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
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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
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© 2004 by the American Mathematical Society. All rights reserved. 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 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 completeness colimit functors homotopically contractible, 10, 36, 39, 83, colimit systems, 58, 86, 160 92, 114 and cocompleteness, 58, 160 homotopy, 4-8, 21-22, see also homotopy categorical uniqueness of, 160 categories homotopical, 87, 169, see also homotopi indexing, 97 cal colimit systems locally small, 4, 23, 79, 89-90, 95 homotopy, 63, 63-65, 73, 86, 160, see model, 3-4, 19-20, 25, 25-29, see also also homotopy colimit systems model categories compatibility n-arrow, 101, see also arrow categories of adjoints writh (co)limit functors, 57, 151 of simplices, 67, 67-72 complete categories, 25, 58, 58, 159, see of weak equivalences, 96 also completeness Reedy, 65 homotopically, 16-17, 73, 160, 169, see Reedy model, 65-72 also homotopical completeness simplicial, 105 completeness, 58, 58, 159-160 small, 23, 79, 89-90, 95 and limit systems, 58, 160 small W-, 22, 90, 94, 94-95 homotopical, 16-17, 62-65, 73, 159-161, W-, 22, 90, 94, 94, 94-95 169, see also homotopical completeness underlying, 96 components we-, 4-5 connected, 102 with cofibrant constants, 67, 67-68 composers, 152 with fibrant constants, 67, 67-68 of cat-systems, 62 category of types, 101 composition functors, 43, 130, 164, see also classical homotopy categories, 31 compositions classifying space, 104 compositions closed model categories, 3-4, 19, 27-28 all or none proposition, 131 closure properties of model categories, 28 of approximations, 120-121, 130-133 cocomplete categories, 25, 58, 58, 159, see of approximations of Quillen functors, 44 also cocompleteness of deformable functors, 52-53, 84-85, 131- homotopically, 16-17, 73, 160, 169, see 133 also homotopical cocompleteness of derived adjunctions, 46, 142 cocompleteness, 58, 58, 159-160 of homotopical colimit functors, 165 and colimit systems, 58, 160 of homotopical Kan extensions, 164 homotopical, 16-17, 56, 62-65, 73, 86-87, of homotopical limit functors, 165 159-161, 169, see also homotopical co- of homotopy colimit functors, 60, 72, 150 completeness of homotopy limit functors, 60, 72, 150 cofibrant constants, 67, 67-68 of Kan extensions, 164 cofibrant fibrant objects, 30 of partial adjunctions, 44, 134 cofibrant objects, 30 conjugate pairs of natural transformations, cofibrations, 25 123, 123, 166 characterization of, 28 deformable, 133, 141 Reedy, 65 connected components, 102 trivial, 26 constant diagram functors, 57, 148 characterization of, 28 contractible categories colim(cat\ 58 categorically, 10, 38, 82, 92, 113 colim15, 57, 148 homotopically, 10, 36, 39, 83, 92, 114 colimw, 57, 149 counits colim
and dinatural transformations, 167 /-deformation retracts, 124 of Kan extensions, 118, 162 maximal, 125 F-defor mat ions, 155 JD-colimit functors, 57, 85, 148, see also co- /-deformations, 7, 124 limit functors homotopical uniqueness of, 125 .D-limit functors, 57, 148, see also limit F-presentations, 169 functors of homotopical Kan extensions along sys deformability result tems, 169 for colimit and limit functors, 56, 61 /-presentations, 163 for Quillen functors, 41 of homotopical Kan extensions, 163 deformable adjunctions, 14-15, 37, 51, 51- factorization axiom, 4, 19, 26 54, 119-122, 133 fibrant constants, 67, 67-68 derived adjunctions of, 8, 14, 36-37, 46, fibrant objects, 30 53, 44-54, 121, 139, 138-143, see also fibrations, 25 derived adjunctions characterization of, 28 Quillen condition for, 38, 54, 121, 143, Reedy, 65 143-145 trivial, 26 deformable adjunctions of systems, 157 characterization of, 28 derived adjunctions of, 158 full subcategories deformable conjugate pairs of natural trans categorically, 38, 82, 92, 112 formations, 133, 141 homotopically, 39, 83, 92, 114 deformable functors, 7, 13-15, 51, 51-54, Fun, 24, 97
