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.

Inhalt

Contributing Authors. Preface. Part I: General surveys. Localizations; W.G. Dwyer. 1. Introduction. 2. Algebra. 3. Homological localization in topology. 4. Localization with respect to a map. 5. Colocalization with respect to an object. 6. Higher invariants of localization. 7. Constructing localizations and colocalizations. References. Generalized sheaf cohomology theories; J.F. Jardine. 1. Simplicial presheaves. 2. Presheaves of spectra. 3. Profinite groups. 4. Generalized Galois cohomology theory. 5. Thomason's descent theorem. References. Axiomatic stable homotopy; N.P. Strickland. 1. Introduction. 2. Axioms. 3. Functors on small objects. 4. Types of subcategories. 5. Quotient categories and Blousfield localization. 6. Versions of the Blousfield lattice. 7. Special types of localization. 8. Nilpotence. 9. Brown representability. References. Part II: Special surveys. (Pre-)sheaves of ring spectra; P.C. Goerss. 1. The realization problem. 2. Moduli spaces and obstruction theory. References. Operads and cosimplicial objects: an introduction; J.E. McClure, J.H. Smith. 1. Introduction. 2. Loop lattices and the little intervals operad. 3. Cosimplicial objects and totalization. 4. A sufficient condition for Tot(X·) to be an AINFINITY space. 5. A reformulation. 6. Operads. 7. A family of cochain operations.8. A sufficient condition for Tot(X.) to be an E8 space. 9. The little n-cubes operad. 10. A sufficient condition for Tot(X.) to be an En space. 11. An extension of Remark 6.3(b). 12. Proof of Theorem 10.6. 13. Applications. 14. The framed little disks operad. 15. Cosimplicial chain complexes. 16. Applications. References. From HAG to DAG: derived moduli stacks; B. Toën, G. Vezzosi. 1. Introduction. 2. The model category of D-stacks. 3. First examples of D stacks. 4. The geometry of D-stacks. 5. Further examples. References. Part III: Motivic homotopy theory. On the motivic &pgr;0 of the sphere spectrum; F. Morel. 1. Introduction. 2. From smooth varieties to 'spaces'. 3. Stable homotopy categories of S1-spectra. 4. The &mathA;1-homotopy t-structure and the stable Hurewicz theorem. 5. Inverting &mathP;1. &pgr;0(S0) and Milnor-Witt K theory of fields. References. Riemann-Roch Theorems for oriented cohomology; I. Panin (after I. Panin, A. Smirnov). 0. Introduction. 1. Oriented cohomology pretheories. 2. Riemann-Roch type theorems. References. Equivariant motivic phenomena; V. Snaith. 1. Introduction. 2. Classical motives. 3. Equivariant Motives. 4. The Gross-Stark conjecture. 5. The fractional ideal