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Book DetailsMathematical Surveys and MonographsVolume: 113; 2004; 181 ppMSC: Primary 18; 55;
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.ReadershipGraduate students and research mathematicians interested in algebraic topology.

Table of Contents

Chapters

I. An overview

II. Model categories and their homotopy categories

III. Quillen functors

IV. Homotopical cocompleteness and completeness of model categories

V. Summary of Part II

VI. Homotopical categories and homotopical functors

VII. Deformable functors and their approximations

VIII. Homotopy colimit and limit functors and homotopical ones


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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.
Graduate students and research mathematicians interested in algebraic topology.

Chapters

I. An overview

II. Model categories and their homotopy categories

III. Quillen functors

IV. Homotopical cocompleteness and completeness of model categories

V. Summary of Part II

VI. Homotopical categories and homotopical functors

VII. Deformable functors and their approximations

VIII. Homotopy colimit and limit functors and homotopical ones