http://dx.doi.org/10.1090/surv/113 Mathematical Surveys and Monographs Volume 113 Homotopy Limit Functors on Model Categories and Homotopical Categories William G. Dwyer Philip S. Hirschhorn Daniel M. Kan Jeffrey H. Smith •\vvEM47y American Mathematical Society EDITORIAL COMMITTEE Jerry L. Bona Peter S. Landweber, Chair Michael G. Eastwood Michael P. Loss J. T. Stafford 2000 Mathematics Subject Classification. Primary 18A99, 18D99, 18G55, 55U35. For additional information and updates on this book, visit www.ams.org/bookpages/surv-113 Library of Congress Cataloging-in-Publication Data Homotopy limit functors on model categories and homotopical categories / William G. Dwyer ... [et al.]. p. cm. — (Mathematical surveys and monographs, ISSN 0076-5376; v. 113) Includes bibliographical references and index. ISBN 0-8218-3703-6 (alk. paper) 1. Homotopy theory. I. Dwyer, William G., 1947- II. Series. QA612.7.H635 2004 514'.24—dc22 2004059481 Copying and reprinting. 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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 09 08 07 06 05 To Sammy Eilenberg Dan Quillen Pete Bousfield Contents Preface vii Part I. Model Categories 1 Chapter I. An Overview 3 1. Introduction 3 2. Slightly unconventional terminology 3 3. Problems involving the homotopy category 5 4. Problem involving the homotopy colimit functors 8 5. The emergence of the current monograph 11 6. A preview of part II 12 Chapter II. Model Categories and Their Homotopy Categories 19 7. Introduction 19 8. Categorical and homotopical preliminaries 22 9. Model categories 25 10. The homotopy category 29 11. Homotopical comments 32 Chapter III. Quillen Functors 35 12. Introduction 35 13. Homotopical uniqueness 38 14. Quillen functors 40 15. Approximations 42 16. Derived adjunctions 44 17. Quillen equivalences 48 18. Homotopical comments 51 Chapter IV. Homotopical Cocompleteness and Completeness of Model Categories 55 19. Introduction 55 20. Homotopy colimit and limit functors 59 21. Homotopical cocompleteness and completeness 62 22. Reedy model categories 65 23. Virtually coflbrant and fibrant diagrams 69 24. Homotopical comments 72 Part II. Homotopical Categories 75 Chapter V. Summary of Part II 77 25. Introduction 77 vi CONTENTS 26. Homotopical categories 78 27. The hom-sets of the homotopy categories 80 28. Homotopical uniqueness 82 29. Deformable functors 83 30. Homotopy colimit and limit functors and homotopical ones 85 Chapter VI. Homotopical Categories and Homotopical Functors 89 31. Introduction 89 32. Universes and categories 93 33. Homotopical categories 96 34. A colimit description of the hom-sets of the homotopy category 101 35. A Grothendieck construction 103 36. 3-arrow calculi 107 37. Homotopical uniqueness 112 38. Homotopically initial and terminal objects 115 Chapter VII. Deformable Functors and Their Approximations 119 39. Introduction 119 40. Deformable functors 123 41. Approximations 126 42. Compositions 130 43. Induced partial adjunctions 133 44. Derived adjunctions 138 45. The Quillen condition 143 Chapter VIII. Homotopy Colimit and Limit Functors and Homotopical Ones 147 46. Introduction 147 47. Homotopy colimit and limit functors 148 48. Left and right systems 152 49. Homotopical cocompleteness and completeness (special case) 159 50. Homotopical colimit and limit functors 161 51. Homotopical cocompleteness and completeness (general case) 166 Index 171 Bibliography 181 Preface This monograph, wThich is aimed at the graduate level and beyond, consists of two parts. In part II we develop the beginnings of a kind of "relative" category theory of what we will call homotopical categories. These are categories with a single distin­ guished class of maps (called weak equivalences) containing all the isomorphisms and satisfying one simple two out of six axiom which states that (*) for every three maps r, s and t for which the two compositions sr and ts are defined and are weak equivalences, the four maps r, s, t and tsr are also weak equivalences, which enables one to define "homotopicar versions of such basic categorical no­ tions as initial and terminal objects, colimit and limit functors, adjunctions, Kan extensions and universal properties. In part I we use the results of part II to get a better