自从开启了代数拓扑现代领域的新纪元，单纯方法已经在计算方面和基本理论方面很系统化并且有效化。随着Quillen的封闭模型类，特别是单纯模型类概念的发展，这个方法已经成为描述非阿贝尔同伦代数的最基本方法，也是表述众多领域，包括K理论的同伦理论观点的重要途径。格兹著的《单纯同伦理论》成为这些观点的一个重要补充表述，强调了模型类理论技巧。

Chapter l 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 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

Chapter 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

Chapter 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 

Chapter V Simplicial groups

 1．Skeleta 

 2．Principal fibrations I：simplicial G-spaces 

 3．Principal fibrations II：classifications 

 4．Universal cocycles and WG 

 5．The loop group construction

 6．Reduced simplicial sets，Milnor’S FK-construction 

 7．Simplicial groupoids

Chapter 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

Chapter VII Reedy model categories

 1．Decomposition of simplicial objects

 2．Reedy model category structures

 3．Geometric realization

 4．Cosimplicial spaces

Chapter VIII Cosimplicial spaces：applications

 1．The homotopy spectral sequence of a cosimplicial space

 2．Homotopy inverse limits

 3．Completions

 4．Obstruction theory

Chapter 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 

Chapter 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

Index