Simplicial homotopy theory
type: Item
instance of: scholarly article
title: Simplicial homotopy theory
zbMATH Open document ID: 0949.55001
author: J. F. Jardine
author: Paul Goerss
published in: Progress in Mathematics
publication date: 1999-08-31T00:00:00Z
review text: 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 \textit{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., [\textit{P. S. Hirschhorn}, \url{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-Friedland…
reviewed by: Marek Golasiński
Mathematics Subject Classification ID: 55-01
Mathematics Subject Classification ID: 18G30
Mathematics Subject Classification ID: 55Q99
Mathematics Subject Classification ID: 55P99
Mathematics Subject Classification ID: 55U35
Mathematics Subject Classification ID: 55U10
zbMATH DE Number: 1329189
zbMATH Keywords: closed model category
zbMATH Keywords: cofibration
zbMATH Keywords: derived functor
zbMATH Keywords: fibration
zbMATH Keywords: homotopy coherence
zbMATH Keywords: localization
zbMATH Keywords: Postnikov tower
zbMATH Keywords: simplicial set
zbMATH Keywords: weak equivalence
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 1329189