83-85, 124, 119-126 Funw, 24, 97 approximations of, 51, 127 functor categories, 22, 24, 97 compositions of, 52-53, 84-85, 131-133 homotopical, 97 deformable systems, 87, 155, see also sys functors tems colimit, 57, 57-58, 85, 148, see also co- deformation retracts, 7, 24, 83, 119, 124 limit functors and model categories, 30 composition, 43, 130, 164, see also com deformations, 7, 24, 83, 119, 124, see also positions /-deformations constant diagram, 57, 148 degree function, 65 D-colimit, 57, 85, 148, see also colimit derived adjunctions, 8, 14, 36-37, 46, 53, functors 44-54, 121, 139, 138-143 D-limit, 57, 148, see also limit functors compositions of, 46, 142 deformable, 7, 13-15, 51, 51-54, 83-85, conjugations between, 141 124, 119-126, see also deformable func of deformable adjunctions of systems, 158 tors of homotopy colimit functors, 59, 149 deformable pairs of, 37, 52, 131, see also of homotopy colimit systems, 161 pairs of functors of homotopy limit functors, 59, 149 forgetful, 97 of homotopy limit systems, 161 homotopical, 12, 24, 24-25, 79-80, 90, 96 derived functors homotopical categories of, 97 total left, 5, 7, 128 homotopical colimit, 15-17, 85-87, 165- total right, 5, 7, 128 166, see also homotopical colimit func diagram categories, 22, 24, 97 tors homotopical, 97 homotopical limit, 15-17, 165-166, see also diagrams, 96 homotopical limit functors restricted, 69, 69-72 homotopical w-colimit, 87, 165, see also virtually cofibrant, 69, 69-72 homotopical colimit functors virtually fibrant, 69, 69-72 homotopical it-limit, 165, see also homo- dinatural transformations, 167, 167-168 topical limit functors and adjunctions of systems, 167 homotopy colimit, 8-11, 15-17, 55-56, 59, and counits of adjunctions, 167 59-62, 72, 85-87, 148-152, see also ho and units of adjunctions, 167 motopy colimit functors homotopy D-colimit, 8-11, 59, 85, 148, embedding CAT in CATW, 98 see also homotopy colimit functors enrichment homotopy D-limit, 59, 148, see also ho Grothendieck, 104 motopy limit functors 174 INDEX
homotopy limit, 15-17, 59, 59-62, 72, 148- homotopical categories, 11-12, 23, 20-25, 152, see also homotopy limit functors 77, 79-80, 90-91, 96, 96-101 homotopy w-colimit, 59, 72, 85, 149, see homotopical equivalences of, 24, 80, 91, also homotopy colimit functors 96 homotopy u-limit, 59, 72, 149, see also homotopy categories of, 12-13, 24, 24-25, homotopy limit functors 80-82, 91-92, 98, 98-107, see also ho induced diagram, 57, 148 motopy categories initial projection, 68, 68-72 locally small, 96 left deformable, 7, 13-15, 37, 51, 51-54, maximal, 80, 97 124, 119-126, see also deformable func minimal, 80, 98 tors of functors, 97 left deformable pairs of, 37, 52, 131, see of homotopical functors, 97 also pairs of functors saturated, 25, 37, 38, 52, 54, 73, 82, 87, left Quillen, 35, 40, 40-54, see also Quillen 92, 99, 121, 122, 132, 144, 150, see also functors saturation limit, 57, 57-58, see also limit functors small, 96 localization, 4, 22, 24, 29 weak equivalences in, 11, 23, 90, 96 locally deformable pairs of, 37, 52, 131, homotopical cocompleteness, 16-17, 56, 62- see also pairs of functors 65, 73, 86-87, 159-161, 169, see also locally left deformable pairs of, 37, 52, homotopical colimit systems, homotopy 131, see also pairs of functors colimit system locally right deformable pairs of, 37, 52, of model categories, 64 131, see also pairs of functors sufficient conditions for, 73, 161, 170 naturally weakly equivalent, 24, 90, 96 homotopical colimit functors, 15-17, 85-87, projection, 68, 68-72 165-166 Quillen, 35, 40, 40-54, see also Quillen and homotopy colimit functors, 165 functors homotopical uniqueness of, 165 right deformable, 7, 13-15, 37, 51, 51- sufficient conditions for composability, 165 54, 124, 119-126, see also deformable sufficient conditions for existence, 165 