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Model Categories, 1999 62 Vladimir I http://dx.doi.org/10.1090/surv/063 Selected Titles in This Series 63 Mark Hovey, Model categories, 1999 62 Vladimir I. Bogachev, Gaussian measures, 1998 61 W. Norrie Everitt and Lawrence Markus, Boundary value problems and symplectic algebra for ordinary differential and quasi-differential operators, 1999 60 Iain Raeburn and Dana P. Williams, Morita equivalence and continuous-trace C*-algebras, 1998 59 Paul Howard and Jean E. Rubin, Consequences of the axiom of choice, 1998 58 Pavel I. Etingof, Igor B. Frenkel, and Alexander A. Kirillov, Jr., Lectures on representation theory and Knizhnik-Zamolodchikov equations, 1998 57 Marc Levine, Mixed motives, 1998 56 Leonid I. Korogodski and Yan S. Soibelman, Algebras of functions on quantum groups: Part I, 1998 55 J. Scott Carter and Masahico Saito, Knotted surfaces and their diagrams, 1998 54 Casper Goffman, Togo Nishiura, and Daniel Waterman, Homeomorphisms in analysis, 1997 53 Andreas Kriegl and Peter W. Michor, The convenient setting of global analysis, 1997 52 V. A. Kozlov, V. G. Maz'ya? and J. Rossmann, Elliptic boundary value problems in domains with point singularities, 1997 51 Jan Maly and William P. Ziemer, Fine regularity of solutions of elliptic partial differential equations, 1997 50 Jon Aaronson, An introduction to infinite ergodic theory, 1997 49 R. E. Showalter, Monotone operators in Banach space and nonlinear partial differential equations, 1997 48 Paul-Jean Cahen and Jean-Luc Chabert, Integer-valued polynomials, 1997 47 A. D. Elmendorf, I. Kriz, M. A. Mandell, and J. P. May (with an appendix by M. Cole), Rings, modules, and algebras in stable homotopy theory, 1997 46 Stephen Lipscomb, Symmetric inverse semigroups, 1996 45 George M. Bergman and Adam O. Hausknecht, Cogroups and co-rings in categories of associative rings, 1996 44 J. Amoros, M. Burger, K. Corlette, D. Kotschick, and D. Toledo, Fundamental groups of compact Kahler manifolds, 1996 43 James E. Humphreys, Conjugacy classes in semisimple algebraic groups, 1995 42 Ralph Freese, Jaroslav Jezek, and J. B. Nation, Free lattices, 1995 41 Hal L. Smith, Monotone dynamical systems: an introduction to the theory of competitive and cooperative systems, 1995 40.3 Daniel Gorenstein, Richard Lyons, and Ronald Solomon, The classification of the finite simple groups, number 3, 1998 40.2 Daniel Gorenstein, Richard Lyons, and Ronald Solomon, The classification of the finite simple groups, number 2, 1995 40.1 Daniel Gorenstein, Richard Lyons, and Ronald Solomon, The classification of the finite simple groups, number 1, 1994 39 Sigurdur Helgason, Geometric analysis on symmetric spaces, 1994 38 Guy David and Stephen Semmes, Analysis of and on uniformly rectifiable sets, 1993 37 Leonard Lewin, Editor, Structural properties of polylogarithms, 1991 36 John B. Conway, The theory of subnormal operators, 1991 35 Shreeram S. Abhyankar, Algebraic geometry for scientists and engineers, 1990 34 Victor Isakov, Inverse source problems, 1990 33 Vladimir G. Berkovich, Spectral theory and analytic geometry over non-Archimedean fields, 1990 (Continued in the back of this publication) To Dan Kan Mathematical Surveys and Monographs Volume 63 Model Categories Mark Hovey American Mathematical Society Editorial Board Georgia M. Benkart Tudor Stefan Raiiu, Chair Peter Land web er Michael Renardy 2000 Mathematics Subject Classification. Primary 55U35; Secondary 13D25, 16W30, 18G55, 20J05. ABSTRACT. This book is a comprehensive study of the relationship between a model category and its homotopy category. We develop the theory of model categories, giving a careful development of the main examples. One highlight of the theory is a proof that the homotopy category of any model category is naturally a closed module over the homotopy category of simplicial sets. The main goal of the book is to give conditions on a model category under which its homotopy category is a stable homotopy category, in an appropriate sense. Research partially supported by an NSF Postdoctoral Fellowship. For additional information and updates on this book, visit www.ams.org/bookpages/surv-63 Library of Congress Cataloging-in-Publication Data Hovey, Mark, 1965- Model categories / Mark Hovey. p. cm. — (Mathematical surveys and monographs, ISSN 0076-5376 ; v. 63) Includes bibliographical references (p. - ) and index. ISBN 0-8218-1359-5 (acid-free paper) AMS softcover ISBN: 978-0-8218-4361-1 1. Model categories. 2. Homotopy theory. 