## 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 2002The 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. |

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### Contents

III | 3 |

V | 7 |

VI | 10 |

VII | 13 |

VIII | 22 |

IX | 24 |

X | 26 |

XI | 29 |

XLVI | 163 |

XLVII | 165 |

XLVIII | 168 |

XLIX | 169 |

L | 173 |

LI | 174 |

LII | 182 |

LIII | 189 |

XII | 30 |

XIII | 43 |

XIV | 51 |

XV | 55 |

XVI | 62 |

XVII | 66 |

XVIII | 69 |

XX | 72 |

XXI | 80 |

XXIII | 84 |

XXIV | 85 |

XXV | 87 |

XXVI | 89 |

XXVII | 91 |

XXVIII | 94 |

XXIX | 101 |

XXX | 102 |

XXXI | 116 |

XXXII | 129 |

XXXIII | 133 |

XXXIV | 135 |

XXXV | 140 |

XXXVI | 143 |

XXXVII | 146 |

XXXVIII | 147 |

XXXIX | 149 |

XL | 151 |

XLI | 154 |

XLII | 157 |

XLIII | 159 |

XLIV | 161 |

XLV | 162 |

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### Common terms and phrases

A'-theory abelian groups algebraic geometry associated Bousfield canonical cdga cdga's chain complex Chern classes closed imbedding cochain cofibre colimit colocalizations conjecture consider construction Corollary corresponding Euler structure cosimplicial space defined Definition denote derived category derived moduli diagram element Euler class exact sequence example finite formal group law functor Galois geometric D-stack given Grothendieck groupoids homology homomorphism homotopy category homotopy groups induced inverse Todd genus invertible isomorphism K-theory Lemma line bundle localisation loop space map f Math model category modules moduli spaces morphism f motivic nilpotent Nisnevich notation object oriented cohomology pretheory perfect integration preprint presheaf presheaf of spectra prime projective bundle Proposition proved pull-back representable Riemann-Roch ring cohomology pretheory ring morphism satisfies schemes Section sheaf sheaves simplicial presheaves simplicial set smooth varieties spectral sequence spectrum stable homotopy theory strongly geometric subcategory symmetric monoidal Todd genus triangulated category vector bundle Voevodsky weak equivalence