Homotopy Limit Functors on Model Categories and Homotopical Categories (Mathematical Surveys and Monographs, 113) - Softcover

William G. Dwyer

Softcover

ISBN 10: 0821839756 ISBN 13: 9780821839751

Publisher: American Mathematical Society, 2005

Synopsis

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 self-contained 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 above-mentioned categorical notions. Those who read part I, and especially its introductory chapter, should also know something about model categories.

Edition details

• Publisher: American Mathematical Society
• Publication date: 2005
• Language: English
• ISBN 10: 0821839756
• ISBN 13: 9780821839751
• Binding: Paperback
• Number of pages: 181
• Editor: William G. Dwyer, Philip S. Hirschhorn, Daniel M. Kan, Jeffrey H. Smith