Abstract Homotopy Theory

本课题的目标是将同伦理论方法应用于代数问题。这里涉及到两条思路。一种是将同伦理论描述为一种拓扑丰富的范畴理论，由Dwyer-Kan等人提出。另一条线索是交换环谱理论（交换环理论的同伦理论推广），特别是环的代数de Rham复形的拓扑版本的发展。同伦理论是拓扑学的一个分支；它起源于对空间某些不变性质的研究，即那些连续变形所留下的不变性质。令人惊讶的是，在空间的同伦理论（似乎是几何和拓扑的）和代数的几个分支（如同伦代数、层论和范畴论）的方法之间有很深的相似之处。例如，我们通过给出生成器和它们之间的关系来描述代数对象（如群或环）的方式，与我们通过给出多面体和它们连接在一起的方式来描述拓扑空间的方式有着密切的相似之处。本课题的目标是在环论和范畴论的背景下发展某些同伦理论方法，并使它们能够应用于传统的代数问题。

DMS-0203936Charles W. RezkThe goal of this project is to apply homotopy theoretic methods to algebraic problems. There are two threads of ideas involved here. One is the description of homotopy theory as a kind of topologically enriched category theory, as has been developed by Dwyer-Kan and many others. Another thread is in the theory of commutative ring spectra (a homotopy theoretic generalization of commutative ring theory), andin particular the development of a topological version of the algebraic de Rham complex of a ring.Homotopy theory is a branch of topology; it arose as the study of certain invariant properties of spaces, namely those left unchanged by continuous deformations. It is a surprising fact that there are deep analogies between the methods of the homotopy theory of spaces (which would seem to be geometric and topological) and those of several branches of algebra (such as homological algebra, sheaf theory, and category theory). For example, there is a close analogy between the way we can describe an algebraic object (such as a group or ring) bygiving generators and the relations between them, and the way we can describe a topological space by giving polyhedra and the way they are attached together to give the space. The goal of this project is to develop certain homotopy theoretic methods in the context of ring theory and of category theory, and to allow them to be applied totraditional algebraic problems.