Nonabelian Algebraic Topology: Filtered Spaces, Crossed Complexes, Cubical Homotopy Groupoids

Contents......Page 5
Preface......Page 13
Prerequisites and reading plan......Page 17
Historical context diagram......Page 19
Sets of base points: Enter groupoids......Page 21
Groupoids in 2-dimensional homotopy theory......Page 22
Crossed modules......Page 23
Filtered spaces......Page 25
Why crossed complexes?......Page 26
Higher Homotopy Seifert–van Kampen Theorem......Page 27
Cubical sets with connections......Page 28
Why cubical homotopy omega-groupoids with connections?......Page 30
Diagram of the relations between the main structures......Page 32
Structure of the book......Page 33
I 1- and 2-dimensional results......Page 37
Introduction to Part I......Page 39
1 History......Page 41
Basic intuitions......Page 42
The fundamental group and homology......Page 44
The search for higher dimensional versions of the fundamental group......Page 46
The origin of the concept of abstract groupoid......Page 48
The Seifert–van Kampen Theorem......Page 50
Proof of the Seifert–van Kampen Theorem (groupoid case)......Page 53
The fundamental group of the circle......Page 61
Higher order groupoids......Page 64
2 Homotopy theory and crossed modules......Page 67
Homotopy groups and relative homotopy groups......Page 69
Whitehead\'s work on crossed modules......Page 74
The 2-dimensional Seifert–van Kampen Theorem......Page 78
The classifying spaces of a group and of a crossed module......Page 82
Cat1-groups......Page 85
The fundamental crossed module of a fibration......Page 88
The category of categories internal to groups......Page 92
3 Basic algebra of crossed modules......Page 100
Introduction......Page 103
Van Kampen diagrams......Page 106
Presentations and identities: 2......Page 110
Precrossed and crossed modules......Page 111
Free precrossed and crossed modules......Page 114
Free crossed module as an adjoint functor......Page 115
Precat1-groups and the existence of colimits......Page 117
Implementation of crossed modules in GAP......Page 118
4 Coproducts of crossed P-modules......Page 122
The coproduct of crossed P-modules......Page 123
The coproduct of two crossed P-modules......Page 125
The coproduct and the 2-dimensional Seifert–van Kampen Theorem......Page 129
Some special cases of the coproduct......Page 134
5 Induced crossed modules......Page 141
Pullbacks of precrossed and crossed modules......Page 143
Induced precrossed and crossed modules......Page 145
Construction of induced crossed modules......Page 148
Induced crossed modules and the Seifert–van Kampen Theorem in dimension 2......Page 149
Calculation of induced crossed modules: the epimorphism case......Page 153
The monomorphism case: inducing from crossed modules over a subgroup......Page 156
On the finiteness of induced crossed modules......Page 160
Inducing crossed modules by a normal inclusion......Page 162
Computation of induced crossed modules......Page 172
6 Double groupoids and the 2-dimensional Seifert–van Kampen Theorem......Page 178
Double categories......Page 180
The category of crossed modules over groupoids......Page 188
The fundamental double groupoid of a triple of spaces......Page 192
Thin structures on a double category: the category of double groupoids......Page 199
Connections in a double category: equivalence with thin structure......Page 206
Equivalence between crossed modules and double groupoids: folding......Page 212
Homotopy Commutativity Lemma......Page 219
Proof of the 2-dimensional Seifert–van Kampen Theorem......Page 227
II Crossed complexes......Page 241
Introduction to Part II......Page 243
7 The basics of crossed complexes......Page 245
The category of filtered topological spaces......Page 247
Modules over groupoids......Page 249
The category of crossed complexes......Page 250
Homotopy and homology groups of crossed complexes......Page 254
The fundamental crossed complex functor......Page 255
Substructures......Page 257
Homotopies of morphisms of crossed complexes......Page 260
Colimits of crossed complexes......Page 264
Computation of colimits of crossed complexes dimensionwise......Page 265
Groupoid modules bifibred over groupoids......Page 266
Crossed modules bifibred over groupoids......Page 267
Free constructions......Page 268
Free modules over groupoids......Page 269
Free crossed modules over groupoids......Page 270
Free crossed complexes......Page 272
Crossed complexes and chain complexes......Page 275
Adjoint module and augmentation module......Page 276
The derived module......Page 280
The derived chain complex of a crossed complex......Page 282
Exactness and lifting properties of the derived functor......Page 283
The right adjoint of the derived functor......Page 286
A colimit in chain complexes with operators......Page 288
8 The Higher Homotopy Seifert–van Kampen Theorem (HHSvKT) and its applications......Page 294
HHSvKT for crossed complexes......Page 295
Coproducts with amalgamation......Page 298
Pushouts......Page 299
Results on pairs of spaces: induced modules and relative homotopy groups......Page 301
Specialisation to pairs......Page 302
Induced modules and homotopical excision......Page 303
Attaching a cone and the Relative Hurewicz Theorem......Page 307
The chain complex of a filtered space and of a CW-complex......Page 310
9 Tensor products and homotopies of crossed complexes......Page 314
Some exponential laws in topology and algebra......Page 316
Monoidal closed structure on the category of modules over groupoids......Page 319
Monoidal closed structure on the category of crossed complexes......Page 323
The internal hom structure......Page 326
The bimorphisms as an intermediate step......Page 330
The tensor product of two crossed complexes......Page 331
The groupoid part of the tensor product......Page 336