functors homotopical colimit systems, 87, 169 right deformable pairs of, 37, 52, see also and homotopy colimit systems, 170 pairs of functors homotopical uniqueness of, 169 right Quillen, 35, 40, 40-54, see also Quillen sufficient conditions for existence, 170 functors terminal projection, 68, 68-72 homotopical compatibility total left derived, 5, 7, 128 of deformable adjoints with homotopy (co)- total right derived, 5, 7, 128 limit functors, 72, 151-152 -u-colimit, 57, 85, 148, see also colimit of Quillen functors with homotopy (co)limit functors functors, 56, 60 it-limit, 57, 148, see also limit functors homotopical completeness, 16-17, 62-65, 73, 159-161, 169, see also homotopical limit 7, 24, 99 systems, homotopy limit system Gr, 103, 103-107 of model categories, 64 Grothendieck construction, 103, 103-107 sufficient conditions for, 73, 161, 170 and simplicial localizations, 105 homotopical diagram categories, 97 colimit description of, 105 3-arrow calculi on, 108 Grothendieck description of homotopy cate and saturation, 25, 99 gories, 104 homotopical equivalences of homotopical cat Grothendieck enrichment, 91, 104 egories, 24, 80, 91, 96 homotopical functor categories, 97 /i-deformation retracts, 126 3-arrow calculi on, 108 ft-deformations, 126 and saturation, 25, 99 hammock localizations, see also simplicial homotopical functors, 12, 24, 24-25, 79-80, localizations 90, 96 higher universes, 94 approximations of, 128, see also approxi Ho, 24, 98, 98-104 mations nomotopic maps, 31, 45 homotopical categories of, 97 homotopical cat-systems, 63 homotopical inverses, 24, 96 INDEX 175 homotopical Kan extensions, 87, 93, 118, full subcategories, 39, 83, 92, 114 163 homotopically initial Kan extensions, 118, and approximations of Kan extensions, 163 163, see also homotopical Kan exten homotopical uniqueness of, 118, 163 sions presentations of, 163 homotopically initial Kan extensions along sufficient conditions for composability, 164 left systems, 168, see also homotopical sufficient conditions for existence, 163 Kan extensions along systems homotopical Kan extensions along systems, homotopically initial objects, 13, 39, 39-40, 168 83, 93, 116, 115-118 and approximations of Kan extensions along homotopical uniqueness of, 40, 93, 115 systems, 169 motivation, 115 homotopical uniqueness of, 168 homotopically terminal Kan extensions, 118, presentations of, 169 163, see also homotopical Kan exten sufficient conditions for existence, 169 sions homotopical limit functors, 15-17, 165-166 homotopically terminal Kan extensions along and homotopj' limit functors, 165 right systems, 168, see also homotopi homotopical uniqueness of, 165 cal Kan extensions along systems sufficient conditions for composability, 165 homotopically terminal objects, 9, 13, 39, sufficient conditions for existence, 165 39-40, 83, 93, 116, 115-118 homotopical limit systems, 169 homotopical uniqueness of, 40, 93, 115 and homotopy limit systems, 170 motivation, 115 homotopical uniqueness of, 169 homotopically unique objects, 10, 35, 39, 83, sufficient conditions for existence, 170 92, 114, see also homotopical unique homotopical structures, 23, 96 ness homotopical subcategories, 96 homotopically universal properties, 40, 115 homotopical systems, 153 homotopy categories, 4-8, 21-22 homotopy systems of, 154 alternate description of, 100 homotopical w-colimit functors, 87, 165, see classical, 31 also homotopical colimit functors colimit description of, 81, 102 homotopical u-limit functors, 165, see also descriptions of, 98-112 homotopical limit functors Grothendieck description of, 104 homotopical uniqueness, 9-10, 39, 39-40, of homotopical categories, 12-13, 24, 24- 82-83, 92-93, 114, 112-118 25, 80-82, 91-92, 98, 98-107 of approximations, 42, 51, 127, 154 of model categories, 29, 29-32 of /-deformations, 125 3-arrow description of, 5, 21, 32, 33, 81, of homotopical colimit functors, 165 91, 109 of homotopical colimit systems, 169 homotopy colimit functors, 