Pursuing stacks (à la poursuite des champs)
Volume I
Alexandre Grothendieck
Georges Maltsiniotis
Société Mathématique de France
Préface
vii
1. Genèse et grandeur de Pursuing Stacksvii
2. Le travail d'édition de Pursuing Stacksxi
3. Le style d'écriture de Grothendieck dans Pursuing Stacksxii
4. La structure de Pursuing Stacks en sections et chapitresxv
5. La chronologie de Pursuing Stacksxvi
6. Remerciements xviii
Introduction
xxi
1. Les ∞-groupoïdes faibles, alias ∞-champs xxii
2. Les catégories test xxxviii
3. Structures homotopiques lxiv
4. Structures d'asphéricité lxxi
5. La théorie homotopique des catégories avant et après Pursuing Stackslxxxii
Résumés section par section
xci
Réflexions Mathématiques - par Alexandre Grothendieck
cix
Pursuing Stacks
cxi
The Modelizing Story
cxv
Table of contents of The Modelizing Story
cxvi
I. The take-off
1. The importance of innocence1
2. A short look into purgatory2
3. « Fundamental ∞-groupoids » as objects of a « model category » ?6
4. A bit of ordering in the mess of « higher order structures »7
5. Jumping over the abyss!9
6. The topological model : hemispheres building up the (tentative) « universal ∞-(co)groupoid »11
7. Gluing hemispheres : the « standard » amalgamations13
8. Description of the universal primitive structure15
9. The main inductive step : just add coherence arrows !
And the abridged story of an (inescapable and irrelevant) ambiguity15
10. Cutting down redundancies - or : « l'embarras du choix »17
11. Back to the topological model.
The canonical functor from spaces to « ∞-groupoids »19
12. About replacing spaces by objects of a « model category »20
13. An urgent reflection on proper names : « stacks » and « coherators »21
II. Test categories25
14. The unnoticed failure (of the foundations of topology)25
15. Overall review on standard descriptions of homotopy types26
16. Stacks over topoi as the unifying concept for homotopical and cohomological algebra30
16¢. Categories as models for hoinotopy types.
First glimpse upon an « impressive bunch » (of modelizers)32
17. The Artin-Mazur cohomological criterion for weak equivalences35
18. Corrections and comments to letter. Bénabou's lonely approach35
19. Beginning of a provisional itinerary (through stacks)37
20. Are model categories sites ?40
21. Further glimpse upon the « bunch » of possible model categories, and a relation between n-complexes and n-stacks41
22. Ordered sets as models for hoinotopy types.43
23. Getting a basic functor M→ (Hot) from a site structure M (the faltering beginning of a systematic reflection)46
24. A bunch of topologies on (Cat)49
25. A tentative equivalence relation for topologies51
26. The dawn of test categories and test functors53
27. Digression on « geometric realization » functors57
28. The « inspiring assumption ». Modelizers58
29. The basic modelizer (Cat).
Provisional definition of test categories and elementary modelizers62
30. Starting the « asphericity game »66
31. The end of the thin air conjecturing : a criterion for test categories68
32. Provisional program of work71
33. Necessity of conditions T1 to T372
34. Examples of test categories77
35. The notion of a modelizing topos.
Need for revising the Cecil-Verdier-Artin-Mazur construction80
36. Characterization of (a particular type of) test functors with values in (Cat)85
37. The « asphericity story » told anew - the « key result » on test functors87
38. Asphericity story retold (continued) : generalized nerve functors92
39. Returning upon terminology: strict test categories, and strict modelizers96
40. Digression on cartesian products of weak equivalences in (Cat) ; weak equivalences relative to a given base object99
41. Role of the « inspiring assumption » and of saturation conditions on « weak equivalences »101
42. Terminology revised (model-preserving functors).
Submodelizers of the basic modelizer (Cat)102
43. The category Δƒof simplices without degeneracies as a weak test category - or « face complexes » as models for homotopy types104
44. Overall review of the basic notions108
a) Weak test categories109
b) Test categories and local test categories111
c) Strict test categories113
d) Weak test functors and test functors (with values in (Cat))114
III. Homotopy structures119
45. It's burning again ! Review of some « recurring striking features » of modelizers. and of standard modelizing functors119
46. Test functors with values in any rnodelizer : an observation, and an inspiring « silly question »123
47. An approach for handling (Cat)-valued test functors, and promise of a « key result » revised. The significance of contractibility125
