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

9703655 McCarthy Tom Goodwillie为同伦函子定义了泰勒塔。它们是从空间(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，级间的纤维是等变同调理论。对于所有空间或光谱，塔不一定恢复原始函子，但对于适当连接的物体则是这样。研究者研究这座塔的一种方式是研究另一座塔，它在给定函子的收敛半径内与泰勒塔一致，但与泰勒塔给出的普适n阶结构相比，它在第n层的定义特征是普适的n次结构。他还将继续通过将精确范畴的函子线性化到谱来研究代数K-理论。称从正合范畴到空间或谱的函子F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应用于取短正合序列的核和余核的正合函数给出的范畴的二重积.有一种通用的结构，称为“线性化”，它把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。当用这些新的塔楼重新解释时，这完全类似于取古德威利意义上的“导数”。代数K-理论本身就是一个自由函子从精确范畴到谱的线性化。拓扑Hochschild同调和拓扑循环同调也是函子线性化的例子，连接这些理论的迹映射可以从这个角度来研究。研究人员将研究通过线性化各种函子而获得的代数K理论的不同结构。希望得到能够有效地研究代数K-理论的新理论。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式。对于函子--一种广义函数--从空间到空间，Tom Goodwillie类似地定义了函子关于空间的泰勒级数展开。在标准分析中，必须假设一个函数有关于一个点的所有导数，以确保泰勒塔的存在，然后可以确定这个对原始函数的近似只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，必须对函子作出假设以确保其泰勒塔的存在，并且即使满足这些假设，也只能对足够接近展开空间的空间得到原始函子的准确估计。研究人员将研究古德威利对函子泰勒塔的原始定义从空间到空间的修改，该修改往往与古德威利对一点收敛半径内空间的定义一致，但总体上不同。这个新塔的两个优点是它更容易定义，而且它可以应用于更多有趣的情况。用这项新技术要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。***

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. ***