8-11, 15-17, 55- of homotopical Kan extensions, 118, 163 56, 59, 59-62, 72, 85-87, 148-152 of homotopical Kan extensions along sys and homotopical colimit functors, 165 tems, 168 compositions of, 60, 72, 150 of homotopical limit functors, 165 derived adjunctions of, 59, 149 of homotopical limit systems, 169 existence on model categories, 59 of homotopically initial objects, 40, 93, homotopical compatibility with left deform- 115 able left adjoints, 72, 151-152 of homotopically terminal objects, 40, 93, homotopical compatibility with left Quillen 115 functors, 56, 60 of homotopy colimit functors, 59, 72, 149 homotopical uniqueness of, 59, 72, 149 of homotopy colimit systems, 64, 160 sufficient conditions for composability, 72, of homotopy limit functors, 59, 72, 149 150 of homotopy limit systems, 64, 160 sufficient conditions for existence, 72, 149 homotopical version, 80, 98 homotopy colimit systems, 63, 63-65, 73, homotopically 86, 160 cocomplete categories, 16-17, 73, 160, 169, and homotopical colimit systems, 170 see also homotopical cocompleteness derived adjunctions of, 161 complete categories, 16-17, 73, 160, 169, homotopical uniqueness of, 64, 160 see also homotopical completeness sufficient conditions for existence, 73, 161 contractible categories, 10, 36, 39, 83, 92, homotopy Z)-colimit functors, 8-11, 59, 85, 114 148, see also homotopy colimit functors 176 INDEX homotopy .D-limit functors, 59, 148, see also homotopical, see also homotopical Kan ex homotopy limit functors tensions along systems homotopy equivalences, 31 sufficient conditions for existence, 169 homotopy inverses, 31 Ken Brown's lemma, 41 homotopy limit functors, 15-17, 59, 59-62, 72, 148-152 latching objects, 66 and homotopical limit functors, 165 left adjoints compositions of, 60, 72, 150 compatibility with colimit functors, 57, 151 derived adjunctions of, 59, 149 of left systems, 157, see also adjunctions existence on model categories, 59 of systems homotopical compatibility with right de- left approximations, 13, 35-37, 42-44, 51, formable right adjoints, 72, 151-152 84, 120-122, 127, 126-130, see also ap homotopical compatibility with right Quillen proximations functors, 60 left cat-systems, 62, 62-65, 73, see also sys homotopical uniqueness of, 59, 72, 149 tems sufficient conditions for composability, 72, left deformable functors, 7, 13-15, 37, 51, 150 51-54, 84, 124, 119-126, see also de sufficient conditions for existence, 72, 149 formable functors homotopy limit systems, 63, 63-65, 73, 160 left deformable left adjoints and homotopical limit systems, 170 homotopical compatibility with homotopy derived adjunctions of, 161 colimit functors, 151-152 homotopical uniqueness of, 64, 160 left deformable natural transformations, 126 sufficient conditions for existence, 73, 161 left deformable pairs of functors, 37, 52, 84, homotopy relations, 30-32 120, 131, see also pairs of functors homotopy systems of homotopical systems, left deformable systems, 87, 155, see also 154 systems homotopy tt-colimit functors, 59, 72, 85, 149, left deformation retracts, 7, 24, 83, 119, 124 see also homotopy colimit functors left deformations, 7, 24, 83, 119, 124 homotopy it-limit functors, 59, 72, 149, see left /-deformation retracts, 124 also homotopy limit functors maximal, 125 left F-deformations, 155 indexing categories, 97 left /-deformations, 7, 124 induced diagram functors, 57, 148 homotopical uniqueness of, 125 initial Kan extensions, 118, 128, 162, see left /i-deformation retracts, 126 also Kan extensions left /i-deformations, 126 initial Kan extensions along left systems, 168, left homotopic maps, 30, 45 see also Kan extensions along systems left lifting property, 26 initial objects left Quillen equivalences, 36-37, 49, 48-50, categorical uniqueness of, 38, 92, 113 see also Quillen equivalences homotopically, 13, 39, 39-40, 83, 116, 115- left Quillen functors, 35, 40, 40-54, see also 118, see also homotopically initial ob Quillen functors