Paul G. Goerss, John F. Jardine

Simplicial Homotopy Theory (eBook, PDF)

• Format: PDF

Produktbeschreibung

Since the beginning of the modern era of algebraic topology, simplicial methods have been used systematically and effectively for both computation and basic theory. With the development of Quillen's concept of a closed model category and, in particular, a simplicial model category, this collection of methods has become the primary way to describe non-abelian homological algebra and to address homotopy-theoretical issues in a variety of fields, including algebraic K-theory. This book supplies a modern exposition of these ideas, emphasizing model category theoretical techniques.

Discussed here are the homotopy theory of simplicial sets, and other basic topics such as simplicial groups, Postnikov towers, and bisimplicial sets. The more advanced material includes homotopy limits and colimits, localization with respect to a map and with respect to a homology theory, cosimplicial spaces, and homotopy coherence. Interspersed throughout are many results and ideas well-known to experts, but uncollected in the literature.

Intended for second-year graduate students and beyond, this book introduces many of the basic tools of modern homotopy theory. An extensive background in topology is not assumed.

Reviews:

"… a book filling an obvious gap in the literature and the authors have done an excellent job on it. No monograph or expository paper has been published on this topic in the last twenty-eight years." - Analele Universitatii din Timisoara

"… is clearly presented and a brief summary preceding every chapter is useful to the reader. The book should prove enlightening to a broad range of readers including prospective students and researchers who want to apply simplicial techniques for whatever reason." - Zentralblatt MATH

"… they succeed. The book is an excellent account of simplicial homotopy theory from a modern point of view […] The bookis well written. […] The book can be highly recommended to anybody who wants to learn and to apply simplicial techniques and/or the theory of (simplicial) closed model categories." - Mathematical Reviews

Produktdetails

• Verlag: Springer Basel
• Erscheinungstermin: 5. Dezember 2009
• Englisch
• ISBN-13: 9783034601894
• Artikelnr.: 37351590

Inhaltsangabe

Simplicial sets.- Model Categories.- Classical results and constructions.- Bisimplicial sets.- Simplicial groups.- The homotopy theory of towers.- Reedy model categories.- Cosimplicial spaces: applications.- Simplicial functors and homotopy coherence.- Localization.

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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.- 3. Closed model structures for bisimplicial sets.- 4. The Bousfield-Friedlander theorem.- 5. Theorem B and group completion.- 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 homotopyspectral 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.- 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.