Sheaves in Homotopy Theory

Background and Motivation 1 1. Sheaves and Stacks 3 1.1. Sheaves 3 1.2. Stacks 4 2. Sheaves on Stacks 5 2.1. The Site of a Stack 5 2.2. The site (Mell)ét 6 2.3. Homotopy Limits and Sheaves of Spectra 7 2.3.1. Limits and Homotopy Limits 7 2.3.2. Derived Limits and Corrected Homotopy Limits 9 2.3.3. Sheaves of Orthogonal Spectra, Symmetric Spectra, and S-Modules 13 3. The Descent Spectral Sequence 15 3.1. Sheaf Cohomology and Cech Cohomology 16 3.2. The Spectral Sequence for a Sheaf of Spectra 17 3.2.1. The Spectral Sequence of a Filtration of Spectra and of a Tower of Spectra 17 3.2.2. The Realization of a Simplicial Spectrum and Tot of a Cosimplicial Spectrum 18 3.2.3. Cosheaves and Simplicial Spectra, and Sheaves and Cosimplicial Spectra 22 3.3. The spectral sequence for π∗TMF 24 Appendix. Degenerate simplicies and codegenerase cosimplicies 25 References 26