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]). There is also [BK72, Part II], which is very good at what it does, but whose emphasis is only on simplicial sets. More recently, there is the expository paper [DS95], which is highly recommended as an introduction. But there is no mention of simplicial sets in that paper, and it does not go very far into the theory. The time seems to be right for a more careful study of model categories from the ground up. Both of the books [DHK] and [Hir97], unfinished as the author writes this, will do this from different perspectives. The book [DHK] overlaps considerably with this one, but concentrates more on homotopy colimits and less on the relationship between a model category and its homotopy category. The book [Hir97] is concerned with localization of model categories, but also contains a significant amount of general theory. There is also the book [GJ97], which con centrates on simplicial examples. All three of these books are highly recommended to the reader.
ix x PREFACE
This book is also an exposition of model categories from the ground up. In particular, this book should be accessible to graduate students. There are very few prerequisites to reading it, beyond a basic familiarity with categories and functors, and some familiarity with at least one of the central examples of chain complexes, simplicial sets, or topological spaces. We also require some familiarity with the basics of set theory, especially ordinal and cardinal numbers. Later in the book we do require more of the reader; in Chapter 7 we use the theory of homotopy limits of diagrams of simplicial sets, developed in [BK72]. However, the reader who gets that far will be well equipped to understand [BK72] in any case. This book is the author's attempt to understand the theory of model categories well enough to answer one question. That question is: when is the homotopy category of a model category a stable homotopy category in the sense of [HPS97]? We do not in the end answer this question in quite as much generality as one would like, though we come fairly close to doing so in Chapter 7. As I tried to answer this question, it became clear that the theory necessary to do so was not in place. After a long period of resistance, I decided it was important to develop the necessary theory, and that the logical and most useful place to do so was in a book that would assume almost nothing of the reader. A book is the logical place because the theory I develop requires a foundation that is simply not in the literature. I think this foundation is beautiful and important, and therefore deserves to be made accessible to the general mathematician. We now provide an overview of the book. See also the introductions to the in dividual chapters. The first chapter of this book is devoted to the basic definitions and results about model categories. In highfalutin language, the main goal of this chapter is to define the 2-category of model categories and show that the homotopy category is part of a pseudo-2-functor from model categories to categories. This is a fancy way, fully explained in Section 1.4, to say that not only can one take the homotopy category of a model category, one can also take the total derived adjunction of a Quillen adjunction, and the total derived natural transformation of a natural transformation between Quillen adjunctions. Doing so preserves compo sitions for the most part, but not exactly. This is the reason for the word "pseudo". In order to reach this goal, we have to adopt a different definition of model category from that of [DHK], but the difference is minor. The definition of [DHK], on the other hand, is considerably different from the original definition of [Qui67], and even from its refinement in [Qui69]. After the theoretical material of the first chapter, the reader is entitled to some examples. We consider the important examples of chain complexes over a ring, topological spaces, and chain complexes of comodules over a commutative Hopf algebra in the second chapter, while the third is devoted to the central example of simplicial sets. Proving that a particular category has a model structure is always difficult. There