Mathematical Surveys and Monographs 2004; 181 pp; softcover Volume: 113 Reprint/Revision History: reprinted 2005 ISBN10: 0821839756 ISBN13: 9780821839751 List Price: US$65 Member Price: US$52 Order Code: SURV/113.S
 The purpose of this monograph, which is aimed at the graduate level and beyond, is to obtain a deeper understanding of Quillen's model categories. A model category is a category together with three distinguished classes of maps, called weak equivalences, cofibrations, and fibrations. Model categories have become a standard tool in algebraic topology and homological algebra and, increasingly, in other fields where homotopy theoretic ideas are becoming important, such as algebraic \(K\)theory and algebraic geometry. The authors' approach is to define the notion of a homotopical category, which is more general than that of a model category, and to consider model categories as special cases of this. A homotopical category is a category with only a single distinguished class of maps, called weak equivalences, subject to an appropriate axiom. This enables one to define "homotopical" versions of such basic categorical notions as initial and terminal objects, colimit and limit functors, cocompleteness and completeness, adjunctions, Kan extensions, and universal properties. There are two essentially selfcontained parts, and part II logically precedes part I. Part II defines and develops the notion of a homotopical category and can be considered as the beginnings of a kind of "relative" category theory. The results of part II are used in part I to obtain a deeper understanding of model categories. The authors show in particular that model categories are homotopically cocomplete and complete in a sense stronger than just the requirement of the existence of small homotopy colimit and limit functors. A reader of part II is assumed to have only some familiarity with the abovementioned categorical notions. Those who read part I, and especially its introductory chapter, should also know something about model categories. Readership Graduate students and research mathematicians interested in algebraic topology. Table of Contents Model categories  An overview
 Model categories and their homotopy categories
 Quillen functors
 Homotopical cocompleteness and completeness of model categories
Homotopical categories  Summary of part II
 Homotopical categories and homotopical functors
 Deformable functors and their approximations
 Homotopy colimit and limit functors and homotopical ones
 Index
 Bibliography
