Nonabelian Algebraic Topology: Filtered Spaces, Crossed Complexes, Cubical Homotopy Groupoids (EMS Tracts in Mathematics)

The main theme of this book is that the use of filtered spaces rather than just topological spaces allows the development of basic algebraic topology in terms of higher homotopy groupoids; these algebraic structures better reflect the geometry of subdivision and composition than those commonly in use. Exploration of these uses of higher dimensional versions of groupoids has been largely the work of the first two authors since the mid 1960s. The structure of the book is intended to make it useful to a wide class of students and researchers for learning and evaluating these methods, primarily in algebraic topology but also in higher category theory and its applications in analogous areas of mathematics, physics, and computer science. Part I explains the intuitions and theory in dimensions 1 and 2, with many figures and diagrams, and a detailed account of the theory of crossed modules. Part II develops the applications of crossed complexes. The engine driving these applications is the work of Part III on cubical $\omega$-groupoids, their relations to crossed complexes, and their homotopically defined examples for filtered spaces. Part III also includes a chapter suggesting further directions and problems, and three appendices give accounts of some relevant aspects of category theory. Endnotes for each chapter give further history and references. A publication of the European Mathematical Society (EMS). Distributed within the Americas by the American Mathematical Society. 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 The main theme of this book is that the use of filtered spaces rather than just topological spaces allows the development of basic algebraic topology in terms of higher homotopy groupoids; these algebraic structures better reflect the geometry of subdivision and composition than those commonly in use. Exploration of these uses of higher dimensional versions of groupoids has been largely the work of the first two authors since the mid 1960s. The structure of the book is intended to make it useful to a wide class of students and researchers for learning and evaluating these methods, primarily in algebraic topology but also in higher category theory and its applications in analogous areas of mathematics, physics and computer science. Part I explains the intuitions and theory in dimensions 1 and 2, with many figures and diagrams, and a detailed account of the theory of crossed modules. Part II develops the applications of crossed complexes. The engine driving these applications is the work of Part III on cubical ω-groupoids, their relations to crossed complexes, and their homotopically defined examples for filtered spaces. Part III also includes a chapter suggesting further directions and problems, and three appendices give accounts of some relevant aspects of category theory. Endnotes for each chapter give further history and references. Sets Of Base Points, Enter Groupoids -- 1-and 2-dimensional Results -- History -- Homotopy Theory And Crossed Modules -- Basic Algebra Of Crossed Modules -- Coproducts Of Crossed P-modules -- Double Groupoids And The 2-dimensional Seifert-van Kampen Theorem -- Crossed Complexes -- The Basics Of Crossed Complexes -- The Higher Homotopy Seifert-van Kampen Theorem (hhsvkt) And Its Applications -- Tensor Products And Homotopies Of Crossed Complexes -- Resolutions -- The Cubial Classifying Space Of A Crossed Complex. Ronald Brown, Philip J.higgins, Rafael Sivera, With Contributions By Christopher D. Wensley And Sergei V. Soloviev. Includes Bibliographical References (p. [615]-642) And Index.