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Homotopy Limit Functors on Model Categories and Homotopical Categories: Mathematical Surveys and Monographs

Editat de William G. Dwyer, Philip S. Hirschhorn, Daniel M. Kan, Jeffrey H. Smith
en Limba Engleză Paperback – iun 2006
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 smallhomotopy 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.
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Specificații

ISBN-13: 9780821839751
ISBN-10: 0821839756
Pagini: 181
Dimensiuni: 175 x 253 x 11 mm
Greutate: 0.36 kg
Editura: American Mathematical Society
Colecția American Mathematical Society
Seria Mathematical Surveys and Monographs

Locul publicării:Oxford, United States