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
Permutative categories as a model of connective stable homotopy
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. We aim to provide a more efficient way of processing multiplicative structure in the passage from permutative categories to spectra. In particular, we develop a multiplicative endomorphism operad for any permutative category, and show that under our passage to spectra, this endomorphism operad maps to the endomorphism operad of the associated spectrum. Together with the result that a permutative category supports bipermutative structure if and only if there is a map from a canonical E∞ operad into the multiplicative endomorphism operad, this gives a relatively simple proof that every bipermutative category gives rise to an E∞ ring spectrum. The other major result of our work so far is the development of a symmetric monoidal product on a certain category of permutative categories which captures all the homotopy information inherent in the category of all permutative categories.
This note describes some aspects of an on-going project to relate various structures on categories to corresponding structures on spectra. This is a preliminary report only; an exposition with full details is still in preparation.
One source of motivation for our project comes from a number of questions that Gunnar Carlsson asked about the K-theory spectra of permutative categories which are naturally raised by the development of associative smash products in [2]. First, we already knew that the spectrum given by a bipermutative category is an E∞ ring spectrum, or equivalently a commutative S-algebra.
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- Chapter
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- Structured Ring Spectra , pp. 19 - 32Publisher: Cambridge University PressPrint publication year: 2004
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