understanding of Quilleir s so useful model categories, which are categories with three distinguished classes of maps (called cofibrations, fibrations and weak equivalences) satisfying a few simple axioms which enable one to "do homotopy theory*'. In particular we show that such model categories are homotopically cocornplete and homotopically complete in a sense which is much stronger than the existence of all small homotopy colimit and limit functors. Both parts are essentially self-contained. A reader of part II is assumed to have some familiarity with the categorical notions mentioned above, wrhile those who read part I (and especially the introductory chapter) should also know something about model categories. In the hope of increasing the local as well as the global readability of this monograph, we not only start each section with some introductory remarks and each chapter with an introductory section, but also each of the two parts with an introductory chapter, with the first chapter of part I serving as motivation for and introduction to the whole monograph and the first chapter of part II summarizing the main results of its other three chapters. Index Page numbers in bold point to the definitions of the indexed terms. adjunctions, 122-123 axiom compatibility with (co)limit functors, 57, factorization, 4, 19, 26 151 lifting, 26 counits of, 122, 122-123, 135-138 limit, 3, 19, 25 deformable, 14-15, 37, 51, 51-54, 119- retract, 19-20, 26 122, 133 two out of six, 26 derived adjunctions of, 8, 14, 36-37, 46, two out of three, 19-20 53, 44-54, 121, 139, 138-143, see also derived adjunctions canonical natural isomorphisms, 45 partial, 44, 53, 133, 133-138 canonical natural transformations, 139, 139- 142 Quillen, 40, see also Quillen adjunctions canonically isomorphic objects, 10, 38, 82, units of, 122, 122-123, 135-138 92, 112, see also categorical uniqueness adjunctions of systems, 157 canonically weakly equivalent objects, 10, and dinatural transformations, 167 39, 83, 92, 114, see also homotopical and Kan extensions along systems, 169 uniqueness categorical uniqueness, 157 CAT, 95, 97-99 counits of, 157 cat, 95 deformable, 157 CAT , 97, 97-99 derived adjunctions of, 158 W cat-systems, see also systems locally deformable, 157 left, 62, 62-65, 73 sufficient conditions for existence, 157 right, 62, 62-65, 73 units of, 157 categorical uniqueness, 38, 38-39, 92-93, 112, all or none proposition for compositions, 131 112-114 alternate description of homotopy categories, of adjoints of systems, 157 100 of colimit systems, 160 approximations, 13, 35-37, 42-44, 51, 84, of initial objects, 38, 92, 113 120-122, 127, 126-130 of limit systems, 160 and total derived functors, 128 of terminal objects, 38, 92, 113 compositions of, 44, 120-121, 130-133 categorically homotopical uniqueness of, 42, 51, 127 contractible categories, 10, 38, 82, 92, 113 of deformable functors, 51, 127 full subcategories, 38, 82, 92, 112 of homotopical functors, 128 unique objects, 10, 38, 82, 92, 112, see of Kan extensions also categorical uniqueness and homotopical Kan extensions, 163 categories, 4, 23, 22-23, 78-79, 89-90, 95, of natural transformations, 129, 129-130 95 of Quillen functors, 42 arrow, 101, 101-112 sufficient conditions for composability, 52 categorically contractible, 10, 38, 82, 92, sufficient conditions for existence, 51, 127 113 approximations of systems, 154 closed model, 3-4, 19, 27-28 homotopical uniqueness of, 154 cocomplete, 25, 58, 58, 159, see also co- sufficient conditions for existence, 155 completeness arrow categories, 101, 101-112 complete, 25, 58, 58, 159, see also com­ T-diagrams of, 102 pleteness 171 172 INDEX connected components of, 102 of Grothendieck constructions, 105 diagram, 22, 24, 97 of homotopy categories, 81, 102 functor, 22, 24, 97 colimit functors, 57, 57-58, 85, 148 homotopical, 11-12, 23, 20-25, 77, 79-80, compatibility with left adjoints, 57, 151 90-91, 96, 96-101 deformability result for, 56, 61 homotopically cocomplete, 16-17, 73, 160, homotopical, 15-17, 85-87, 165-166, see 169, see also homotopical cocomplete- also homotopical colimit functors ness homotopy, 8-11, 15-17, 55-56, 59, 59-62, homotopically complete, 16-17, 73, 160, 72, 85-87, 148-152, see also homotopy 169, see also homotopical