### 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

- Basic definitions
- Realization
- Kan complexes
- Anodyne extensions
- Function complexes
- Simplicial homotopy
- Simplicial homotopy groups
- Fundamental groupoid
- Categories of fibrant objects
- Minimal fibrations
- The closed model structure

- Chapter 2: Model Categories

- Homotopical algebra
- Simplicial categories
- Simplicial model categories
- Detecting weak equivalences
- The existence of simplicial model category structures
- Examples of simplicial model categories
- A generalization of Theorem 4.1
- Quillen's total derived functor theorem
- Homotopy cartesian diagrams

- Chapter 3: Classical Results and Constructions

- The fundamental groupoid, revisited
- Simplicial abelian groups: the Dold-Kan correspondence
- The Hurewicz map
- The $Ex^{\infty}$-functor
- The Kan suspension

- Chapter 4: Bisimplicial Sets

- Bisimplicial sets: first properties
- Bisimplicial abelian groups
- Closed model structures for bisimplicial sets
- The Bousfield-Friedlander theorem
- Theorem B and group completion

- Chapter 5: Simplicial Groups

- Skeleta
- Principal fibrations I: simplicial G-spaces
- Principal fibrations II: classifications
- Universal cocycles and $\overline{W}G$
- The loop group construction
- Reduced simplicial sets, Milnor's FK construction
- Simplicial groupoids

- Chapter 6: The Homotopy Theory of Towers

- A model category structure for towers of spaces
- Postnikov towers
- Local coefficients and equivariant cohomology
- Generalities: equivariant cohomology
- On k-invariants
- Nilpotent spaces

- Chapter 7: Cosimplicial Spaces

- Decomposition of simplicial objects
- Reedy model category structures
- Geometric realization
- Cosimplicial spaces
- The total space of a cosimplicial space
- The homotopy spectral sequence of a tower of fibrations
- The homotopy spectral sequence of a cosimplicial space
- Obstruction theory

- Chapter 8: Simplicial Functors and Homotopy Coherence

- Simplicial functors
- The Dwyer-Kan theorem
- Homotopy coherence
- Realization theorems

- Chapter 9: Localization

- Localization with respect to a map
- The closed model category structure
- Bousfield localization
- Localization in simplicial model categories
- Localization in diagram categories
- A model for the stable homotopy category

P.G. Goerss <pgoerss@math.washington.edu>

J.F. Jardine <jardine@jardine.math.uwo.ca>