**Mathematical Surveys and Monographs**

Volume: 113;
2004;
181 pp;
Softcover

MSC: Primary 18; 55;

Print ISBN: 978-0-8218-3975-1

Product Code: SURV/113.S

List Price: $69.00

Individual Member Price: $55.20

**Electronic ISBN: 978-1-4704-1340-8
Product Code: SURV/113.S.E**

List Price: $69.00

Individual Member Price: $55.20

#### Supplemental Materials

# Homotopy Limit Functors on Model Categories and Homotopical Categories

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*William G. Dwyer; Philip S. Hirschhorn; Daniel M. Kan; Jeffrey H. Smith*

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.

#### Table of Contents

# Table of Contents

## Homotopy Limit Functors on Model Categories and Homotopical Categories

- Contents v6 free
- Preface vii8 free
- Part I. Model Categories 110 free
- Part II. Homotopical Categories 7584
- Chapter V. Summary of Part II 7786
- Chapter VI. Homotopical Categories and Homotopical Functors 8998
- 31. Introduction 8998
- 32. Universes and categories 93102
- 33. Homotopical categories 96105
- 34. A colimit description of the hom-sets of the homotopy category 101110
- 35. A Grothendieck construction 103112
- 36. 3-arrow calculi 107116
- 37. Homotopical uniqueness 112121
- 38. Homotopically initial and terminal objects 115124

- Chapter VII. Deformable Functors and Their Approximations 119128
- Chapter VIII. Homotopy Colimit and Limit Functors and Homotopical Ones 147156

- Index 171180
- Bibliography 181190

#### Readership

Graduate students and research mathematicians interested in algebraic topology.