Greenlees, J. P. C. (ed.), Axiomatic, enriched and motivic homotopy theory. Proceedings of the NATO Advanced Study Institute, Cambridge, UK, September 9–20, 2002. Dordrecht: Kluwer Academic Publishers (ISBN 1-4020-1833-9/hbk). NATO Science Series II: Mathematics, Physics and Chemistry 131, 29-68 (2004).

The paper is an expanded version of notes for a set of lectures given at the Isaac Newton Institute for Mathematical Sciences (September 23 and 24, 2002). Its aim is to present some of the basic features of the homotopy theory of simplicial presheaves and the stable homotopy theory of presheaves of spectra, by specifying their use in applications. The sections of this paper are the following: 1. Simplicial presheaves; 2. Presheaves of spectra; 3. Profinite groups; 4. Generalized Galois cohomology theory; 5. Thomason’s descent theorem.

##### MSC:

55N30 | Sheaf cohomology (algebraic topology) |

55N20 | Generalized homology and cohomology theories (algebraic topology) |

55N35 | Other homology theories (algebraic topology) |

55P42 | Stable homotopy theory, spectra |