Abstract Homotopy Theory

University Of Illinois At Urbana-Champaign, Urbana IL

Investigators

Abstract

DMS-0203936 Charles W. Rezk The 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), and in 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) by giving 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 to traditional algebraic problems.