is, however, a standard method, introduced by Quillen [Qui67] but formalized in [DHK]. This method is an elaboration of the small object argu ment and is known as the theory of cofibrantly generated model categories. After examining this theory in Section 2.1, we consider the category of modules over a Probenius ring, where projective and injective modules coincide. This is perhaps the simplest nontrivial example of a model category, as every object is both cofi- brant and fibrant. Nevertheless, the material in this section seems not to have appeared before, except in Georgian [Pir86]. Then we consider chain complexes of modules over an arbitrary ring. Our treatment differs somewhat from the standard PREFACE xi
one in that we do not assume our chain complexes are bounded below. We then move on to topological spaces. Here our treatment is the standard one, except that we offer more details than are commonly provided. The model category of chain complexes of comodules over a commutative Hopf algebra, on the other hand, has not been considered before. It is relevant to the recent work in modular represen tation theory of Benson, Carlson, Rickard and others (see, for example [BCR96]), as well as to the study of stable homotopy over the Steenrod algebra [Pal97]. The approach to simplicial sets given in the third chapter is substantially the same as that of [GJ97]. In the fourth chapter we consider model categories that have an internal tensor product making them into closed monoidal categories. Almost all the standard model categories are like this: chain complexes of abelian groups have the tensor product, for example. Of course, one must require the tensor product and the model structure to be compatible in an appropriate sense. The resulting monoidal model categories play the same role in the theory of model categories that ordinary rings do in algebra, so that one can consider modules and algebras over them. A module over the monoidal model category of simplicial sets, for example, is the same thing as a simplicial model category. Of course, the homotopy category of a monoidal model category is a closed monoidal category in a natural way, and similarly for modules and algebras. The material in this chapter is all fairly straightforward, but has not appeared in print before. It may also be in [DHK], when that book appears. The fifth and sixth chapters form the technical heart of the book. In the fifth chapter, we show that the homotopy category of any model category has the same good properties as the homotopy category of a simplicial model category. In our highfalutin language, the homotopy pseudo-2-functor lifts to a pseudo-2-functor from model categories to closed Ho SSet-modules, where HoSSet is the homo topy category of simplicial sets. This follows from the idea of framings developed in [DK80]. This chapter thus has a lot of overlap with [DHK], where framings are also considered. However, the emphasis in [DHK] seems to be on using framings to develop the theory of homotopy colimits and homotopy limits, whereas we are more interested in making Ho SSet act naturally on the homotopy category. There is a nagging question left unsolved in this chapter, however. We find that the homotopy category of a monoidal model category is naturally a closed algebra over HoSSet, but we are unable to prove that it is a central closed algebra. In the sixth chapter we consider the homotopy category of a pointed model category. As was originally pointed out by Quillen [Qui67], the apparently minor condition that the initial and terminal objects coincide in a model category has profound implications in the homotopy category. One gets a suspension and loop functor and cofiber and fiber sequences. In the light of the fifth chapter, however, we realize we get an entire closed HoSSet*-action, of which the suspension and loop functors are merely specializations. Here Ho SSet* is the homotopy category of pointed simplicial sets. We prove that the cofiber and fiber sequences are com patible with this action in an appropriate sense, as well as reproving the standard facts about cofiber and fiber sequences. We then get a notion of pre-triangulated categories, which are closed Ho