Axiomatic, Enriched and Motivic Homotopy Theory

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