jects left systems, 152, see also systems initial projection functors, 68, 68-72 length inverses of a zigzag, 98 homotopical, 24, 96 lifting axiom, 26 homotopy, 31 lim(cat\ 58 invertibility property lim°, 57, 148 weak, 23, 96 limu, 57, 149 lim^'u), 57 Kan extensions, 87, 118, 128, 162 limit axiom, 3, 19, 25 counits of, 118, 162 limit functors, 57, 57-58, 148 homotopical, 87, 93, 118, 163, see also compatibility with right adjoints, 57, 151 homotopical Kan extensions deformability result for, 56, 61 sufficient conditions for composability, 164 homotopical, 15-17, 165-166, see also ho sufficient conditions for existence, 162 motopical limit functors units of, 118, 162 homotopy, 15-17, 59, 59-62, 72, 148-152, Kan extensions along systems, 168 see also homotopy limit functors and adjunctions of systems, 169 limit systems, 58, 160 INDEX 177
and completeness, 58, 160 Reedy, 65-72 categorical uniqueness of, 160 saturation of, 21, 31 homotopical, 169, see also homotopical 3-arrow calculi of, 6, 34 limit systems weak equivalences in, 20, 25 homotopy, 63, 63-65, 73, 160, see also model structures, 25 homotopy limit systems maximal, 29 local left F-deformations, 155 minimal, 29 local right F-deformations, 155 Reedy, 65, 65-68 localization, 4, 22 simplicial, 105 n-arrow categories, 101, see also arrow cat localization functors, 4, 22, 24, 29, 99 egories locally deformable adjunctions of systems, natural transformations 157, see also adjunctions of systems approximations of, 129, 129- 130 locally left deformable pairs of functors, 37, canonical, 139, 139-142 52, 84, 120, 131, see also pairs of func conjugate pairs of, 123, 123, 166 tors deformable, 133 locally left deformable systems, 87, 155, see deformable, 126 also systems di-, 167, 167-168 locally right deformable pairs of functors, 37, natural weak equivalences, 24, 90, 96 52, 84, 120, 131, see also pairs of func naturally weakly equivalent functors, 24, 90, tors 96 locally right deformable systems, 155, see nerve, 104 also systems objects locally small canonically isomorphic, 10, 38, 82, 92, 112, categories, 4, 23, 79, 89-90, 95 see also categorical uniqueness homotopical categories, 96 canonically weakly equivalent, 10, 39, 83, 92, 114, see also homotopical unique maps ness between cat-systems, 63 categorically unique, 10, 38, 82, 92, 112, between left systems, 153 see also categorical uniqueness between right systems, 153 cofibrant, 30 homotopic, 31, 45 cofibrant fibrant, 30 left homotopic, 30, 45 fibrant, 30 right homotopic, 31, 45 homotopically initial, 13, 39, 39-40, 83, matching objects, 66 93, 116, 115-118, see also homotopi maximal cally initial objects /-deformation retracts, 125 homotopically terminal, 9, 13, 39, 39-40, homotopical categories, 80, 97 83, 93, 116, 115-118, see also homo model structures, 29 topically terminal objects structure functors, 97 homotopically unique, 10, 35, 39, 83, 92, minimal 114, see also homotopical uniqueness homotopical categories, 80, 98 latching, 66 model structures, 29 matching, 66 structure functor, 98 weakly equivalent, 23, 96 model categories, 3-4, 19-20, 25, 25-29 and deformation retracts, 30 pairs of functors closed, 3-4, 19, 27-28 deformable, 37, 52, 84, 120, 131 closure properties, 28 locally deformable, 37, 52, 84, 120, 131 colimit systems on, 64 sufficient conditions for deformability, 52, deformability result for colimit and limit 132 functors, 56, 61 partial adjunction functors, 133 homotopical cocompleteness of, 64 partial adjunction isomorphisms, 44, 53, 134, homotopical completeness of, 64 140-143 homotopy categories of, 29, 29-32 partial adjunctions, 44, 53, 133, 133-138 homotopy colimit functors on, 59 compositions of, 44, 134 homotopy limit functors on, 59 naturality of, 134 Ken Brown's lemma, 41 presentations limit systems on, 64 F-, 169 178 INDEX
