Simplicial Homotopy Theory, by P.G. Goerss and J.F. Jardine

This book is a modern presentation of the homotopy theory of simplicial sets. The manuscript is complete and online, in the form of dvi and postscript files which can be found here.

Table of Contents

• Chapter 1: 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

• Chapter 2: Model Categories
  1. Homotopical algebra
  2. Simplicial categories
  3. Simplicial model categories
  4. Detecting weak equivalences
  5. The existence of simplicial model category structures
  6. Examples of simplicial model categories
  7. A generalization of Theorem 4.1
  8. Quillen's total derived functor theorem
  9. Homotopy cartesian diagrams

• Chapter 3: Classical Results and Constructions
  1. The fundamental groupoid, revisited
  2. Simplicial abelian groups: the Dold-Kan correspondence
  3. The Hurewicz map
  4. The $Ex^{\infty}$-functor
  5. The Kan suspension

• Chapter 4: Bisimplicial Sets
  1. Bisimplicial sets: first properties
  2. Bisimplicial abelian groups
  3. Closed model structures for bisimplicial sets
  4. The Bousfield-Friedlander theorem
  5. Theorem B and group completion

• Chapter 5: Simplicial Groups
  1. Skeleta
  2. Principal fibrations I: simplicial G-spaces
  3. Principal fibrations II: classifications
  4. Universal cocycles and $\overline{W}G$
  5. The loop group construction
  6. Reduced simplicial sets, Milnor's FK construction
  7. Simplicial groupoids

• Chapter 6: The Homotopy Theory of Towers
  1. A model category structure for towers of spaces
  2. Postnikov towers
  3. Local coefficients and equivariant cohomology
  4. Generalities: equivariant cohomology
  5. On k-invariants
  6. Nilpotent spaces

• Chapter 7: Cosimplicial Spaces
  1. Decomposition of simplicial objects
  2. Reedy model category structures
  3. Geometric realization
  4. Cosimplicial spaces
  5. The total space of a cosimplicial space
  6. The homotopy spectral sequence of a tower of fibrations
  7. The homotopy spectral sequence of a cosimplicial space
  8. Obstruction theory

• Chapter 8: Simplicial Functors and Homotopy Coherence
  1. Simplicial functors
  2. The Dwyer-Kan theorem
  3. Homotopy coherence
  4. Realization theorems

• Chapter 9: Localization
  1. Localization with respect to a map
  2. The closed model category structure
  3. Bousfield localization
  4. Localization in simplicial model categories
  5. Localization in diagram categories
  6. A model for the stable homotopy category

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