RUI: Koszul duality of operads and the calculus of functors

该项目结合了同伦理论目前研究的两个主要领域：古德威利的同伦函子演算和运算理论。总的目标是了解函子的古德威利导数所具有的普遍结构，由此可以重建函子的泰勒塔。PI和Greg Arone已经证明了这样一个结构的存在性，其形式为某个comonad上的余代数。在谱值函子的情况下，这个余元与各种操作数（包括小圆盘操作数）上的右模相关的余元之间有密切的关系。现在我们研究空间值函子的相应余子，以及任意基对象的导数。我们还将我们以前的理论应用于计算，如代数K理论的泰勒塔。谱操作的Koszul对偶在这个理论中起着关键作用，我们在PI和John E.哈珀。这里的主要目标是得到一个等价的范畴之间的代数在一个operad和除幂余代数在Koszul对偶operad。拓扑学是研究性质的形状和空间在任何数量的层面。一个特别的目标是制定方法来衡量这些空间的方面，通常被认为是定性的。例如，一个基本的问题是给出地球表面（球体）和百吉饼表面（环面）形成的形状之间的差异的精确描述。这种区别在直觉上是显而易见的--百吉饼上有一个洞--但是对形状上的“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，直觉是不太可靠的。其中一些计算（所谓的“球面同伦群”）被证明是极其复杂的，并已被广泛研究。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。我们的中心目标是了解如何以自然的方式从简单的部分构建更复杂的计算。对这些部分的良好理解应该为我们提供更好的工具来进行困难的计算。我们的方法的基础性质适合于在一系列数学领域的应用，包括同调代数和表示论，除了拓扑。

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.