Nonabelian algebraic topology

Contents......Page 2
Preface......Page 11
Introduction......Page 15
I 1 and 2-dimensional results......Page 21
Introduction to Part I......Page 23
1 History......Page 25
1.1 Basic intuitions......Page 26
1.2 The fundamental group and homology......Page 27
1.3 The search for higher dimensional versions of the fundamental group......Page 29
1.4 The origin of the concept of abstract groupoid......Page 30
1.5 The van Kampen Theorem......Page 32
1.6 Proof of the van Kampen Theorem (groupoid case)......Page 34
1.7 The fundamental group of the circle......Page 40
1.8 Higher order groupoids......Page 42
2 Homotopy theory and crossed modules......Page 45
2.1 Homotopy groups and relative homotopy groups......Page 46
2.2 Whitehead's work on crossed modules......Page 51
2.3 The 2-dimensional van Kampen Theorem......Page 54
2.4 The classifying spaces of a group and of a crossed module......Page 57
2.5 Cat1-groups.......Page 60
2.6 The fundamental crossed module of a fibration......Page 62
2.7 The category of categories internal to groups......Page 66
3 Basic algebra of crossed modules......Page 71
3.1 Presentation of groups and identities among relations.......Page 73
3.2 van Kampen diagrams......Page 77
3.3 Precrossed and crossed modules......Page 80
3.4 Free precrossed and crossed modules......Page 83
3.5 Precat1-groups and the existence of colimits......Page 86
3.7 Notes......Page 87
4 Coproducts of crossed P-modules......Page 89
4.1 The coproduct of crossed P-modules......Page 90
4.2 The coproduct of two crossed P-modules......Page 91
4.3 The coproduct and the 2-dimensional van Kampen Theorem......Page 95
4.4 Some special cases of the coproduct......Page 99
4.5 Notes......Page 103
5 Induced crossed modules......Page 105
5.1 Pullbacks of precrossed and crossed modules.......Page 107
5.2 Induced precrossed and crossed modules......Page 109
5.3 Induced crossed modules: Construction in general.......Page 111
5.4 Induced crossed modules and the 2-dimensional van Kampen Theorem......Page 113
5.5 Calculation of induced crossed modules: the epimorphism case.......Page 115
5.6 The monomorphism case. Inducing from crossed modules over a subgroup......Page 118
5.7 On the finiteness of some induced crossed modules......Page 122
5.8 Inducing crossed modules by a normal inclusion......Page 123
5.9 Computational issues for induced crossed modules......Page 131
5.10 Notes......Page 134
6 Double groupoids and the 2-dimensional van Kampen Theorem.......Page 135
6.1 Double categories......Page 137
6.2 The category XMod of crossed modules of groupoids......Page 143
6.3 The fundamental double groupoid of a triple of spaces.......Page 146
6.4 Thin structures on a double category. The category DGpds of double groupoids.......Page 153
6.5 Connections in a double category: equivalence with thin structure.......Page 159
6.6 Equivalence between XMod and DGpds: folding.......Page 164
6.7 Homotopy commutativity lemma.......Page 170
6.8 Proof of the 2-dimensional van Kampen Theorem.......Page 177
II Crossed complexes......Page 185
Introduction to Part II......Page 187
7 The basics of crossed complexes......Page 193
7.1.1 The category FTop of filtered topological spaces.......Page 195
7.1.2 The category Crs of crossed complexes.......Page 197
7.1.3 The fundamental crossed complex functor.......Page 199
7.1.4 Homotopy and homology groups of crossed complexes.......Page 200
7.1.5 Homotopies of morphisms of crossed complexes......Page 201
7.2.1 Groupoids bifibred over sets......Page 203
7.2.2 Groupoid modules bifibred over groupoids......Page 204
7.2.3 Crossed modules bifibred over groupoids......Page 206
7.3.1 n-truncated crossed complexes.......Page 209
7.3.2 Colimits of crossed complexes.......Page 212
7.4.1 Free groupoids.......Page 214
7.4.2 Free crossed modules, and free modules, over groupoids.......Page 215
7.4.3 Free crossed complexes.......Page 216
7.5 Crossed complexes and chain complexes......Page 219
7.5.1 Adjoint module and augmentation module.......Page 220
7.5.2 The derived module......Page 224
7.5.3 The derived chain complex of a crossed complex......Page 225
7.5.4 Exactness and lifting properties of......Page 226
7.5.5 The right adjoint of the derived functor......Page 229
7.5.6 Some colimits in chain complexes.......Page 231
7.6 Notes......Page 232
8 The Higher Homotopy van Kampen Theorem (HHvKT) and its applications......Page 233
8.1 HHvKT for crossed complexes......Page 234
8.2.1 Coproducts with amalgamation......Page 236
8.2.2 Pushouts......Page 237
8.3.1 Specialisation to pairs......Page 239
8.3.3 Induced modules and homotopical excision......Page 240
8.3.4 Attaching a cone, and the Relative Hurewicz Theorem......Page 243
8.4 The chain complex of a filtered space and of a CW-complex.......Page 245
9 Tensor products and homotopies of crossed complexes......Page 247
9.1 Some exponential laws in topology and algebra......Page 248
9.2 Monoidal closed structure on Mod.......Page 252
9.3 Monoidal closed structure on Crs......Page 254
9.3.1 The internal hom structure in Crs......Page 256
9.3.2 The bimorphisms as an intermediate step......Page 260
9.3.3 The tensor product of two crossed complexes......Page 261
9.4.1 The groupoid part of the tensor product.......Page 265
9.4.2 The crossed module part of the tensor product.......Page 267
