Tom Goodwillie）类似地定义了函数围绕一个空间的泰勒级数扩展。 在标准分析中，必须假设一个函数具有大约一个点的所有衍生物，以确保存在泰勒塔，然后可以确保对原始函数的近似值仅在围绕该点的收敛半径内准确。 类似地，对于空间的函子，必须对函子进行假设，以确保其泰勒塔存在，即使满足这些函数，也只能获得原始函数的准确估计，仅对于足够接近扩展空间的空间。 研究人员将研究对泰勒塔的原始定义的修改，从空间到空间的泰勒塔，这些定义倾向于与他对点融合半径范围内的空间的定义一致，但一般而言。 这个新塔的两个优点是，它更容易定义，并且可以应用于更多有趣的情况。 该新技术要探索的一个领域是代数K理论，从这个角度来看，这仅仅是一个特别容易的函子的派生。 ***