SSet*-modules with cofiber and fiber sequences satisfying many axioms. The seventh chapter is devoted to the stable situation. We define a pre- triangulated category to be triangulated if the suspension functor is an equivalence xii PREFACE of categories. This is definitely not the same as the usual definition of triangulated categories, but it is closer than one might think at first glance. We also argue that it is a better definition. Every triangulated category that arises in nature is the homotopy category of a model category, so will be triangulated in our stronger sense. We also consider generators in the homotopy category of a pointed model category. These generators are extremely important in the theory of stable homo topy categories developed in [HPS97]. Our results are not completely satisfying, but they do go a long way towards answering our original question: when is the homotopy category of a model category a stable homotopy category? Finally, we close the book with a brief chapter containing some unsolved or partially solved problems the author would like to know more about. Note that bold-faced page numbers in the index are used to indicate pages containing the definition of the entry. Ordinary page numbers indicate a textual reference. I would like to acknowledge the help of several people in the course of writing this book. I went from knowing very little about model categories to writing this book in the course of about two years. This would not have been possible without the patient help of Phil Hirschhorn, Dan Kan, Charles Rezk, Brooke Shipley, and Jeff Smith, experts in model categories all. I wish to thank John Palmieri for count less conversations about the material in this book. Thanks are also due Gaunce Lewis for help with compactly generated topological spaces, and Mark Johnson for comments on early drafts of this book. And I wish to thank my family, Karen, Grace, and Patrick, for the emotional support so necessary in the frustrating en terprise of writing a book. Bibliography
[AFH97] L. Avramov, H. Foxby, and S. Halperin, Differential graded homological algebra, preprint, 1997. [BBD82] A. A. Beilinson, J. Bernstein, and P. Deligne, Faisceaux pervers, Asterisque, vol. 100, Soc. Math France, Paris, 1982. [BCR96] D. J. Benson, Jon F. Carlson, and J. Rickard, Complexity and varieties for infinitely generated modules. II, Math. Proc. Cambridge Philos. Soc. 120 (1996), 597-615. [BF78] A. K. Bousfield and E. M. Friedlander, Homotopy theory of V-spaces, spectra, and bisimplicial sets, Geometric applications of homotopy theory (Proc. Conf., Evanston, 111., 1977), II (M. G. Barratt and M. E. Mahowald, eds.), Lecture Notes in Math., vol. 658, Springer, Berlin, 1978, pp. 80-130. [BK72] A. K. Bousfield and D. M. Kan, Homotopy limits, completions, and localizations, Lecture Notes in Mathematics, vol. 304, Springer-Verlag, Berlin-New York, 1972, v+348 pp. [BK90] A. I. Bondal and M. M. Kapranov, Framed triangulated categories, Mat. Sb. 181 (1990), no. 5, 669-683. [BL96] Jonathan Block and Andrey Lazarev, Homotopy theory and generalized duality for spectral sheaves, Internat. Math. Res. Notices (1996), no. 20, 983-996. [Bou79] A. K. Bousfield, The localization of spectra with respect to homology, Topology 18 (1979), 257-281. [Bro62] Edgar H. Brown, Jr., Cohomology theories, Ann. of Math. (2) 75 (1962), 467-484. [CR88] Charles W. Curtis and Irving Reiner, Representation theory of finite groups and associative algebras, Wiley Classics Library, John Wiley & Sons Inc., New York, 1988, Reprint of the 1962 original, A Wiley-Interscience Publication. [DHK] W. G. Dwyer, P. S. Hirschhorn, and D. M. Kan, Model categories and general abstract homotopy theory, in preparation. [DK80] W. G. Dwyer and D. M. Kan, Function complexes in homotopical algebra, Topology 19 (1980), 427-440. [DM97] Marius Dadarlat and James McClure, When are two commutative C* -algebras stably homotopy equivalent?, preprint, 1997. [DS95] W. G. Dwyer and J. Spalinski, Homotopy theories and model categories, Handbook of algebraic topology (Amsterdam), North-Holland, Amsterdam, 1995, pp. 73-126. [EKMM97] A. D. Elmendorf, I. Kriz, M. A. Mandell, and J. P. May, Rings, modules, and algebras