The crossed module part of the tensor product......Page 337
Monoidal closed structure on chain complexes......Page 341
Crossed complexes and chain complexes: relations between the internal homs......Page 343
The tensor product of free crossed complexes is free......Page 345
The monoidal closed category of filtered spaces......Page 347
Tensor products and the fundamental crossed complex......Page 349
The Homotopy Addition Lemma for a simplex......Page 351
Simplicial sets and crossed complexes......Page 355
Covering morphisms of crossed complexes......Page 360
Coverings of free crossed complexes......Page 364
Existence, examples......Page 365
Standard free crossed resolution......Page 367
Uniqueness up to homotopy......Page 368
Some more complex examples: Free products with amalgamation and HNN-extensions......Page 372
Home for a contracting homotopy: chain complexes......Page 377
Computing a free crossed resolution......Page 378
Acyclic models......Page 389
The Acyclic Model Theorem......Page 390
Simplicial sets and normalisation......Page 393
Cubical sets and normalisation......Page 394
The Eilenberg–Zilber–Tonks Theorem......Page 396
Excision......Page 398
The cubical site......Page 404
The category of cubical sets......Page 405
Geometric realisation of a cubical set......Page 407
Monoidal closed structure on the category of cubical sets......Page 408
Tensor product of cubical sets......Page 409
Homotopies of cubical maps......Page 411
The internal hom functor on cubical sets......Page 413
Fibrant cubical sets......Page 415
Fibrations of cubical sets......Page 418
Homotopy......Page 421
An equivalence of cubical and topological homotopy sets......Page 422
The fundamental crossed complex of a cubical set......Page 424
The cubical nerve of a crossed complex......Page 425
The Homotopy Classification Theorem......Page 427
The pointed case......Page 428
Introduction......Page 432
Fibrations of crossed complexes......Page 433
Fibrations of crossed complexes and cubical nerves......Page 437
Long exact sequences of a fibration of crossed complexes......Page 439
Homotopy classification of morphisms......Page 440
Homotopy classification of maps of spaces......Page 444
Local coefficients and local systems......Page 451
Cohomology of a groupoid......Page 454
The cohomology of a cover of a space......Page 456
Dimension 2 cohomology of a group......Page 458
Crossed n-fold extensions and cohomology......Page 463
Concluding remarks to Part II......Page 469
III Cubical omega-groupoids......Page 475
Introduction to Part III......Page 477
13 The algebra of crossed complexes and cubical omega-groupoids......Page 479
Connections and compositions in cubical sets......Page 481
omega-groupoids......Page 486
The crossed complex associated to an omega-groupoid......Page 488
Folding operations......Page 491
n-shells: coskeleton and skeleton......Page 499
The equivalence of -groupoids and crossed complexes......Page 505
The Homotopy Addition Lemma and properties of thin elements......Page 508
14 The cubical homotopy omega-groupoid of a filtered space......Page 516
The cubical homotopy groupoid of a filtered space......Page 518
The fibration and deformation theorems......Page 523
The HHSvKT Theorem for omega-groupoids......Page 528
The HHSvKT for crossed complexes......Page 532
Realisation properties of -groupoids and crossed complexes......Page 534
Free properties......Page 536
Homology and homotopy......Page 538
Relative Hurewicz Theorem: dimension 1......Page 540
Absolute Hurewicz Theorem and Whitehead\'s exact sequence......Page 541
The cubical Dold–Kan Theorem......Page 544
15 Tensor products and homotopies......Page 549
Monoidal closed structure on omega-groupoids......Page 550
Relations between the internal homs for cubes and for omega-groupoids......Page 554
The monoidal closed structure on crossed complexes revisited......Page 556
The internal hom on crossed complexes......Page 557
Bimorphisms on crossed complexes......Page 561
The tensor product of crossed complexes......Page 565
Another description of the internal hom in Crs......Page 566
Crossed complexes and cubical sets......Page 567
The Eilenberg–Zilber natural transformation......Page 568
The symmetry of tensor products......Page 570
The pointed case......Page 572
Dense subcategories......Page 573
Application to the tensor product of covering morphisms......Page 575
16 Future directions?......Page 580
Problems and questions......Page 581
Appendices......Page 589
A resumé of some category theory......Page 591
Notation for categories......Page 592
Representable functors......Page 593
Slice and comma categories......Page 594
Colimits and limits......Page 595
Generating objects and dense subcategories......Page 599
Adjoint functors......Page 600
Adjoint functors, limits and colimits......Page 602
Abelianisations of groupoids......Page 604
Coends and ends......Page 605
Simplicial objects......Page 607
Crossed complexes, omega-groupoids and simplicial sets......Page 610
Fibrations of categories......Page 613
Cofibrations of categories......Page 616
Pushouts and cocartesian morphisms......Page 619
Crossed squares and triad homotopy groups......Page 622
Groupoids bifibred over sets......Page 624
Free groupoids......Page 625
Covering morphisms of groupoids......Page 626
Model categories for homotopy theory......Page 630
Products of categories and coherence......Page 634
The internal hom for categories and groupoids......Page 636
The monoid of endomorphisms in the case of groupoids......Page 638
The symmetry groupoid and the actor of a groupoid......Page 641
The case of a group......Page 642
Monoidal and monoidal closed categories......Page 644
Crossed modules and quotients of groups......Page 648
Bibliography......Page 651
Glossary of Symbols......Page 679
Index......Page 689