3. Complexes. I. Title. II. Series: Mathematical surveys and monographs : no. 63. QA169.H68 1998 514/.24—dc21 98-34539 CIP Copying and reprinting. Individual readers of this publication, and nonprofit libraries acting for them, are permitted to make fair use of the material, such as to copy a chapter for use in teaching or research. Permission is granted to quote brief passages from this publication in reviews, provided the customary acknowledgment of the source is given. Republication, systematic copying, or multiple reproduction of any material in this publication is permitted only under license from the American Mathematical Society. Requests for such permission should be addressed to the Acquisitions Department, American Mathematical Society, 201 Charles Street, Providence, Rhode Island 02904-2294, USA. Requests can also be made by e-mail to [email protected]. © 1999 by the American Mathematical Society. All rights reserved. Reprinted with corrections by the American Mathematical Society, 2007. The American Mathematical Society retains all rights except those granted to the United States Government. Printed in the United States of America. @ The paper used in this book is acid-free and falls within the guidelines established to ensure permanence and durability. Visit the AMS home page at http: //www. ams. org/ 10 9 8 7 6 5 4 3 2 1 12 11 10 09 08 07 Contents Preface ix Chapter 1. Model categories 1 1.1. The definition of a model category 2 1.2. The homotopy category 7 1.3. Quillen functors and derived functors 13 1.3.1. Quillen functors 14 1.3.2. Derived functors and naturality 16 1.3.3. Quillen equivalences 19 1.4. 2-categories and pseudo-2-functors 22 Chapter 2. Examples 27 2.1. Cofibrantly generated model categories 28 2.1.1. Ordinals, cardinals, and transfinite compositions 28 2.1.2. Relative /-cell complexes and the small object argument 30 2.1.3. Cofibrantly generated model categories 34 2.2. The stable category of modules 36 2.3. Chain complexes of modules over a ring 40 2.4. Topological spaces 49 2.5. Chain complexes of comodules over a Hopf algebra 60 2.5.1. The category of B-comodules 60 2.5.2. Weak equivalences 65 2.5.3. The model structure 67 Chapter 3. Simplicial sets 73 3.1. Simplicial sets 73 3.2. The model structure on simplicial sets 79 3.3. Anodyne extensions 81 3.4. Homotopy groups 83 3.5. Minimal fibrations 88 3.6. Fibrations and geometric realization 95 Chapter 4. Monoidal model categories 101 4.1. Closed monoidal categories and closed modules 101 4.2. Monoidal model categories and modules over them 107 4.3. The homotopy category of a monoidal model category 115 Chapter 5. Framings 119 5.1. Diagram categories 120 5.2. Diagrams over Reedy categories and framings 123 5.3. A lemma about bisimplicial sets 128 viii CONTENTS 5.4. Function complexes 129 5.5. Associativity 133 5.6. Naturality 136 5.7. Framings on pointed model categories 144 Chapter 6. Pointed model categories 147 6.1. The suspension and loop functors 147 6.2. Cofiber and fiber sequences 152 6.3. Properties of cofiber and fiber sequences 156 6.4. Naturality of cofiber sequences 163 6.5. Pre-triangulated categories 170 6.6. Pointed monoidal model categories 173 Chapter 7. Stable model categories and triangulated categories 177 7.1. Triangulated categories 177 7.2. Stable homotopy categories 186 7.3. Weak generators 187 7.4. Finitely generated model categories 190 Chapter 8. Vistas 195 Bibliography 201 Index 205 Errata 211 Preface Model categories, first introduced by Quillen in [Qui67], form the foundation of homotopy theory. The basic problem that model categories solve is the following. Given a category, one often has certain maps (weak equivalences) that are not isomorphisms, but one would like to consider them to be isomorphisms. One can always formally invert the weak equivalences, but in this case one loses control of the morphisms in the quotient category. The morphisms between two objects in the quotient category may not even be a set. If the weak equivalences are part of a model structure, however, then the morphisms in the quotient category from X to Y are simply homotopy classes of maps from a cofibrant replacement of X to a fibrant replacement of Y. Because this idea of inverting weak equivalences is so central in mathematics, model categories are extremely important. However, so far their utility has been mostly confined to areas historically associated with algebraic topology, such as homological algebra, algebraic if-theory, and algebraic topology itself. The author is certain that this list will be expanded to cover other areas of mathematics in the near future. For example, Voevodksy's work [Voe97] is certain to make model categories part of every algebraic geometer's toolkit. These examples should make it clear that model categories are really funda­ mental. However, there is no systematic study of model categories in the literature. Nowhere can the author find a definition of the category of model categories, for example. Yet one of the main lessons of twentieth century mathematics is that to study a structure, one must also study the maps that preserve that structure. In addition, there is no excellent source for information about model categories. The standard reference [Qui67] is difficult to read, because there is no index and because the definitions are not ideal (they were changed later in [Qui69]).
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