in stable homotopy theory. With an appendix by M. Cole, Mathematical Surveys and Monographs, vol. 47, American Mathematical Society, Providence, RI, 1997, xii-f-249 pp. [Fra96] J. Franke, Uniqueness theorems for certain triangulated categories possessing an Adams spectral sequence, preprint, 1996. [GJ97] P. F. Goerss and J. F. Jardine, Simplicial homotopy theory, preprint, 1997. [Gra74] John W. Gray, Formal category theory: adjointness for 2-categories, Lecture Notes in Mathematics, vol. 391, Springer-Verlag, 1974. [Gra95] Marco Grandis, Homotopical algebra and triangulated categories, Math. Proc. Cam bridge Philos. Soc. 118 (1995), no. 2, 259-285. [Gro97] J. Grodal, unpublished notes, 1997. [GZ67] P. Gabriel and M. Zisman, Calculus of fractions and homotopy theory, Springer- Verlag, New York, 1967. [Hir97] P. S. Hirschhorn, Localization, cellularization, and homotopy colimits, preprint, 1997. [Hop] M. J. Hopkins, lectures at M. I. T.
201 202 BIBLIOGRAPHY
[Hov98a] Mark Hovey, Monoidal model categories, preprint, 1998. [Hov98b] Mark Hovey, Stabilization of model categories, preprint, 1998. [HPS97] Mark Hovey, John H. Palmieri, and Neil P. Strickland, Axiomatic stable homotopy theory, Mem. Amer. Math. Soc. 128 (1997), no. 610, x+114. [HSS98] Mark Hovey, Brooke Shipley, and Jeff Smith, Symmetric spectra, preprint, 1998. [Jac89] Nathan Jacobson, Basic Algebra II, second ed., W. H. Freeman and Company, New York, 1989. [Jec78] Thomas Jech, Set theory, Pure and applied mathematics, vol. 79, Academic Press, New York, 1978. [KM96] I. Kriz and J. P. May, Operads, algebras, modules, and motives, Asterisque, vol. 233, Societe Mathematique de France, 1996. [KS74] G. M. Kelly and R. H. Street, Review of the elements of 2-categories, Category Semi nar, Sydney 1972/73, Lecture Notes in Mathematics, vol. 420, Springer-Verlag, 1974, pp. 75-103. [Lew78] L. G. Lewis, Jr., The stable category and generalized Thorn spectra, Ph.D. thesis, University of Chicago, 1978. [LMS86] L. G. Lewis, Jr., J. P. May, and M. Steinberger, Equivariant stable homotopy theory, Lecture Notes in Mathematics, vol. 1213, Springer-Verlag, 1986. [Mar83] H. R. Margolis, Spectra and the Steenrod algebra. Modules over the Steenrod alge bra and the stable homotopy category, North-Holland Mathematical Library, vol. 29, North-Holland Publishing Co., Amsterdam-New York, 1983. [May67] J. P. May, Simplicial objects in algebraic topology, Mathematical Studies, vol. 11, Van Nostrand, 1967. [May72] J. P. May, The geometry of iterated loop spaces, Lecture Notes in Mathematics, vol. 271, Springer, 1972. [ML71] S. Mac Lane, Categories for the working mathematician, Graduate Texts in Mathe matics, vol. 5, Springer-Verlag, 1971. [Mun75] James R. Munkres, Topology: a first course, Prentice-Hall, 1975. [Mun84] James R. Munkres, Elements of algebraic topology, Addison-Wesley Publishing Com pany, Menlo Park, Calif., 1984. [Nee92] Amnon Neeman, Some new axioms for triangulated categories, J. Algebra 139 (1992), 221-255. [Pal97] John H. Palmieri, Stable homotopy over the Steenrod algebra, preprint (100 pages), 1997. [Pir86] T. I. Pirashvili, Closed model category structures on abelian categories, Soobshch. Akad. Nauk Gruzin. SSR 124 (1986), no. 2, 265-268. [Pup62] Dieter Puppe, On the formal structure of stable homotopy theory, Colloq. Alg. Topol ogy, Aarhus Univ., 1962, pp. 65-71. [Qui67] Daniel G. Quillen, Homotopical algebra, Lecture Notes in Mathematics, vol. 43, Springer-Verlag, 1967. [Qui68] Daniel G. Quillen, The geometric realization of a Kan fibration is a Serre fibration, Proc. Amer. Math. Soc. 19 (1968), 1499-1500. [Qui69] Daniel Quillen, Rational homotopy theory, Ann. of Math. (2) 90 (1969), 205-295. [Rez96] C. W. Rezk, Spaces of algebra structures and cohomology of operads, Ph.D. thesis, Massachusetts Institute of Technology, 1996. [Ric97] Jeremy Rickard, Idempotent modules in the stable category, J. London Math. Soc. (2) 56 (1997), no. 1, 149-170. [Spa81] Edwin H. Spanier, Algebraic topology, Springer-Verlag, 1981, Corrected reprint of the 1966 original. [SS97] Stefan Schwede and Brooke Shipley, Algebras and modules in monoidal model cate gories, preprint, 1997. [Sta98] Don Stanley, Determining closed model category structures, preprint, 1998. [Str72] Arne Strom, The homotopy category is a homotopy category, Arch. Math (Basel) 23 (1972), 435-441. [Ver77j J.-L. Verdier, Categories derivees, Cohomologie Etale (SGA 4|) (P. Deligne, ed.), 1977, Springer Lecture Notes in Mathematics 569, pp. 262-311. [Voe97] Vladimir Voevodsky, The Milnor conjecture, preprint, 1997. BIBLIOGRAPHY 203