48. A journey through abstract homotopy notions (in terms of a set W of « weak equivalences »)128
49. Contractible objects. Multiplicative intervals136
50. Reflection on some main impressions.
The foresight of an « idyllic picture » (of would-be « canonical modelizers »)139
51. The four basic « pure » homotopy notions and their interplay (a fugue with variations)140
A) Homotopy relation between maps141
13) Homotopisms, and homotopism structures143
C) Homotopy interval structures145
D) Contractibility structures147
E) Generating sets of homotopy intervals. Two standard ways of getting multiplicative intervals. Contractibility of Hom(X, V)'s149
F) The canonical homotopy structure : preliminaries on п0152
52. Inaccuracies rectified154
53. Compatibility of a functor u : M → N with a homotopy structure on M159
54. Compatibility of a homotopy structure h with a set W of « weak equivalences ». The homotopy structure hW161
55. Maps between homotopy structures164
56. Another glimpse upon canonical modelizers.
Provisional working plan - and recollection of some questions165
57. Relation of hoinotopy structures to 0-connectedness and π0.
The canonical hoinotopy structure hM of a category M168
58. Case of a totally 0-connected category M. The category (Cat) of (small) categories and homotopy classes of functors173
59. Case of the « next best » modelizer (Spaces) - and need of introducing the π0-functor as an extra structure on a would-be modelizer M175
60. Case of a strictly totally aspheric topos.
A timid start on axiomatizing the set W of weak equivalences in (Cat)177
61. Remembering about the promised « key result » at last !180
62. An embarrassing case of hasty over-axiomatization.
The unexpected richness185
63. Review of terminology (provisional)188
64. Review of properties of the « basic localizer » W(cat)192
65. Still another review of the test notions (relative to a given basic localizer)197
A) Total W-asphericity199
B) Weak W-test categories200
C) W-test categories201
D) Strict W-test categories203
E) Weak W-test functors, and W-test functors.
The « key result » in the long last ! (first version)204
F) W-test functors A→ (Cat) of strict W-test categories207
66. Revising (and fixing ? ) terminology again208
IV. Asphericity structures215
67. Back to the asphericity game : the categories (HotA)215
68. Digression on a « new continent »218
69. Digression on six weeks' scratchwork : derivators, and integration of homotopy types222
70. Digression on scratchwork (2) : cohomological properties of maps in (Cat) and in Â.
Does any topos admit a « dual » topos ? Kan fibrations rehabilitated228
71. Working program and rambling questions (group objects as models, Dold-Puppe theorem ...)234
72. Back to asphericity : criteria for a map in (Cat)240
73. Asphericity criteria (continued)247
74. Application to products of test categories249
75. Asphericity structures : a bunch of useful tautologies258
76. Examples. Totally aspheric asphericity structures264
77. The canonical functor HotM→ (Hot)266
78. Test functors and modelizing asphericity structures : the outcome (at last !) of an early « observation »268
79. Asphericity structure generated by a contractibility structure : the final shape of the « awkward main result » on test functors273
80. Reminders and questions around canonical modelizers280
81. Contractibility as the common expression of homotopy, asphericity and 0-connectedness notions.
(An overall review of the notions met with so far.)283
82. Proof of injectivity of α : Cont(M) →W-Asph(M).
Application to Hom objects and to products of aspheric functors A → M290
83. Tautologies on Im α, and related questions296
84. A silly (provisional) answer to the « silly question » (of section 46)
and the new perplexity ƒ! (Mas) ⊂ M'as ?297
85. Digression on left exactness properties of ƒ! functors, application to the inclusion i : Δ→ (Cat)304
86. Bimorphisms of coiitractibility structures as the (final ?) answer.
Does the notion of a map of asphericity structures exist ?308
87. Comments on Thomason's paper on closed model structure of (Cat)313
88. Review of pending questions and topics (questions (1) to (5), including characterizing canonical modelizers)315
89. Digression (continued) on left exactness properties of ƒ! functors322
90. Review of questions (continued) : 6) Existence of test functors and related questions.
Digression on strictly generating subcategories324
91. Review of questions (continued) : 7) Homotopy types of finite type, 8) test categories with boundary operations, 9) miscellaneous334
95. Contractors341
96. « Vertical » and « horizontal » topoi... (afterthoughts on terminology)348
97. « Projective » topoi. Morphisms and bimorphisms of contractors351
98. Sketch of proof of Δƒ being a weak test category and perplexities about its being aspheric !355
Définition des principales notations
359
Index des notations
365
Index des principaux thèmes
367
Index terminologique
369
Index des personnes citées par Grothendieck
381
Commentaires
383
1. La catégorie globulaire ou hémisphérique383
2. La catégorie cubique standard et ses variantes389
3. Les topologies test394
4. Un ensemble ordonné qui est une catégorie test faible399
5. Les notions purement homotopiques ; un contre-exemple401
6. L'adjoint à gauche du nerf ne respecte pas les objets asphériques403
7. Un procédé de construction de foncteurs test407
Glossaire des principaux termes
411
1. Les ;-groupoïdes faibles, alias ?-champs411
2. Notions de saturation d'une classe de flèches413
3. Segments414
4. Les localisateurs de base415
5. Les notions d'asphéricité dans (Cat) et les catégories de préfaisceaux416
6. Les modélisateurs et les catégories test418
7. Les foncteurs test à valeurs dans (Cat)420
8. Les notions purement homotopiques422
9. Précontracteurs et contracteurs424
10. Structures d'asphéricité424
Bibliographie
429