##
**Simplicial homotopy theory.**
*(English)*
Zbl 0949.55001

Progress in Mathematics (Boston, Mass.). 174. Basel: Birkhäuser. xv, 510 p. (1999).

The category of topological spaces is equivalent to simplicial sets. But the methods and ideas of simplicial homotopy theory exist outside any topological context. This point of view has been encoded by D. G. Quillen [Homotopical algebra, Lect. Notes Math. 43 (1967; Zbl 0168.20903)] in the notion of a closed model category. The homotopy theories associated to Quillen’s model categories can be interpreted as a purely algebraic enterprise having broad substantial applications in many mathematical areas.

Some subtle aspects of this theory avoided in the book under review are available in the literature or via the Internet e.g., [P. S. Hirschhorn, http://www-math.mit.edu/~psh]. There are ten chapters in all; everything depends on the first two chapters and the remaining material often reflects the original nature of the project. Chapter I contains fundamental organizing theorems of the category of simplicial sets. The foundations of abstract homotopy theory given by Quillen are presented in Chapter II (consisting of only eight but not nine sections as it is announced in the Introduction). Various equivalent formulations are presented in Chapter III which is a further repository of things used later. Homotopy theory of bisimplicial sets and bisimplicial abelian groups are discussed in Chapter IV. Chapter V offers a discussion of skeleta in the category of simplicial groups. Towers of fibrations, nilpotent spaces and homotopy spectral sequences are the subjects of Chapter VI. Chapter VII deals with the model structure for the category of simplicial objects in a closed model category. The main purpose of Chapter VIII is to define and discuss the homotopy spectral sequence, and outline some of its applications. Simplicial model structures return in Chapter IX in the context of homotopy coherence. Chapter X takes up one of the recent versions of the localization theory. At the end, Bousfield’s approach to localization leads to the Bousfield-Friedlander model in the stable category.

All material of the monograph 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.

Some subtle aspects of this theory avoided in the book under review are available in the literature or via the Internet e.g., [P. S. Hirschhorn, http://www-math.mit.edu/~psh]. There are ten chapters in all; everything depends on the first two chapters and the remaining material often reflects the original nature of the project. Chapter I contains fundamental organizing theorems of the category of simplicial sets. The foundations of abstract homotopy theory given by Quillen are presented in Chapter II (consisting of only eight but not nine sections as it is announced in the Introduction). Various equivalent formulations are presented in Chapter III which is a further repository of things used later. Homotopy theory of bisimplicial sets and bisimplicial abelian groups are discussed in Chapter IV. Chapter V offers a discussion of skeleta in the category of simplicial groups. Towers of fibrations, nilpotent spaces and homotopy spectral sequences are the subjects of Chapter VI. Chapter VII deals with the model structure for the category of simplicial objects in a closed model category. The main purpose of Chapter VIII is to define and discuss the homotopy spectral sequence, and outline some of its applications. Simplicial model structures return in Chapter IX in the context of homotopy coherence. Chapter X takes up one of the recent versions of the localization theory. At the end, Bousfield’s approach to localization leads to the Bousfield-Friedlander model in the stable category.

All material of the monograph 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.

Reviewer: Marek Golasiński (Toruń)

### MathOverflow Questions:

A notion of fibration on bisimplicial setsReedy fibrancy and composition in Segal spaces

### MSC:

55-01 | Introductory exposition (textbooks, tutorial papers, etc.) pertaining to algebraic topology |

18G30 | Simplicial sets; simplicial objects in a category (MSC2010) |

55Q99 | Homotopy groups |

55P99 | Homotopy theory |

55U35 | Abstract and axiomatic homotopy theory in algebraic topology |

55U10 | Simplicial sets and complexes in algebraic topology |