[Vog71] Rainer M. Vogt, Convenient categories of topological spaces for homotopy theory. Arch. Math. (Basel) 22 (1971), 545-555. [Wei97] Charles A. Weibel, The mathematical enterprises of Robert Thomason, Bull. Amer. Math. Soc. (N.S.) 34 (1997), no. 1, 1-13. [Wyl73] Oswald Wyler, Convenient categories for topology, General Topology and Appl. 3 (1973), 225-242. Index
2-category, 15, 22, 23, 24-26 <9A[n], 75 model, 195, 198 C-Quillen functor, 114 of C-algebras, 105 C-algebra, 104 of C-model categories, 114 C-algebra functor, 104 of C-modules, 104 C-algebra natural transformation, 104 of categories, 18, 24 C-algebra structure, 104 of categories and adjunctions, 24 C-model category, 101, 114, 118 of central C-algebras, 106 C-module, 104 of central monoidal C-algebras, 115 C-module functor, 104 of closed monoidal categories, 106 C-module natural transformation, 104 of closed monoidal pre-triangulated cate C-module structure, 104 gories, 174 K, see also k-spaces of model categories, 18, 24, 195 K*, see also /c-spaces, pointed of monoidal C-model categories, 115 A[n], 75 of monoidal categories, 103 7-filtered cardinal, see also cardinal, of monoidal model categories, 113 7-filtered of pointed model categories, 145 | — |, see also geometric realization of pre-triangulated categories, 173 Ar[n], 75 of stable homotopy categories, 187 K-small, 29 of stable model categories, 178 A-sequence, see also lambda sequence of symmetric C-algebras, 105 SSet, see also simplicial sets of symmetric monoidal C-algebras, 115 SSet*, see also simplicial sets, pointed of symmetric monoidal categories, 104 Top, see also topological spaces of triangulated categories, 178 Top*, see also topological spaces, pointed 2-functor, 24 T, see also topological spaces, compactly A*
Catad, see also 2-category, of categories and anodyne extension, 79, 80-83, 109 adjunctions associativity isomorphism, 134 Ch(B), see also chain complexes, of comod- ules basepoint, 4 Ch( 205 206 INDEX cardinal, 28 cosmall, 35 7-filtered, 28 cotriple, 197 cardinality argument, 45 cube lemma, 126, 153, 158, 161, 166, 168 category of simplices cylinder object, 8, 9, 127, 153 of a simplicial set, 75, 124 central C-algebra, 105 deformation retract, 53 central C-algebra functor, 105 degeneracy central monoidal C-model category, 118 of a simplex, 74 chain complex degeneracy map, 73 acyclic, 41 degree function, 124 chain complexes derived adjunction, 19 of abelian groups, 114, 143-144, 146, 196 derived functor of comodules, 27-28, 60-72, 112, 114, 178, total left, 16, 17-22 191, 198 total right, 16 of modules, 27, 40, 41-49, 111-112, 114, diagonal functor, 128-129, 132 178, 186, 191, 198 dimension chain homotopy, 43 of a simplex, 73 chain map, 40 direct category, 119, 120, 121-124 closed Ti inclusion, 50 dual model category, 4 dosed monoidal category, 106 duality 2-functor, 24, 107, 115, 118, 122, closed monoidal functor, 106 126, 127, 171, 173 closed monoidal structure, 106 Dwyer, Bill, 1, 119 closed symmetric monoidal category, 63, 77 cofiber, 147 equivalence cofiber sequence, 147, 152, 156, 157-165, in a 2-category, 25 exact adjunction, 173, 184 170, 179-185 cofibrant, 4 face cofibrant replacement functor, 5 of a simplex, 74 cofibration, 3 face map, 73 generating, 34 fiber, 148 trivial, 3 fiber homotopy equivalence, 89, 90-91 generating, 34 fiber sequence, 147, 152, 156, 157-165, 170, cofinality 179-185 of a cardinal, 29 fibrant, 4 cogroup object, 151 fibrant replacement functor, 5 cogroup structure, 151 fibration, 3 on EX, 151 Kan, 79 cohomology functor, 187, 199 locally trivial, 89, 92-93, 95-97 colimit minimal, 89, 91, 92-95, 97 homotopy, 188, 199 trivial, 3 sequential, 188, 199 finite, 29, 190 commutative monoid, 196-197 framing, 119, 123, 127, 128-129, 131-146 comoduie, 61 Franke, Jens, 199 cofree, 64 Probenius ring, 36, 37-40, 72, 112 injective, 