The final prices may differ from the prices shown due to specifics of VAT rules About this book Since the beginning of the modern era of algebraic topology, simplicial methods have been used systematically and effectively for both computation and basic 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.

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The final prices may differ from the prices shown due to specifics of VAT rules About this book Since the beginning of the modern era of algebraic topology, simplicial methods have been used systematically and effectively for both computation and basic 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. 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.

The book is an excellent account of simplicial homotopy theory from a modern point of view […] The book is well written.

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## Simplicial set

These theorems assert that the category of simplicial sets satisfies Quillens axioms for a closed model category, and that the associated homotopy category is equivalent to that arising from topological spaces. They are delicate but central results, and are the basis for all that follows. Chapter I contains the definition of a closed model category. The foundations of abstract homotopy theory, as given by Quillen, start to appear in the first section of Chapter II. The simplicial model structure that most of the closed model structures appearing in nature exhibit is discussed in Sections A simplicial model structure is an enrichment of the underlying category to simplicial sets which interacts with the closed model structure, like function spaces do for simplicial sets; the category of simplicial sets with function spaces is a standard example.

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## Simplicial Homotopy Theory

The geometric realization is functorial on sSet. It is significant that we use the category CGHaus of compactly-generated Hausdorff spaces, rather than the category Top of topological spaces, as the target category of geometric realization: like sSet and unlike Top, the category CGHaus is cartesian closed ; the categorical product is defined differently in the categories Top and CGHaus, and the one in CGHaus corresponds to the one in sSet via geometric realization. This definition is analogous to a standard idea in singular homology of "probing" a target topological space with standard topological n-simplices. Furthermore, the singular functor S is right adjoint to the geometric realization functor described above, i. Intuitively, this adjunction can be understood as follows: a continuous map from the geometric realization of X to a space Y is uniquely specified if we associate to every simplex of X a continuous map from the corresponding standard topological simplex to Y, in such a fashion that these maps are compatible with the way the simplices in X hang together. Homotopy theory of simplicial sets[ edit ] In order to define a model structure on the category of simplicial sets, one has to define fibrations, cofibrations and weak equivalences. One can define fibrations to be Kan fibrations.

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## Simplicial homotopy theory

This is a seminar jointly organized by Moritz Groth and Urs Schreiber. Time and place: Tuesday first session: February 14 , Hilbert space The notion of a simplicial set is a powerful combinatorial tool for studying topological spaces up to weak homotopy equivalence. Simplicial sets have fundamental applications throughout mathematics, whenever homotopy theory plays a role.

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## Goerss, Jardine - Simplicial Homotopy Theory

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