Algebraic Homotopy

CONTENTS......Page 7
Preface......Page 11
Introduction......Page 13
List of symbols......Page 16
I Axioms for homotopy theory and examples of eofibration categories......Page 21
0 Cofibrations and fibrations in topology......Page 22
1 Cofibration categories......Page 25
la Appendix: fibration categories......Page 30
lb Appendix: proof of (1.4)......Page 32
2 The axioms of Quillen......Page 34
3 Categories with a natural cylinder......Page 38
3a Appendix: categories with a natural path object......Page 47
4 The category of topological spaces......Page 49
4a Appendix: the relative homotopy lifting property......Page 51
4b Appendix: the cube theorem......Page 53
5 Examples of cofibration categories......Page 54
6 The category of chain complexes......Page 59
7 The category of chain algebras......Page 65
8 The category of commutative cochain algebras......Page 78
9 The category of chain Lie algebras......Page 94
1 Some properties of a cofibration category......Page 103
2 Sets of homotopy classes......Page 112
3 The homotopy category of fibrant and cofibrant objects......Page 119
4 Functors between cofibration categories......Page 122
5 The groupoid of homotopies......Page 124
5a Appendix: homotopies and functors......Page 130
6 Homotopy groups......Page 135
6a Appendix: homotopy groups and functors......Page 138
7 Relative homotopy groups and the exact homotopy sequence of a pair......Page 139
8 Principal cofibrations and the cofiber sequence......Page 145
9 Induced maps on cofibers......Page 151
10 Homotopy groups of function spaces......Page 155
10a Appendix: homotopy groups of function spaces and functors......Page 161
11 The partial and functional suspensions......Page 162
12 The difference operator......Page 165
13 Double mapping cones......Page 167
14 Homotopy theory in a fibration category......Page 172
15 Whitehead products......Page 174
15a Appendix: Whitehead products, co-Whitehead products and cup products in topology......Page 180
16 Examples on the classification of maps in topology......Page 184
17 Homotopy groups in the category of chain algebras......Page 189
III The homotopy spectral sequences in a cofibration category......Page 199
1 Homotopy groups of homotopy limits......Page 200
2 The homotopy spectral sequence......Page 209
3 Based complexes and cohomology......Page 214
4 Complexes and twisted cohomology......Page 219
5 Cohomology with local coefficients of CW-complexes......Page 226
5a Appendix: admissible classes of complexes......Page 231
6 Eilenberg-Mac Lane spaces and cohomology with local coefficients......Page 233
7 Postnikov towers......Page 235
8 Nilpotency of function spaces in topology......Page 236
9 Homotopy groups of function spaces in topology......Page 238
10 The relative homotopy spectral sequence......Page 239
11 The relative homotopy spectral sequence for CW-complexes......Page 247
1 Detecting functors......Page 249
2 Group actions on categories......Page 251
3 Linear extensions of categories......Page 256
4 Linear coverings of categories and exact sequences for functors......Page 259
5 The cohomology of a small category......Page 265
6 Classification of linear extensions......Page 271
7 Classification of linear coverings......Page 275
1 Group actions on the category PAIR......Page 278
2 Principal and twisted maps between mapping cones......Page 280
3 A linear group action......Page 283
3a Appendix: the homotopy category of Moore spaces in degree n......Page 287
4 A quadratic group action......Page 289
5 The equivalence problem......Page 291
6 Maps between fiber spaces in a fibration category......Page 295
7 The homotopy type of a mapping cone in topology......Page 297
7a Appendix: proof of the general suspension theorem under D......Page 303
8 Example: homotopy theory of the 2-stem and of the 3-stem......Page 306
9 Example: The group of homotopy equivalence of the connected sum (S1 x S3)#(S2 x S2)......Page 310
10 The homotopy type of a fiber space in topology......Page 315
10a Appendix: proof of the general loop theorem over D......Page 322
1 Crossed chain complexes......Page 326
2 The functional suspension and the Peiffer-group for 2-dimensional CWcomplexes......Page 335
3 The homotopy category of 2-dimensional CW-complexes under D......Page 339
4 The tower of categories for the homotopy category of crossed chain complexes......Page 342
5 Homotopy systems of order n, n ^ 3......Page 356
6 The tower of categories for the homotopy category of CW-complexes under D......Page 366
7 Small models and obstructions to finiteness for CW-complexes under D......Page 376
8 n-dimensional CW-complexes and (n ?a 1)-types......Page 384
1 Twisted maps between complexes......Page 391
2 Twisted maps and 1-homotopies......Page 405
3 Examples of complexes in topology......Page 417
4 Complexes in the category of chain algebras......Page 421
1 The tower of categories for Postnikov decompositions......Page 424
2 The category of K(A, n)-fibrations over D......Page 434
3 The de Rham theorem for cohomology groups with local coefficients......Page 436
4 The tower of categories for commutative cochain algebras under D......Page 444
5 The category of A(F,n)-cofibrations under D......Page 446
6 Integral homotopy theory of spaces over D and minimal models......Page 447
1 Reduced complexes......Page 456
2 The tower of categories for reduced complexes......Page 458
3 Functors on reduced complexes and the Quillen equivalence of rational homotopy categories......Page 463
4 Whitehead's classification of simply connected 4-dimensional CWcomplexes and the corresponding result for reduced complexes......Page 467
Bibliography......Page 475
Index......Page 481