64-65 Noetherian, 40 simple, 62 function complex, 128 comodules, 102, 105 of simplicial sets, 77 comonad, 197 functor compactly closed, 58 monoidal, see also monoidal functor compactly open, 58 functorial factorization, 2, 15, 16, 28 correspondence of homotopies, 150 geometric realization, 77, 78-81, 85, 95-99, cosimplicial frame, 127, 128, 130-140, 144, 102, 114, 118 167-169 preserves fibrations, 97 cosimplicial frames preserves finite limits, 80 map of, 127 preserves products, 77 cosimplicial identities, 73 Goerss, Paul, 197 cosimplicial object, 73, 76 group object, 151 INDEX 207 group ring, 37 matching space, 120, 122, 124 group structure, 151 May, Peter, 187, 197 on QX} 151 McClure, Jim, 196 minimal fibration, see also fibration, mini Hirschhorn, Phil, x, 1 mal homology model category, 3 of a chain complex, 41 cofibrantly generated, 27, 34, 35-36, 108, homotopic maps, 9 123, 186, 187, 189 homotopy fibrantly generated, 35 of continuous maps, 50 finitely generated, 34, 177, 187, 190-193 of simplicial maps, 86 pointed, 4, 14, 15, 21, 26, 36, 115, 144- of vertices, 84 175, 187-193, 199 homotopy category, 7 simplicial, 101, 114, 119, 128, 136, 138- homotopy equivalence, 9, 12 140, 143-144, 146, 198 homotopy groups pointed, 115 of a chain complex stable, 177, 178, 188, 198 of comodules, 65 model structure, 3 of a fibrant simplicial set, 83, 85, 86-89, product, 4, 7, 14, 19 97-99 module of a topological space, 50 over a monoidal category, see also homotopy pullback square, 184 C-module homotopy pushout square, 184 modules Hopf algebra, 37, 60, 112 over a Frobenius ring, 27 Hopkins, Mike, 197 mnnajcL, W. horizontal composition monoidal C-model category, 115, 118 in a 2-category, 23 monoidal C-Quillen functor, 115 of natural transformations, 18, 23 monoidal category, 101, 102 Hurewicz cofibration, 195 monoidal functor, 102 Hurewicz fibration, 96, 196 monoidal model category, 101, 108, 109- 119, 140-146 inclusion pointed, 146, 173-175 of topological spaces, 49 stable, 178 injective model structure, 45, 112 symmetric, 109-118 inverse category, 119, 120, 121-124 monoidal natural transformation, 103 Johnson, Mark, x monoidal Quillen adjunction, 113 monoidal Quillen functor, 113, 141 Kan, Dan, x, 1, 3, 119, 198 monoidal structure, 102 Kelley space, see also /c-space Ken Brown's lemma, 6, 11-12, 14, 131-132, natural transformation 137 monoidal, see also monoidal natural trans formation lambda sequence, 28 total derived, 16, 18, 22 latching space, 120, 122, 124 non-degenerate left exact, 173 simplex, 74 left homotopy, 9 between right homotopies, 149 octahedral axiom, 160-162, 170, 179 left lifting property, 3 operads, 197 Lewis, Gaunce, x ordinal, 28, 120 limit homotopy, 177, 188, 189, 199 Palmieri, John, x linear extension, 120 paracompact, 96 localizing subcategory, 187, 199 path component locally trivial fibration, see also fibration, lo of a simplicial set, 85 cally trivial path object, 9 loop functor, 148, 149-175, 178-179, 181- pre-triangulated category, 147, 170, 171- 186 175, 186, 198 closed central monoidal, 174 mapping cylinder, 166, 167 closed monoidal, 173, 174 double, 165 closed symmetric monoidal, 174 208 INDEX pre-triangulation, 170 of modules, 37 pseudo-2-functor, 18, 22, 25, 101, 115, 117- stable homotopy category, 199 118, 136, 138, 140-141, 143, 145, 173- algebraic, 177, 187 175, 187, 199 without duality, 187 homotopy, 26 ordinary, 198 natural isomorphism of, 137 Steenrod algebra, 61 pushout product, 107, 109 strongly dualizable, 187 superclass, 23 Quillen, Daniel, 1-199 suspension functor, 148, 149-175, 177-186, Quillen adjunction, 14, 15-22, 36, 71, 131, 188-190 136-139, 163, 195 symmetric C-algebra, 105 of two variables, 107 symmetric C-algebra functor, 105 Quillen bifunctor, 107, 108-116 symmetric monoidal C-model category, 118 Quillen equivalence, 13, 19, 20-22, 26, 48, symmetric monoidal category, 103 98, 195 symmetric monoidal functor, 103 Quillen functor, 36, 48, 123, 128, 163-165 symmetric monoidal model category, 109, left, 14, 15-19 196-197 monoidal, see also monoidal Quillen func symmetric monoidal structure, 103 tor symmetric spectra, 178, 196, 198 right, 14, 15-19 tame module, 62 Reedy category, 119, 123, 124, 125-127, 199 Tate