Inhalt

I Simplicial sets.- 1. Basic definitions.- 2. Realization.- 3. Kan complexes.- 4. Anodyne extensions.- 5. Function complexes.- 6. Simplicial homotopy.- 7. Simplicial homotopy groups.- 8. Fundamental groupoid.- 9. Categories of fibrant objects.- 10. Minimal fibrations.- 11. The closed model structure.- II Model Categories.- 1. Homotopical algebra.- 2. Simplicial categories.- 3. Simplicial model categories.- 4. The existence of simplicial model category structures.- 5. Examples of simplicial model categories.- 6. A generalization of Theorem 4.1.- 7. Quillen's total derived functor theorem.- 8. Homotopy cartesian diagrams.- III Classical results and constructions.- 1. The fundamental groupoid, revisited.- 2. Simplicial abelian groups.- 3. The Hurewicz map.- 4. The Ex? functor.- 5. The Kan suspension.- IV Bisimplicial sets.- 1. Bisimplicial sets: first properties.- 2. Bisimplicial abelian groups.- 2.1. The translation object.- 2.2. The generalized Eilenberg-Zilber theorem.- 3. Closed model structures for bisimplicial sets.- 3.1. The Bousfield-Kan structure.- 3.2. The Reedy structure.- 3.3. The Moerdijk structure.- 4. The Bousfield-Friedlander theorem.- 5. Theorem B and group completion.- 5.1. The Serre spectral sequence.- 5.2. Theorem B.- 5.3. The group completion theorem.- V Simplicial groups.- 1. Skeleta.- 2. Principal fibrations I: simplicial G-spaces.- 3. Principal fibrations II: classifications.- 4. Universal cocycles and $$ \bar W $$G.- 5. The loop group construction.- 6. Reduced simplicial sets, Milnor's FK-construction.- 7. Simplicial groupoids.- VI The homotopy theory of towers.- 1. A model category structure for towers of spaces.- 2. The spectral sequence of a tower of fibrations.- 3. Postnikov towers.- 4. Local coefficients and equivariant cohomology.- 5. On k-invariants.- 6. Nilpotent spaces.- VII Reedy model categories.- 1. Decomposition of simplicial objects.- 2. Reedy model category structures.- 3. Geometric realization.- 4. Cosimplicial spaces.- VIII Cosimplicial spaces: applications.- 1. The homotopy spectral sequence of a cosimplicial space.- 2. Homotopy inverse limits.- 3. Completions.- 4. Obstruction theory.- IX Simplicial functors and homotopy coherence.- 1. Simplicial functors.- 2. The Dwyer-Kan theorem.- 3. Homotopy coherence.- 3.1. Classical homotopy coherence.- 3.2. Homotopy coherence: an expanded version.- 3.3. Lax functors.- 3.4. The Grothendieck construction.- 4. Realization theorems.- X Localization.- 1. Localization with respect to a map.- 2. The closed model category structure.- 3. Bousfield localization.- 4. A model for the stable homotopy category.- References.