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Axiomatic, Enriched and Motivic Homotopy Theory : Proceedings of the NATO Advanced Study Institute on Axiomatic, Enriched and Motivic Homotopy Theory Cambridge, United Kingdom 9-20 September 2002 - John Greenlees

Axiomatic, Enriched and Motivic Homotopy Theory

Proceedings of the NATO Advanced Study Institute on Axiomatic, Enriched and Motivic Homotopy Theory Cambridge, United Kingdom 9-20 September 2002

By: John Greenlees (Editor)

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Published: 31st January 2004
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The NATO Advanced Study Institute "Axiomatic, enriched and rna- tivic homotopy theory" took place at the Isaac Newton Institute of Mathematical Sciences, Cambridge, England during 9-20 September 2002. The Directors were J.P.C.Greenlees and I.Zhukov; the other or- ganizers were P.G.Goerss, F.Morel, J.F.Jardine and V.P.Snaith. The title describes the content well, and both the event and the contents of the present volume reflect recent remarkable successes in model categor- ies, structured ring spectra and homotopy theory of algebraic geometry. The ASI took the form of a series of 15 minicourses and a few extra lectures, and was designed to provide background, and to bring the par- ticipants up to date with developments. The present volume is based on a number of the lectures given during the workshop. The ASI was the opening workshop of the four month programme "New Contexts for Stable Homotopy Theory" which explored several themes in greater depth. I am grateful to the Isaac Newton Institute for providing such an ideal venue, the NATO Science Committee for their funding, and to all the speakers at the conference, whether or not they were able to contribute to the present volume. All contributions were refereed, and I thank the authors and referees for their efforts to fit in with the tight schedule. Finally, I would like to thank my coorganizers and all the staff at the Institute for making the ASI run so smoothly. J.P.C.GREENLEES.

Contributing Authors
Preface
General surveys
Localizations
Introduction
Algebra
Homological localization in topology
Localization with respect to a map
Colocalization with respect to an object
Higher invariants of localization
Constructing localizations and colocalizations
References
Generalized sheaf cohomology theories
Simplicial presheaves
Presheaves of spectra
Profinite groups
Generalized Galois cohomology theory
Thomason's descent theorem
References
Axiomatic stable homotopy
Introduction
Axioms
Functors on small objects
Types of subcategories
Quotient categories and Blousfield localization
Versions of the Blousfield lattice
Special types of localization
Nilpotence
Brown representability
References
Special surveys. (Pre-)sheaves of ring spectra
The realization problem
Moduli spaces and obstruction theory
References
Operads and cosimplicial objects: an introduction
Introduction
Loop lattices and the little intervals operad
Cosimplicial objects and totalization
A sufficient condition for Tot(X) to be an AINFINITY space
A reformulation
Operads
A family of cochain operations
A sufficient condition for Tot(X.) to be an E8 space
The little n-cubes operad
A sufficient condition for Tot(X.) to be an En space
An extension of Remark 6.3 (b)
Proof of Theorem 10.6
Applications
The framed little disks operad
Cosimplicial chain complexes
Applications
References
From HAG to DAG: derived moduli stacks
Introduction
The model category of D-stacks
First examples of D-stacks
The geometry of D-stacks
Further examples
References
Motivic homotopy theory
On the motivic &pgr;0 of the sphere spectrum
Introduction
From smooth varieties to "spaces"
Stable homotopy categories of S1-spectra
The &mathA;1-homotopy t-structure and the stable Hurewicz theorem
Inverting &mathP
&pgr;0(S0) and Milnor-Witt K theory of fields
References
Riemann-Roch Theorems for oriented cohomology
Introduction
Oriented cohomology pretheories
Riemann-Roch type theorems
References
Equivariant motivic phenomena
Introduction
Classical motives
Equivariant Motives
The Gross-Stark conjecture
The fractional ideal rF
New conjectures
Supporting evidence
Annihilators and K0(Z[G], Ql)
The Lichtenbaum conjecture
Relation to Vandiver's conjecture
References
A road map of motivic homotopy and homology theory
The road map
Motivic homotopy theory
Motivic homology theory
Stable motivic complexes
The Hurewicz functor
References
Table of Contents provided by Publisher. All Rights Reserved.

ISBN: 9781402018336
ISBN-10: 1402018339
Series: NATO Science Series II
Audience: Professional
Format: Hardcover
Language: English
Number Of Pages: 392
Published: 31st January 2004
Publisher: Springer-Verlag New York Inc.
Country of Publication: US
Dimensions (cm): 23.5 x 15.5  x 2.54
Weight (kg): 1.65