resolution, 72 Reedy model structure, 126, 127, 130, 133, Thomason, Robert, 2 145, 167 topological spaces, 27, 49-60, 77-81, 102, Reedy scheme, 199 106, 110, 191, 195-196 reflect, 21 compactly generated, 58, 106, 110, 114 relative /-cell complex, 30, 31-36 pointed, 111, 114 retract argument, 5 Hurewicz model category of, 34 Rezk, Charles, x, 195, 197 pointed, 58, 172, 178 right exact, 173 topology right homotopy, 9 /c-space, 58 between right homotopies, 149 total derived natural transformation, see also right lifting property, 3 natural transformation, total derived right proper, 57 total left derived functor, see also derived functor, total left Schwede, Stefan, 49, 109, 198 total right derived functor, see also derived Serre fibration, 51, 96, 97 functor, total right Shipley, Brooke, x transfinite composition, 28 Sierpinski space, 49 triangle, 179, 180 simplicial category, 73, 119, 123-125 left, 170 simplicial frame, 127, 128, 130-133, 144, right, 170 164 triangulated category, 173, 177, 178, 179- simplicial frames 187, 198 map of, 127 classical, 177, 180, 182-183 simplicial identities, 74 closed monoidal, 178 simplicial object, 73 closed symmetric monoidal, 185 simplicial set triangulation finite, 74 classical, 179 simplicial sets, 73, 74-101, 109, 114, 118, triple, 197 119, 128-146, 191, 196 trivial cofibration, see also cofibration, triv pointed, 98, 100, 110, 114, 118, 178, 186, ial 188-189 trivial fibration, see also fibration, trivial singular functor, 77, 98 two out of three property small, 29, 177, 186, 190-193 of Quillen equivalences, 21 small object argument, 28, 32, 189 of weak equivalences, 3 Smith, Jeff, x, 107, 109 stable category Verdier's octahedral axiom, see also octahe of modules, 37 dral axiom stable equivalence vertical composition INDEX in a 2-category, 23 of natural transformations, 23 Voevodsky, Vladimir, 196 weak equivalence, 3 weak generators, 177, 186, 187-190 weak Hausdorff, 58 Errata to Model Categories by Mark Hovey Thanks to Mike Cole, Ed Enochs, Georges Maltsiniotis, Doug Ravenel, and Don Stanley for noticing and/or fixing many of these errors. (1) p.2,1.-6: The definition 1.1.1 (2) of functorial factorization is not as strong as I intended, nor as strong as the small object argument implies. Given a category 6, let d,c: Map 6 —> £ denote the domain and codomain functors. We need the following equalities of functors: doa = d,coa = do/3,and c o /? = c. (2) p. 10, 1.-4: Should be "define t as the map" instead of "define s as the map". (3) p.ll, 1.3: Should be "trivial fibration" instead of "trivial cofibration". (4) p.ll, 1.4: Should be "cylinder object for £". (5) p.12, 1.-7: The cylinder object Af needs to be both cofibrant and fibrant. The simplest way to do this is to take A' to be the functorial cylinder object on A, using Corollary 1.2.6 to get the left homotopy from A'. (6) p.17, 1.2: The phrase "weak left Quillen functors" should be replaced by "left Quillen functors". (7) p.21, Corollary 1.3.16: There is some ambiguity in the phrase "reflects weak equivalences between cofibrant objects". This means that if / is a map between cofibrant objects such that Ff is a weak equivalence, then / is a weak equivalence. (8) p.21, 1.27: Should be /: X -* Y. (9) p.33, 1.-22: Should be /? < A. (10) p.33,1.-21: At the end of the line, pp+\ should be pp+v (11) p.36 and the rest of the book: What I have called Probenius rings are actually called quasi-Probenius rings in the literature. (12) p.37,1.19: Should be "the surjection M 0 P -> M." (13) p.38, Lemma 2.2.8: Should be "injective", not "projective". (14) p.39, Theorem 2.2.12: Every quasi-Probenius ring is left and right Noe- therian, so the Noetherian hypothesis is unnecessary. (15) p.49,1.20: J should be X (16) p.49,1.22: Again, J should be 1 (17) p.49, 1.24: The paragraph that begins here is wrong; the map from the Sierpinski space factors through every Xa. Here is a better example, from Don Stanley. Let X = {0,1} with the indiscrete topology, let A be a limit ordinal, and for a < A let Xa — [a, A) x X, with topology consisting of the sets Vp for /3 G [a, A) together with the empty set. Here Vp = [a, A) x {0} U [/?, A) x {1}. For a < a', there is a continuous map 211 212 ERRATA TO MODEL CATEGORIES Xa —> Xa> that sends (/3,x) to (a',x) if (3 < a! and sends (/?, x) to itself otherwise. The colimit of the Xa consists of the two points (A,0) and (A, 1) with the indiscrete topology, so is homeomorphic to X, but there is no continuous map X —• Xa. Thus X is not small. (18) p.51, Definition 2.4.3: One must also explicitly define the identity map of the empty set as a weak equivalence. (19) p.51, 1.10: Should be V instead of /. (20) p.51, 1.