Directed Algebraic Topology: Models of Non-Reversible Worlds

Cover......Page 1
Half-title......Page 3
Series-title......Page 4
Title......Page 5
Copyright......Page 6
Dedication......Page 7
Contents......Page 9
1 Aims and applications......Page 13
2 Some examples......Page 14
3 Directed spaces and other directed structures......Page 15
4 Formal foundations for directed algebraic topology......Page 17
5 Interactions with category theory......Page 18
7 From directed to weighted algebraic topology......Page 19
8 Terminology and notation......Page 20
9 Acknowledgements......Page 21
Part I First-order directed homotopy and homology......Page 23
1 Directed structures and first-order homotopy properties......Page 25
1.1.0 The structure of the classical interval......Page 26
1.1.1 The cylinder......Page 27
1.1.2 The cocylinder......Page 29
1.1.3 Preordered topological spaces......Page 30
1.1.4 The basic structure of directed homotopies......Page 32
1.1.5 Breaking the symmetries of classical algebraic topology......Page 33
1.1.6 Categories and directed homotopy......Page 35
1.1.7 Dioids......Page 36
1.1.9 A digression on dioids......Page 38
1.2.1 The basic setting, I......Page 40
1.2.2 The basic setting, II......Page 41
1.2.4 Concrete structures......Page 43
1.2.5 Monoidal and cartesian structures......Page 46
1.2.6 Functors and preservation of homotopies......Page 47
1.2.7 Dualities......Page 49
1.2.9 A setting based on directed homotopies......Page 50
1.3.1 Future and past homotopy equivalence......Page 52
1.3.2 Contractible and co-contractible objects......Page 53
1.3.3 Coarse d-homotopy equivalence......Page 54
1.3.4 Examples......Page 55
1.3.5 Homotopy pushouts......Page 57
1.3.6 Examples......Page 58
1.3.8 Lemma......Page 60
1.3.9 Lemma (Special pasting lemma)......Page 61
1.4.0 Spaces with distinguished paths......Page 62
1.4.1 Limits and colimits......Page 64
1.4.3 Standard models......Page 65
1.4.4 Remarks......Page 67
1.4.5 Comparing preordered spaces and d-spaces......Page 68
1.4.6 Directed paths......Page 69
1.4.7 Vortices, discs and cones......Page 70
1.4.8 Theorem (Exponentiable d-spaces)......Page 71
1.4.9 Lemma......Page 72
1.5.1 The directed structure......Page 73
1.5.2 Homotopies......Page 74
1.5.3 Homotopy functors......Page 75
1.5.5 Pointed directed homotopies......Page 76
1.6 Cubical sets......Page 77
1.6.0 The singular cubical set of a space......Page 78
1.6.1 Cubical sets......Page 79
1.6.2 Subobjects and quotients......Page 80
1.6.3 Tensor product......Page 81
1.6.4 Standard models......Page 82
1.6.5 Elementary directed homotopies......Page 83
1.6.6 The classical geometric realisation......Page 86
1.6.7 A directed geometric realisation......Page 87
1.6.8 Sets with distinguished cubes......Page 88
1.6.9 Pointed cubical sets......Page 90
1.7.0 Homotopical categories via cylinders......Page 91
1.7.1 Lemma (The preservation of h-pushouts)......Page 92
1.7.2 Mapping cones, cones and suspension......Page 93
1.7.4 Cones and suspension for d-spaces......Page 95
1.7.6 Cones and suspension for cubical sets......Page 97
1.7.7 Differential and comparison......Page 98
1.7.8 The cofibre sequences of a map......Page 99
1.7.9 Theorem and definition (The cofibre diagram)......Page 100
1.8 First-order homotopy theory by the path functor......Page 101
1.8.1 Homotopy pullbacks......Page 102
1.8.2 Homotopical categories via cocylinders......Page 103
1.8.3 Cocones and loop objects......Page 104
1.8.4 Examples of cocones and loop objects......Page 105
1.8.5 Fibre sequences......Page 106
1.8.7 The fibre–cofibre sequence......Page 107
1.8.9 Homotopy pullbacks of categories......Page 108
1.9 Other topological settings......Page 109
1.9.1 Inequilogical spaces......Page 110
1.9.2 Locally transitive spaces......Page 112
1.9.4 Sets with distinguished cubes......Page 113
1.9.5 Bitopological spaces......Page 114
1.9.6 Metrisability......Page 115
2 Directed homology and non-commutative geometry......Page 117
2.1.1 Directed chain complexes......Page 118
2.1.2 Directed homology of cubical sets......Page 120
2.1.3 Preordered coefficients......Page 121
2.1.4 Elementary computations......Page 123
2.1.5 Relative directed homology......Page 124
2.1.6 Cohomology......Page 125