/-, 163 right deformable pairs of functors, 37, 52, of homotopical Kan extensions, 163 84, 120, 131, see also pairs of functors of homotopical Kan extensions along sys right deformable right adjoints tems, 169 homotopical compatibility with homotopy projection functors, 68, 68-72 limit functors, 151-152 property right deformable systems, 155, see also sys homotopically universal, 40, 115 tems left lifting, 26 right deformation retracts, 7, 24, 83, 119, right lifting, 26 124 two out of six, 10-11, 19, 23, 79, 90, 96, right deformations, 7, 24, 83, 119, 124 110, 117, 125 right /-deformation retracts, 124 two out of three, 4, 11, 23, 79, 90, 96 maximal, 125 universal, 38, 113 right F-deformations, 155 weak invertibility, 23, 79, 96 right /-deformations, 7, 124 homotopical uniqueness of, 125 Quillen adjunctions, 40 right /i-deformation retracts, 126 and Reedy model structures, 66 right /i-deformations, 126 derived adjunctions of, 46 right nomotopic maps, 31, 45 Quillen condition for, 49 right lifting property, 26 Quillen conditions for, 37 right Quillen equivalences, 36-37, 49, 48-50, Quillen condition see also Quillen equivalences for deformable adjunctions, 38, 54, 121, right Quillen functors, 35, 40, 40-54, see 143,143-145 also Quillen functors for Quillen adjunctions, 37, 49 right systems, 152, see also systems Quillen equivalences, 36-37, 49, 48-50 Quillen condition for, 37, 49 saturated homotopical categories, 25, 37, 38, Quillen functors, 35, 40, 40-54 52, 54, 73, 82, 87, 92, 99, 121, 122, 132, approximations of, 42 144, 150, see also saturation compositions of approximations, 44 saturated systems, 153 deformability result for, 41 saturation, 5, 25, 50, 99, see also saturated existence of approximations, 42 homotopical categories homotopical compatibility with homotopy and homotopical diagram categories, 25, (co)limit functors, 56, 60 99 Reedy categories, 65 and homotopical functor categories, 25, 99 Reedy cofi brat ions, 65 and 3-arrow calculi, 11, 34, 82, 92, 110 Reedy fibrations, 65 of model categories, 21, 31 Reedy model categories, 65-72 sets, 23, 79, 95 Reedy model structures, 65, 65-68 simplicial, 104 and Quillen adjunctions, 66 small, 23, 79, 95 explicit description of, 66 U-, 22, 89, 94 implicit description of, 66 simplices, 104 Reedy weak equivalences, 65 categories of, 67, 67-72 restricted diagrams, 69, 69-72 simplicial categories, 105 restricted zigzags, 81, 98, 101 simplicial localizations, 105 retract axiom, 19-20, 26 and Grothendieck construction, 105 right adjoints simplicial sets, 104 compatibility with limit functors, 57, 151 small categories, 23, 79, 89-90, 95 of right systems, 157, see also adjunctions small homotopical categories, 96 of systems small sets, 23, 79, 95 right approximations, 13, 35-37, 42-44, 51, small ^-categories, 22, 90, 94, 94-95 84, 120-122, 127, 126-130, see also ap structures proximations homotopical, 23, 96 right cat-systems, 62, 62-65, 73, see also model, 25 systems Reedy model, 65, 65-68 right deformable functors, 7, 13-15, 37, 51, subcategories 51-54, 84, 124, 119-126, see also de categorically full, 38, 82, 92, 112 formable functors homotopical, 96 right deformable natural transformations, 126 homotopically full, 39, 83, 92, 114 INDEX 179 successor universes, 23, 90, 94 homotopical Kan extensions along, 168, sufficient conditions for see also homotopical Kan extensions a- homotopical cocompleteness, 73, 161, 170 long systems homotopical compatibility of deformable homotopical limit, 169, see also homo- adjoints with homotopy (co)limit func topical limit systems tors, 72, 151-152 homotopy colimit, 63, 63-65, 73, 86, 160, homotopical completeness, 73, 161, 170 see also homotopy colimit systems sufficient conditions for composability of homotopy limit, 63, 63-65, 73, 160, see approximations, 52 also homotopy limit systems derived adjunctions, 142 Kan extensions along, 168, see also Kan of homotopy colimit functors, 60, 72, extensions along systems left, 152 150 left cat-, 62, 62-65, 73 of homotopy limit functors, 60, 72, 150 left deformable, 87, 155 homotopical colimit functors, 165 limit, 58, 160, see also