RUI: Koszul duality of operads and the calculus of functors

本项目结合了当前同伦理论研究的两个主要领域：古德威利的同伦函子演算和操作理论。总体目标是理解函子的古德威利导数所具有的普遍结构，由此可以重构函子的泰勒塔。PI和Greg Arone证明了这种结构在一定公数上以协代数的形式存在。在谱值函子的情况下，这个公函数与各种操作数（包括小圆盘操作数）上的右模相关的公函数之间有密切的关系。我们现在研究空间值函子的公式，以及在任意基对象上的导数。我们也将以前的理论应用于计算，如代数k理论的泰勒塔。谱算子的Koszul对偶在这个理论中起着关键作用，我们在PI和John E. Harper之前的工作的基础上，研究了这种对偶性。这里的主要目标是得到一个运算符上的代数范畴和Koszul对偶运算符上的除幂代数范畴之间的等价性。拓扑学是研究任意维数的形状和空间性质的学科。一个特定的目标是开发方法来衡量这些空间的各个方面，这些方面通常被认为是定性的。例如，一个基本问题是精确描述地球表面（一个球体）和百吉饼表面（一个环面）形成的形状之间的差异。从直觉上看，两者的区别很明显——百吉饼上有一个洞——但对形状上的“洞”给出一个精确的定义，可以让我们在更高的维度上进行计算，而在更高的维度上，直觉不太可靠。其中一些计算（所谓的“球的同伦群”）被证明是极其复杂的，并且已经被广泛研究过。这个项目关注的是这些计算的系统近似，类似于本科微积分的泰勒级数。我们的中心目标是理解如何以一种自然的方式，从更简单的部分构建更复杂的计算。对这些部分的良好理解将为我们提供更好的工具来进行困难的计算。我们的方法的基本性质使其能够应用于一系列数学领域，包括同调代数和表示理论，以及拓扑学。

This project combines two principal areas of current research in homotopy theory: Goodwillie's calculus of homotopy functors and operad theory. The overall goal is to understand the universal structure possessed by the Goodwillie derivatives of a functor, from which the Taylor tower of the functor can be reconstructed. The PI and Greg Arone have proved the existence of such a structure in the form of a coalgebra over a certain comonad. In the case of spectrum-valued functors, there is a close relationship between this comonad and those associated to right modules over various operads, including the little disc operads. We now study the corresponding comonads for space-valued functors, as well as for derivatives at arbitrary base objects. We also apply our previous theory to calculations such as the Taylor tower of algebraic K-theory. The Koszul duality of operads of spectra plays a key role in this theory, and we study this duality in its own right, building on previous work of the PI and John E. Harper. The main goal here is to get an equivalence between the categories of algebras over one operad and of divided power coalgebras over the Koszul dual operad.Topology is the study of properties of shapes and spaces in any number of dimensions. One particular goal is to develop ways to measure aspects of these spaces that are usually considered qualitative. For example, a basic problem is to give a precise description of the difference between the shapes formed by the surface of the Earth (a sphere), and the surface of a bagel (a torus). The difference is intuitively clear - the bagel has a hole - but giving a precise definition of what we mean by a 'hole' in a shape allows us to make calculations in higher dimensions, where intuition is less reliable. Some of these calculations (the so-called 'homotopy groups of spheres') turn out to be extremely complicated and have been studied extensively. This project is concerned with systematic approximations to these calculations that are analogous to the Taylor series of undergraduate calculus. Our central goal is to understand how more complex calculations can be built, in a natural way, from simpler pieces. A good understanding of the pieces should then provide us with better tools for making the hard computations. The foundational nature of our approach lends itself to applications in a range of areas of mathematics, including homological algebra and representation theory, in addition to topology.