2.2.1 Theorem (Homotopy invariance)......Page 126
2.2.2 Theorem (The Mayer-Vietoris sequence)......Page 127
2.2.3 Theorem and definition (Excision)......Page 128
2.2.5 Directed homology of d-spaces......Page 129
2.2.6 Proposition (Symmetry versus preorder)......Page 130
2.3.1 The interest of pointed objects in the directed case......Page 132
2.3.2 Pointed homotopies......Page 133
2.3.3 Pointed homology......Page 135
2.3.5 Theorem (Homology of a suspension)......Page 136
2.3.6 Theorem (The homology cofibre sequence of a map)......Page 137
2.3.8 An elementary example......Page 138
2.4.1 Basics......Page 139
2.4.2 Lemma (Free actions)......Page 140
2.4.4 Corollary (Free actions on acyclic spaces)......Page 141
2.5 Interactions with non-commutative geometry......Page 142
2.5.1 Rotation algebras......Page 143
2.5.2 Irrational rotation structures......Page 144
2.5.3 The non-commutative two-dimensional torus......Page 145
2.5.4 Higher foliations of codimension 1......Page 146
2.5.5 Higher foliations......Page 147
2.5.7 Lemma......Page 148
2.5.8 Theorem (The homology of irrational rotation c-sets)......Page 150
2.6.1 Directed singular homology of inequilogical spaces......Page 152
2.6.2 Axioms for directed homology......Page 153
2.6.3 Homology sequences and perfect theories......Page 154
2.6.5 The singular cubical set......Page 155
3 Modelling the fundamental category......Page 157
3.1.1 An example......Page 158
3.1.2 Directed homotopy invariance......Page 160
3.1.3 The higher structure of the cylinder......Page 161
3.1.4 The higher structure of the path functor......Page 162
3.1.6 Homotopies of d-spaces......Page 163
3.1.7 Lemma (From pushouts to pullbacks)......Page 164
3.2.1 Double homotopies and 2-homotopies......Page 165
3.2.2 Constructing double homotopies......Page 166
3.2.3 The fundamental category......Page 168
3.2.4 Theorem (The fundamental category)......Page 169
3.2.5 Homotopy monoids......Page 171
3.2.6 Pasting theorem......Page 172
3.2.7 Elementary computations......Page 174
3.2.8 Remarks......Page 176
3.3 Future and past equivalences of categories......Page 177
3.3.1 Future equivalences......Page 178
3.3.2 Lemma (Cancellation properties of future equivalences)......Page 180
3.3.3 The relation of future equivalence......Page 181
3.3.4 Full reflective subcategories as future retracts......Page 182
3.3.5 Theorem (Future equivalence and reflective subcategories)......Page 183
3.3.6 Definition and proposition (Strong contractibility)......Page 185
3.3.8 Lemma (Maximal points)......Page 186
3.3.9 Elementary examples......Page 187
3.4.1 Pf-equivalences......Page 189
3.4.2 Composition of pf-equivalences......Page 190
3.4.3 Lemma (Pf-coherence)......Page 191
3.4.4 Injections and projections......Page 192
3.4.5 Theorem (The middle model)......Page 193
3.4.6 Split pf-injections......Page 194
3.4.7 Split pf-projections......Page 195
3.4.8 Proposition (Split pf-projections)......Page 196
3.5.1 Main definitions......Page 197
3.5.3 Theorem and definition (Pf-presentations and injective models)......Page 198
3.5.5 Factorisation of adjunctions......Page 200
3.5.7 Theorem and defininition (Pf-presentations and projective models)......Page 202
3.5.8 From injective to projective models......Page 204
3.6.1 Ordinary skeleta......Page 205
3.6.2 Minimal models......Page 206
3.6.4 Injective and projective contractibility......Page 207
3.6.5 The model of the ordered line......Page 208
3.6.6 The model of the directed circle......Page 209
3.6.7 Minimal faithful models......Page 210
3.7.1 Future regularity......Page 211
3.7.2 Lemma (Future regular morphisms)......Page 212
3.7.3 Theorem (Future equivalence and regular morphisms)......Page 213
3.7.5 Lemma......Page 215
3.7.6 Definition (Future regularity equivalence)......Page 216
3.7.7 Theorem (Future equivalence and branching points)......Page 217
3.8.0 Least future retracts......Page 218
3.8.1 Spectra......Page 219
3.8.2 Theorem (Properties of the future spectrum)......Page 220
3.8.3 Lemma (Characterisation of future spectra)......Page 221
3.8.5 Spectral presentations......Page 222
3.8.7 Theorem (Preservation of future spectra and pf-spectra)......Page 223
3.8.9 Theorem (Uniqueness of future spectra, II)......Page 225