9.5.1 Monoidal closed structure on chain complexes......Page 270
9.5.2 Abelianisation and the closed category structure......Page 271
9.6 The tensor product of free crossed complexes is free......Page 273
9.7 The monoidal closed category of filtered spaces......Page 275
9.8 Tensor products and the fundamental crossed complex......Page 276
9.9 The homotopy addition lemma for a simplex.......Page 278
9.10 Notes......Page 282
10.1 The cubical site......Page 283
10.1.2 The category Cub of cubical sets.......Page 284
10.1.3 Geometric realisation of a cubical set......Page 285
10.2 Monoidal closed structure on Cub......Page 286
10.2.1 Tensor product of cubical sets......Page 287
10.2.2 Homotopies of cubical maps......Page 289
10.2.3 The internal hom functor on Cub......Page 290
10.3.1 Kan cubical sets......Page 292
10.3.2 Kan fibrations of cubical sets......Page 295
10.3.3 Homotopy......Page 296
10.3.4 Relation with simplicial sets......Page 297
10.3.5 The equivalence of homotopy categories......Page 298
10.4 Cubical sets and crossed complexes......Page 299
10.4.1 The fundamental crossed complex of a cubical set......Page 300
10.4.2 The cubical nerve of a crossed complex......Page 301
10.4.3 The homotopy classification theorem......Page 302
10.5 Fibrations of crossed complexes......Page 303
10.6 The pointed case......Page 306
10.7 Applications......Page 308
11.1.1 Existence, examples and uniqueness......Page 313
11.1.2 Some more complex examples: Free products with amalgamation and HNN-extensions......Page 318
11.2 Construction of free crossed resolutions of groupoids......Page 322
11.2.1 Covering morphisms of groupoids......Page 323
11.2.2 Covering morphisms of crossed complexes......Page 325
11.2.3 Coverings of free crossed complexes......Page 327
11.2.4 A computational procedure......Page 328
11.3 Acyclic models......Page 336
11.3.1 The Acyclic Model Theorem......Page 337
11.3.2 Simplicial sets and normalisation......Page 339
11.3.3 Cubical sets and normalisation......Page 340
11.3.4 Relating simplicial and cubical by acyclic models......Page 341
11.3.6 Excision......Page 342
11.4 Notes......Page 343
12.1 The exact sequences of a fibration of crossed complexes......Page 345
12.2 Homotopy classification of morphisms......Page 346
12.3 Generalisation of abstract kernel theory......Page 348
12.4 Homotopy classification of maps of spaces......Page 349
12.5 Cohomology of a group......Page 350
12.6 Cohomology of groups as classes of crossed extensions......Page 351
12.7 Local systems......Page 356
12.8 Dimension 2 cohomology with coefficients in a crossed module and extension theory......Page 357
12.9 Notes......Page 359
III -groupoids......Page 361
Introduction to Part III......Page 363
13 The category -Gpds of -groupoids......Page 365
13.1 Connections and compositions in cubical complexes......Page 367
13.2 -groupoids......Page 371
13.3 The crossed complex associated to an -groupoid.......Page 372
13.4 Folding operations......Page 375
13.5 n-shells: coskeleton and skeleton......Page 381
13.6 The equivalence of -Gpds and Crs......Page 386
13.7 The HAL and properties of thin elements......Page 389
14 The cubical homotopy -groupoid of a filtered space......Page 395
14.1 Construction of the homotopy -groupoid of a filtered space......Page 397
14.2 The fibration and deformation theorems......Page 400
14.3 The HHvKT theorem for -groupoids......Page 405
14.4 The HHvKT for crossed complexes......Page 408
14.5 Realisation properties of -groupoids and crossed complexes......Page 410
14.6 Free properties......Page 411
14.7 Homology and homotopy......Page 413
14.8.1 Notes to section 14.7......Page 416
15 Tensor products and homotopies......Page 417
15.1 The monoidal closed structure on omega-groupoids......Page 418
15.1.1 Relations between the internal homs for cubes and for omega-groupoids.......Page 421
15.2 The monoidal closed structure on crossed complexes revisited......Page 423
15.2.1 The internal hom on crossed complexes......Page 424
15.2.2 Bimorphisms on crossed complexes......Page 427
15.2.3 The tensor product of crossed complexes......Page 430
15.2.5 Crossed complexes and cubical sets......Page 431
15.3 The Eilenberg-Zilber natural transformation......Page 432
15.4 The symmetry of tensor products......Page 433
15.5 The pointed case......Page 435
15.6 Dense subcategories......Page 436
15.7 Fibrations and coverings of -groupoids......Page 437
15.8 Application to the tensor product of covering morphisms......Page 438
16.1 Problems......Page 439
A.2 Representable functors......Page 445
A.3 Colimits and limits......Page 446
A.4 Generating objects and dense subcategories......Page 449
A.5 Adjoint functors......Page 450
A.7 Fibrations of categories......Page 452
A.8 Cofibrations of categories......Page 456
A.9 Pushouts and cocartesian morphisms......Page 458
A.10 Groupoids bifibred over sets......Page 460
B.1 Products of categories and coherence......Page 463
B.2 Cartesian closed categories......Page 464
B.3 The internal hom for categories and groupoids......Page 465
B.4 The monoid of endomorphisms in the case of groupoids......Page 467
B.5 The symmetry groupoid and the actor of a groupoid......Page 469
B.6 The case of a group......Page 470
B.7 Crossed modules and quotients of groups......Page 471