PURSUINGSTACKS (À la poursuite des Champs) First episode: THEMODELIZINGSTORY (histoire de modèles) by Alexander Grothendieck 1983 ii Pursuing Stacks (À la poursuite des Champs) First episode: The modelizing story (histoire de modèles) by Alexander Grothendieck (* March 28, 1918 in Berlin; † November 13, 2014 Lasserre, Ariège) Original typescript written 1983 This 2015 edition (including all source files) by the scrivener is in the public domain. Typeset with LATEX and the fabulous memoir class. Composed with Bitstream Charter for the text type and Paul Pichaureau’s mathdesign package for the math type. All diagrams and illustrations were made with Till Tantau’s TikZ (TikZ ist kein Zeichenprogramm) and the tikz-cd package. Short contents Short contents iii Contents iv Scrivener’s preface xi I The Take-off (a letter to Daniel Quillen)1 II Test categories and test functors 21 III Grinding my way towards canonical modelizers 93 IV Asphericity structures and canonical modelizers 171 V Homology and cohomology (abelianization of homotopy types) 253 VI Schematization 371 VII Linearization of homotopy types (2) 459 Bibliography 489 iii Contents Short contents iii Contents iv Scrivener’s preface xi I The Take-off (a letter to Daniel Quillen)1 1 The importance of innocence.1 2 A short look into purgatory.2 3 “Fundamental -groupoids” as objects of a “model category”?4 4 A bit of ordering1 in the mess of “higher order structures”.5 5 Jumping over the abyss!7 6 The topological model: hemispheres building up the (tentative) “universal -(co)groupoid”.9 7 Gluing hemispheres: the “standard amalgamations”. 10 8 Description of the universal1 primitive structure. 11 9 The main inductive step: just add coherence arrows! The abridged story of an (inescapable and irrelevant) ambiguity 12 10 Cutting down redundancies – or: “l’embarras du choix”. 14 11 Returning to the topological model (the canonical functor from spaces to “ -groupoids”). 15 12 About replacing spaces by objects of a “model category”. 151 13 An urgent reflection on proper names: “Stacks” and “coherators”. 16 II Test categories and test functors 21 14 The unnoticed failure (of the present foundations of topology for expressing topological intuition). 21 15 Overall review of standard descriptions of homotopy types. 22 16 Stacks over topoi as unifying concept for homotopical and cohomological algebra. 24 16bis Categories as models for homotopy types. First glimpse upon an “impressive bunch” (of modelizers). 25 17 The Artin-Mazur cohomological criterion for weak equivalence. 27 iv Contents v 18 Corrections and contents to letter. Bénabou’s lonely approach. 28 19 Beginning of a provisional itinerary (through stacks). 29 20 Are model categories sites? 31 21 Further glimpse upon the “bunch” of possible model categories and a relation between n-complexes and n-stacks. 32 22 Oriented sets as models for homotopy types. 33 23 Getting a basic functor M (Hot) from a site structure M (altering beginning of a systematic! reflection). 35 24 A bunch of topologies on (Cat). 37 25 A tentative equivalence relation for topologists. 39 26 The dawn of test categories and test functors. 40 27 Digression on “geometric realization” functors. 43 28 The “inspiring assumption”. Modelizers. 44 29 The basic modelizer (Cat). Provisional definition of test categories and elementary modelizers. 47 30 Starting the “asphericity game”. 50 31 The end of the thin air conjecturing: a criterion for test categories. 52 32 Provisional program of work. 54 33 Necessity of conditions T1 to T3, and transcription in terms of elementary modelizers. 55 34 Examples of test categories. 59 35 The notion of a modelizing topos. Need for revising the Cech-Verdier-Artin-Mazurˇ construction. 61 36 Characterization of a particular type of test functors with values in (Cat). 65 37 The “asphericity story” told anew – the “key result”. 68 38 Asphericity story retold (cont’d): generalized nerve functors. 72 39 Returning upon terminology: strict test categories, and strict modelizers. 75 40 Digression on cartesian products of weak equivalences in (Cat); 4 weak equivalences relative to a given base object. 77 41 Role of the “inspiring assumption”, and of saturation conditions on “weak equivalences”. 78 42 Terminology revised (model preserving functors). Submodelizers of the basic modelizer (Cat). 79 43 The category f of simplices without degeneracies as a weak test category – or “face complexes” as models for homotopy types. 81 44 Overall review of the basic notions. 84 a) Weak test categories. 85 b) Test categories and local test categories. 87 c) Strict test categories. 88 d) Weak test functors and test functors (with values in (Cat)). 89 III Grinding my way towards canonical modelizers 93 vi Contents 45 It’s burning again! Review of some “recurring striking features” of modelizers and standard modelizing functors. 93 46 Test functors with values in any modelizer: an observation, with an inspiring “silly question”. 96 47 An approach for handling (Cat)-valued test functors, and promise of a “key result” revised. The significance of contractibility. 