3.9.1 Future spectra of preorders......Page 226
3.9.2 Modelling an ordered space......Page 227
3.9.3 The projective model......Page 230
3.9.4 Variations......Page 231
3.9.6 The Swiss flag......Page 233
3.9.8 Faithful and non-faithful spectra......Page 235
3.9.9 Some hints at applications......Page 236
Part II Higher directed homotopy theory......Page 239
4 Settings for higher order homotopy......Page 241
4.1.0 The basic setting......Page 242
4.1.1 Double homotopies and 2-homotopies......Page 243
4.1.2 The main preservation property......Page 244
4.1.3 Theorem (The higher property of h-pushouts)......Page 245
4.1.4 Symmetric dI1-categories......Page 246
4.1.5 Cylinder functor and homotopies......Page 247
4.1.6 Theorem (The h-pushout functor on homotopies)......Page 248
4.1.7 Theorem (Homotopy invariance of the cone and suspension functors)......Page 249
4.1.8 External transposition......Page 250
4.2.1 Connections and transposition......Page 251
4.2.2 Concatenation and dI3-categories......Page 252
4.2.3 Concatenating homotopies......Page 254
4.2.5 dI4-categories......Page 256
4.2.6 Homotopical categories......Page 258
4.2.7 Functors and subcategories......Page 259
4.2.8 The structure of the directed interval......Page 260
4.3 Examples, I......Page 262
4.3.2 Categories......Page 263
4.3.3 Reflexive graphs......Page 264
4.3.4 Cubical sets......Page 265
4.4.1 Chain complexes......Page 266
4.4.2 The path functor......Page 267
4.4.3 The higher structure......Page 268
4.4.4 Positive chain complexes......Page 270
4.4.5 Directed chain complexes......Page 271
4.4.6 Singular directed chains......Page 272
4.4.7 The monoidal structure......Page 273
4.5 Double homotopies and the fundamental category......Page 274
4.5.2 Theorem (Double concatenations)......Page 275
4.5.3 Transposition of concatenations......Page 277
4.5.5 The homotopy 2-category......Page 278
4.5.6 Theorem (Weak regularity of concatenation)......Page 279
4.5.7 The fundamental category in the concrete case......Page 280
4.6.1 Proposition (Special invariance)......Page 281
4.6.2 Pasting theorem for h-pushouts......Page 282
4.6.3 Cofibrations and fibrations......Page 284
4.6.4 Theorem (Pushouts of cofibrations as h-pushouts)......Page 285
4.6.5 Extended acceleration......Page 286
4.6.6 Theorem (h-pushouts and cofibrations)......Page 287
4.6.7 Theorem (Factorisations via (co)fibrations)......Page 288
4.7.1 Theorem (The higher property of h-cokernels)......Page 289
4.7.2 The suspension functor......Page 290
4.7.4 Lemma (The comparison square)......Page 291
4.7.5 Theorem (Higher properties of the cofibre diagram)......Page 293
4.7.6 Theorem (Homology theories)......Page 294
4.8 The cone monad......Page 295
4.8.1 Theorem (The second order cone)......Page 296
4.8.2 Examples......Page 297
4.8.3 Theorem (The cone monad)......Page 298
4.8.4 Theorem (The transposition of the cone)......Page 299
4.8.6 Preadditive h-categories......Page 300
4.9.1 Reversing homotopies and 2-homotopies......Page 302
4.9.3 Theorem (Homotopy preservation)......Page 303
4.9.4 Invariance theorem......Page 304
4.9.5 Baues cofibration categories......Page 305
4.9.6 Theorem......Page 306
5 Categories of functors and algebras, relative settings......Page 308
5.1.1 Categories of diagrams......Page 309
5.1.2 Theorem (Diagrams and homotopy)......Page 310
5.1.3 Sheaves on a site......Page 311
5.1.4 Theorem (Sheaves and cocylinder)......Page 312
5.2.1 Some remarks on pushouts......Page 313
5.2.2 Bilateral slice categories and reversors......Page 314
5.2.3 Theorem and Definition (Lifting functors to slice categories)......Page 315
5.2.4 Theorem (First order homotopy structure for slice categories)......Page 317
5.2.5 Theorem (Higher homotopy structure for slice categories)......Page 319
5.2.6 Topological examples......Page 320
5.3 Algebras for a monad and the path functor......Page 321
5.3.1 Lifting functors to algebras......Page 322
5.3.3 Monads and path-functors......Page 324
5.3.4 The functor of consecutive pair of paths......Page 325
5.3.5 The remaining second-order structure......Page 326
5.3.7 Theorem (Lifting adjunctions to algebras)......Page 327
5.3.8 Formal remarks......Page 330