-4: Should be V instead of /. (21) p.52, Corollary 2.4.6: Should be J' instead of J. (22) p.71, 1.13: The definition of U is wrong. Instead, one can construct U using Preyd's adjoint functor theorem, or by defining U(B'®V) = B (37) p.124, Definition 5.2.1: The degree function d is not a functor on the Reedy category, just a map on the objects. (38) p.133, 1.13: At the beginning of the line, it should be Ho C. (39) p. 142, Conjecture 5.6.6: This conjecture has been proved by Cisinski in Corollaire 6.7 of "Proprietes universelles et extensions de Kan derivees" (preprint, 2002). (40) p.191, 1.-10: Should be "have", not "hve". (41) p. 196, Problem 8.4: This problem has been addressed by Joachim and Johnson in pp. 163-197, Contemp. Math. 399, Amer. Math. Soc. 2006. (42) p. 198, Problem 8.9: This problem has been solved by Dugger (Trans. Amer. Math. Soc. 353 (2001), no. 12, 5003-5027). Dugger needs some conditions on the model category, but they are certainly general enough to also solve Problem 8.10. (43) p. 198, Problem 8.11: This problem has also been solved by Dugger (Ho mology, Homotopy Appl. 8 (2006), no. 1, 1-30). (44) p. 199, Problem 8.13: This problem is quite close to Grothendieck's notion of a derivateur, and has been studied by Cisinski (Ann. Math. Blaise Pascal 10 f2003V no. 2. 195-244V Selected Titles in This Series (Continued from the front of this publication) 32 Howard Jacobowitz, An introduction to CR structures, 1990 31 Paul J. Sally, Jr. and David A. Vogan, Jr., Editors, Representation theory and harmonic analysis on semisimple Lie groups, 1989 30 Thomas W. Cusick and Mary E. Flahive, The Markoff and Lagrange spectra, 1989 29 Alan L. T. Paterson, Amenability, 1988 28 Richard Beals, Percy Deift, and Carlos Tomei, Direct and inverse scattering on the line, 1988 27 Nathan J. Fine, Basic hypergeometric series and applications, 1988 26 Hari Bercovici, Operator theory and arithmetic in H°°, 1988 25 Jack K. Hale, Asymptotic behavior of dissipative systems, 1988 24 Lance W. Small, Editor, Noetherian rings and their applications, 1987 23 E. H. Rothe, Introduction to various aspects of degree theory in Banach spaces, 1986 22 Michael E. Taylor, Noncommutative harmonic analysis, 1986 21 Albert Baernstein, David Drasin, Peter Duren, and Albert Marden, Editors, The Bieberbach conjecture: Proceedings of the symposium on the occasion of the proof, 1986 20 Kenneth R. Goodearl, Partially ordered abelian groups with interpolation, 1986 19 Gregory V. Chudnovsky, Contributions to the theory of transcendental numbers, 1984 18 Frank B. Knight, Essentials of Brownian motion and diffusion, 1981 17 Le Baron O. Ferguson, Approximation by polynomials with integral coefficients, 1980 16 O. Timothy O'Meara, Symplectic groups, 1978 15 J. Diestel and J. J. Uhl, Jr., Vector measures, 1977 14 V. Guillemin and S. Sternberg, Geometric asymptotics, 1977 13 C. Pearcy, Editor, Topics in operator theory, 1974 12 J. R. Isbell, Uniform spaces, 1964 11 J. Cronin, Fixed points and topological degree in nonlinear analysis, 1964 10 R. Ayoub, An introduction to the analytic theory of numbers, 1963 9 Arthur Sard, Linear approximation, 1963 8 J. Lehner, Discontinuous groups and automorphic functions, 1964 7.2 A. H. Clifford and G. B. Preston, The algebraic theory of semigroups, Volume II, 1961 7.1 A. H. Clifford and G. B. Preston, The algebraic theory of semigroups, Volume I, 1961 6 C. C. Chevalley, Introduction to the theory of algebraic functions of one variable, 1951 5 S. Bergman, The kernel function and conformal mapping, 1950 4 O. F. G. Schilling, The theory of valuations, 1950 3 M. Marden, Geometry of polynomials, 1949 2 N. Jacobson, The theory of rings, 1943 1 J. A. Shohat and J. D. Tamarkin, The problem of moments, 1943