98 48 A journey through abstract homotopy notions (in terms of a set W of “weak equivalences”). 101 49 Contractible objects. Multiplicative intervals. 108 50 Reflection on some main impressions. The foresight of an “idyllic picture” (of would-be “canonical modelizers”). 110 51 The four basic “pure” homotopy notions with variations. 111 A) Homotopy relation between maps 112 B) Homotopisms, and homotopism structures 113 C) Homotopy interval structures 115 D) Contractibility structures 116 E) Generating sets of homotopy intervals. Two standard ways of generating multiplicative intervals. Contractibility of Hom(X , Y )’s 118 F) The canonical homotopy structure: preliminaries on π0. 121 52 Inaccuracies rectified. 122 53 Compatibility of a functor u : M N with a homotopy structure on M. 127 ! 54 Compatibility of a homotopy structure with a set W of “weak equivalences”. The homotopy structure hW . 128 55 Maps between homotopy structures. 130 56 Another glimpse upon canonical modelizers. Provisional working plan – and recollection of some questions. 131 57 Relation of homotopy structures to 0-connectedness and π0. The canonical homotopy structure hM of a category M. 134 58 Case of totally 0-connected category M. The category (Cat) of (small) categories and homotopy classes of functors. 137 59 Case of the “next best” modelizer (Spaces) – and need of introducing the π0-functor as an extra structure on a would-be modelizer M. 139 60 Case of strictly totally aspheric topos. A timid start on axiomatizing the set W of weak equivalences in (Cat). 141 61 Remembering about the promised “key result” at last! 143 62 An embarrassing case of hasty over-axiomatization. The unexpected riches. 147 63 Review of terminology (provisional). 150 W 64 Review of properties of the “basic localizer” (Cat). 152 65 Still another review of the test notions (relative to given basic localizer W). 156 A) Total asphericity. 158 B) Weak W-test categories. 159 C) W-test categories. 160 D) Strict W-test categories. 161 Contents vii E) Weak W-test functors and W-test functors. 161 F) W-test functors A (Cat) of strict W-test categories. 164 ! 66 Revising (and fixing?) terminology again. 165 IV Asphericity structures and canonical modelizers 171 67 Setting out for the asphericity game again: variance of the category (HotA), for arbitrary small category A and aspheric functors. 171 68 Digression on a “new continent”. 174 69 Digression on six weeks’ scratchwork: derivators, and integration of homotopy types. 177 70 Digression on scratchwork (2): cohomological properties of maps in (Cat) and in Aˆ. Does any topos admit a “dual” topos? Kan fibrations rehabilitated. 182 71 Working program and rambling questions (group objects as models, Dold-Puppe theorem. ). 186 72 Back to asphericity: criteria for a map in (Cat). 191 73 Asphericity criteria (continued). 197 74 Application to products of test categories. 198 75 Asphericity structures: a bunch of useful tautologies. 205 76 Examples. Totally aspheric asphericity structures. 211 77 The canonical functor (HotM ) (Hot). 213 78 Test functors and modelizing asphericity! structures: the outcome (at last!) of an easy “observation”. 214 79 Asphericity structure generated by a contractibility structure: the final shape of the “awkward main result” on test functors. 218 80 Reminders and questions around canonical modelizers. 224 81 Contractibility as the common expression of homotopy, asphericity and 0-connectedness notions. (An overall review of the notions met with so far.) 227 82 Proof of injectivity of α : Contr(M) , W-Asph(M). Application to Hom objects and to products! of aspheric functors A M. 233 83 Tautologies! on Im α, and related questions. 238 84 A silly (provisional) answer to the “silly question” – and the new perplexity f!(Mas) Mas0 ? 239 85 Digression on left exactness⊂ properties of f! functors, application to the inclusion i : , (Cat). 244 86 Bimorphisms of contractibility structure! as the (final?) answer. Does the notion of a map of asphericity structures exist? 248 V Homology and cohomology (abelianization of homotopy types) 253 87 Comments on Thomason’s paper on closed model structure on (Cat). 253 88 Review of pending questions and topics (questions 1) to 5), including characterizing canonical modelizers). 254 89 Digression (continued) on left exactness properties of f! functors. 260 viii Contents 90 Review of questions (continued): 6) Existence of test functors and related questions. Digression on strictly generating subcategories. 261 91 Review of questions (continued): 7) Homotopy types of finite type, 8) test categories with boundary operations, 9) miscellaneous. 269 92 Short range working program, and an afterthought on abelianization of homotopy types: a handful of questions around the Whitehead and Dold-Puppe theorems. 274 93 The afterthought continued: abelianizators, and “standard” abelianizators for categories with boundary operators. 283 94 Afterthought (continued): retrospective on the “De Rham complex with divided powers” and on some wishful thinking about linearization of homotopy types and arbitrary ground-ring extension in homotopy types.