limit systems homotopical Kan extensions, 164 locally deformable, 87, 155 homotopical limit functors, 165 locally left deformable, 87, 155 homotopy colimit functors, 72, 150 locally right deformable, 155 homotopy limit functors, 72, 150 maps between, 153 Kan extensions, 164 right, 152 partial adjunctions, 44, 134 right cat-, 62, 62-65, 73 sufficient conditions for deformability of right deformable, 155 pairs of functors, 52 saturated, 153 systems, 157 sufficient conditions for deformability, 157 sufficient conditions for existence of weak equivalences between, 153 adjoints of systems, 157 approximations, 51, 127 T, 101 approximations of systems, 155 T-diagrams of arrow categories, 102 derived adjunctions of homotopy colimit terminal Kan extensions, 118, 128, 162, see functors, 149 also Kan extensions derived adjunctions of homotopy limit func terminal Kan extensions along right systems, tors, 149 168, see also Kan extensions along sys homotopical colimit functors, 165 tems homotopical colimit systems, 170 terminal objects homotopical Kan extensions, 163 categorical uniqueness of, 38, 92, 113 homotopical Kan extensions along systems, homotopically, 9, 13, 39, 39-40, 83, 116, 169 115-118, see also homotopically termi nal objects homotopical limit functors, 165 terminal projection functors, 68, 68-72 homotopical limit systems, 170 3-arrow calculi, 6, 33, 81, 91, 107, 107-112 homotopy colimit functors, 72, 149 and saturation, 11, 34, 82, 92, 110 homotopy colimit systems, 73, 161 and 3-arrow description of homotopy cat homotopy limit functors, 72, 149 egories, 5, 33, 81, 91, 109 homotopy limit systems, 73, 161 of model categories, 6, 34 Kan extensions, 162 on homotopical diagram categories, 108 Kan extensions along systems, 169 on homotopical functor categories, 108 systems, 152, 157 3-arrow description of homotopy categories, adjunctions of, 157, see also adjunctions 5, 21, 32, 33, 81, 91, 109 of systems and 3-arrow calculi, 33 approximations of, 154, see also approxi total left derived functors, 5, 7, 128 mations of systems and left approximations, 128 colimit, 58, 86, 160, see also colimit sys total right derived functors, 5, 7, 128 tems and right approximations, 128 deformable, 87, 155 trivial cofibrations, 26 homotopical, 153 characterization of, 28 homotopy systems of, 154 trivial fibrations, 26 homotopical colimit, 87, 169, see also ho characterization of, 28 motopical colimit systems two out of six axiom, 26 180 INDEX two out of six property, 10-11, 19, 23, 79, 90, 96, 110, 117, 125 two out of three axiom, 19-20 two out of three property, 4, 11, 23, 79, 90, 96 types category of, 101 types of zigzags, 81, 101
^-categories, 22, 90, 94, 94-95 small, 22, 90, 94, 94-95 it-colimit functors, 57, 85, see also colimit functors it-limit functors, 57, 148, see also limit func tors Z^-sets, 22, 89, 94 underlying categories, 96 uniqueness categorical, 38, 38-39, 92-93, 112, 112- 114, see also categorical uniqueness homotopical, 9-10, 39, 39-40, 82-83, 92- 93, 114, 112-118, see also homotopical uniqueness units of adjunctions, 122, 122-123, 135-138 of adjunctions of systems, 157 and dinatural transformations, 167 of Kan extensions, 118, 162 universal properties, 38, 113 homotopically, 40, 115 universes, 22, 22-23, 78-79, 89-90, 94, 94- 95 basic assumption, 94 higher, 94 successor, 23, 90, 94 virtually cofibrant diagrams, 69, 69-72 virtually fibrant diagrams, 69, 69-72 we-categories, 4-5 weak equivalences between cat-systems, 63 between systems, 153 categories of, 96 in homotopical categories, 11, 23, 90, 96 in model categories, 20, 25 in we-categories, 4 natural, 24, 90, 96 Reedy, 65 weak invertibility property, 23, 79, 96 weakly equivalent objects, 23, 96 canonically, 39, 83, 92, 114 homotopically, 10 zigzags, 98, 101-112 length of, 98 restricted, 81, 98, 101 type of, 101 Bibliography
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181 Titles in This Series
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