5.4.1 Directed topological semigroups and monoids......Page 331
5.4.2 The homotopy structure of directed topological semigroups......Page 332
5.4.3 Directed topological groups......Page 334
5.4.4 Equivariant directed homotopy......Page 335
5.4.5 Algebras for pointed d-spaces......Page 337
5.4.7 Objects under A as algebras......Page 338
5.5 The path functor of differential graded algebras......Page 339
5.5.1 Generalities......Page 340
5.5.3 The path functor......Page 341
5.5.5 Homotopy pullbacks......Page 343
5.5.6 The fibre sequence......Page 344
5.5.7 Forgetting multiplication......Page 345
5.6 Higher structure and cylinder of dg-algebras......Page 346
5.6.1 Second order paths......Page 347
5.6.3 The transposition......Page 348
5.6.4 Tensor product......Page 349
5.6.5 The co-interval......Page 350
5.6.7 The cylinder functor......Page 351
5.6.8 Free dg-algebras......Page 352
5.6.9 Unital dg-algebras......Page 353
5.7.1 The standard reversion......Page 354
5.7.2 The skew structure of cochain complexes......Page 355
5.7.4 The monad of cochain algebras......Page 356
5.8.1 The main definitions......Page 357
5.8.2 Lemma......Page 358
5.8.3 Theorem (Homology theories in the relative setting)......Page 359
5.8.4 Differential graded algebras......Page 360
5.8.7 Inequilogical spaces......Page 361
6 Elements of weighted algebraic topology......Page 363
6.1 Generalised metric spaces......Page 364
6.1.1 Real weights......Page 365
6.1.2 Directed metrics......Page 366
6.1.3 Lipschitz maps......Page 367
6.1.5 Standard models......Page 368
6.1.6 The symmetric case......Page 369
6.1.7 Proposition (Symmetrisation and products)......Page 370
6.1.8 Definition and Proposition (The length of paths)......Page 371
6.1.9 The associated topology and direction......Page 373
6.2.1 Elementary and extended paths......Page 374
6.2.2 The elementary cylinder......Page 375
6.2.3 The Lipschitz cylinder......Page 376
6.2.4 Homotopies......Page 377
6.3.1 Weighted categories......Page 378
6.3.2 Proposition (The monoidal closed structure)......Page 379
6.3.3 The elementary cylinder of weighted categories......Page 380
6.3.4 The Lipschitz cylinder of weighted categories......Page 381
6.3.5 Lemma......Page 382
6.3.6 The fundamental weighted category......Page 383
6.3.7 Geodesics......Page 384
6.4 Minimal models......Page 385
6.4.1 The fundamental weighted category of a square annulus......Page 386
6.4.2 Future equivalence of weighted categories......Page 387
6.5 Spaces with weighted paths......Page 388
6.5.1 Main definitions......Page 389
6.5.2 The weight of a map......Page 390
6.5.3 Limits......Page 391
6.5.4 Theorem (Quotients of w-spaces)......Page 392
6.5.5 Standard models......Page 393
6.5.7 Theorem (Exponentiable w-spaces)......Page 394
6.5.8 Elementary and extended paths......Page 397
6.5.9 Proposition......Page 398
6.6.1 Linear w-spaces......Page 399
6.6.2 Span-metrisable w-spaces......Page 400
6.6.3 The length adjunction......Page 401
6.6.5 Directed spaces......Page 402
6.7.1 Irrational rotation w-spaces......Page 403
6.7.2 Theorem......Page 404
6.7.4 Theorem (Lipschitz isomorphic classification)......Page 405
6.8 Tentative formal settings for the weighted case......Page 406
6.8.2 Examples......Page 407
6.8.3 A defective case......Page 408
A1.1 Smallness......Page 409
A1.2 Basic terminology......Page 410
A1.3 Universal properties, products and equalisers......Page 411
A1.4 Duality, sums and coequalisers......Page 412
A1.5 Isomorphism and equivalence of categories......Page 413
A1.6 A digression on mathematical structures and categories......Page 414
A1.7 Categories of functors......Page 415
A1.9 Universal arrows......Page 416
A2.1 Main definition......Page 417
A2.2 Particular cases......Page 418
A3.1 Main definitions......Page 419
A3.2 Remarks......Page 420
A3.4 Reflective and coreflective subcategories......Page 421
A4.2 Exponentiable objects and internal homs......Page 422
A4.4 Monads and adjunctions......Page 423
A4.5 Algebras for a monad......Page 424
A4.6 Additive categories......Page 425
A5.1 Sesquicategories......Page 426
A5.3 Natural transformations and mates......Page 427
A5.4 Mates and limits......Page 428
References......Page 430
Glossary of symbols......Page 436
Index......Page 439