Book contents
- Frontmatter
- Contents
- Preface
- The development of structured ring spectra
- Compromises forced by Lewis's theorem
- Permutative categories as a model of connective stable homotopy
- Morita theory in abelian, derived and stable model categories
- Higher coherences for equivariant K-theory
- (Co-)Homology theories for commutative (S-)algebras
- Classical obstructions and S-algebras
- Moduli spaces of commutative ring spectra
- Cohomology theories for highly structured ring spectra
- Index
The development of structured ring spectra
Published online by Cambridge University Press: 23 October 2009
- Frontmatter
- Contents
- Preface
- The development of structured ring spectra
- Compromises forced by Lewis's theorem
- Permutative categories as a model of connective stable homotopy
- Morita theory in abelian, derived and stable model categories
- Higher coherences for equivariant K-theory
- (Co-)Homology theories for commutative (S-)algebras
- Classical obstructions and S-algebras
- Moduli spaces of commutative ring spectra
- Cohomology theories for highly structured ring spectra
- Index
Summary
Abstract. The problem of giving a succinct description of multiplicative structure on spectra was recognized almost as soon as the idea of a spectrum was formulated. This paper aims to describe the major features of the historical precursors to the S-module approach of [2]. In particular, we consider the purely homotopical notion of a ring spectrum, May's concepts of external smash product and its internalization, the Lewis-May twisted half-smash product, and this product's use in formulating May and Quinn's notion of an E∞ ring spectrum. We then describe how three essentially trivial (but crucial) observations led to the idea of an v-spectrum, and soon thereafter to S-modules. We conclude by describing the good formal and homotopical properties of the category of S-modules.
The aim of this paper is to give some historical background to the first of the modern treatments of structured ring spectra: the S-module approach of [2]. There have been subsequent models developed as well; I'd like to mention in particular the symmetric spectra originally developed by Jeff Smith [3] and the orthogonal spectra of Mandell and May ([6] and [7]). There has been quite a lot of work done relating these various approaches, but this paper is concerned with the S-module approach only.
The invention of spectra, in the sense of algebraic topology, is usually credited to Lima in the late 1950's, although the first definition in print appears to be Spanier's [10].
- Type
- Chapter
- Information
- Structured Ring Spectra , pp. 7 - 14Publisher: Cambridge University PressPrint publication year: 2004