Homological Algebra: The Interplay of Homology with Distributive Lattices and Orthodox Semigroups

Contents......Page 10
Preface......Page 8
0.1 Homological algebra in a non-abelian setting......Page 14
0.2 The coherence problem for subquotients......Page 15
0.3 The transfer functor......Page 16
0.4 Distributivity and coherence......Page 17
0.5 Universal models and crossword chasing......Page 19
0.6 Outline......Page 21
0.7 Further extensions......Page 22
0.9 Acknowledgements......Page 23
1.1.1 Monomorphisms and epimorphisms......Page 25
1.1.2 Lattices......Page 26
1.1.3 Distributive and modular lattices......Page 27
1.2 Coherence and distributive lattices......Page 28
1.2.1 Subquotients and regular induction......Page 29
1.2.2 Relations of abelian groups......Page 30
1.2.3 Induced relations and canonical isomorphisms......Page 32
1.2.4 Examples of incoherence......Page 34
1.2.5 Coherent systems of isomorphisms......Page 35
1.2.7 Coherence Theorem of homological algebra (Reduced form)......Page 36
1.3.1 Representing a bifiltered object......Page 38
1.3.2 Extending the representation......Page 40
1.3.3 Preparing a further extension......Page 41
1.3.4 The complete representation......Page 43
1.3.5 The Jordan-Holder Theorem......Page 44
1.3.6 Representing a sequence of morphisms......Page 46
1.4 Coherence and representations of spectral sequences......Page 48
1.4.1 The universal model of the filtered complex......Page 49
1.4.2 The spectral sequence......Page 51
1.4.3 The spectral sequence, continued......Page 52
1.4.4 Transgressions......Page 53
1.4.5 A non-distributive structure......Page 54
1.5 Introducing p-exact categories......Page 55
1.5.1 Some terminology......Page 56
1.5.3 Kernels and cokernels......Page 57
1.5.4 Exact categories and exact functors......Page 58
1.5.5 Smallness......Page 59
1.5.6 Examples......Page 60
1.5.7 Galois connections......Page 61
1.5.8 Modular lattices and modular connections......Page 63
1.6.1 Direct and inverse images of abelian groups......Page 65
1.6.2 The transfer functor......Page 66
1.6.3 Distributivity and coherence......Page 67
1.6.4 The category of sets and partial bijections......Page 68
1.6.5 Generalisations......Page 69
1.7.1 The Birkhoff Theorem (finite case)......Page 71
1.7.2 The Birkhoff Theorem (general case)......Page 73
2 Puppe-exact categories......Page 75
2.1.1 Additive categories and biproducts......Page 76
2.1.3 Theorem and Definition (Semiadditive categories)......Page 77
2.1.4 Additive categories......Page 78
2.1.5 Theorem and definition (Abelian categories)......Page 79
2.1.6 Biproducts in abelian categories......Page 81
2.1.8 Examples of split products......Page 82
2.2 Subobjects, quotients and the transfer functor......Page 83
2.2.2 Exact sequences......Page 84
2.2.3 Theorem (Modular lattices)......Page 86
2.2.4 Lemma (Pullbacks and pushouts in p-exact categories)......Page 87
2.2.5 Direct and inverse images......Page 89
2.2.6 Theorem and Definition (The transfer functor)......Page 90
2.2.8 Further remarks on modular lattices......Page 93
2.2.9 Lemma (Noether isomorphisms)......Page 95
2.3.1 The associated projective category......Page 96
2.3.2 Proposition (The projective congruence of vector spaces)......Page 97
2.3.3 Projective spaces and projective maps......Page 98
2.3.4 Remarks......Page 99
2.3.5 Lemma (The lack of products)......Page 100
2.4 Categories with a regular involution......Page 101
2.4.2 Projections and monomorphisms......Page 102
2.4.3 Bicommutative squares......Page 103
2.4.4 Lemma (Bicommutative squares of monos)......Page 104
2.4.5 Involution and order......Page 105
2.4.6 Domination......Page 106
2.4.8 The modular operation......Page 107
2.5 Relations for p-exact categories......Page 108
2.5.2 The construction of relations......Page 109
2.5.3 Theorem (Functors and relations)......Page 111
2.5.5 Proposition (Monorelations and isomorphisms)......Page 112
2.5.6 Other factorisations......Page 113
2.5.7 Special subobjects and quotients......Page 114
2.5.8 The proof of associativity [T2]......Page 115
2.5.9 Lemma (Epi-mono factorisation of a pullback)......Page 117
2.6.1 Exact squares......Page 118
2.6.2 Theorem (Exact squares, I)......Page 119
2.6.3 Theorem (Exact squares, II)......Page 120
2.6.5 Subquotients and monorelations......Page 122
2.6.7 Induction on subquotients......Page 123
2.6.8 Canonical isomorphisms......Page 124
2.6.9 Regular induction......Page 125
2.7 Coherence, distributivity, orthodoxy......Page 126
2.7.2 Theorem (Distributive properties)......Page 127
2.7.3 Orthodox and inverse semigroups......Page 128
2.7.4 Orthodox and inverse categories......Page 129
2.7.5 Theorem (Coherence for involutive categories)......Page 130
2.7.6 Theorem (Coherence Theorem for homological algebra)......Page 133
2.7.7 Corollary (Distributivity and coherence of induction)......Page 137
2.7.9 Lemma (Quasi inverse involutive semigroups)......Page 138
2.8.1 Weak induction......Page 140
2.8.3 The inverse symmetrisation......Page 141
2.8.5 Theorem (The distributive expansion)......Page 142
2.8.6 Lemma (Bicommutative squares in inverse categories)......Page 144
2.8.7 Theorem (Boolean p-exact categories)......Page 145
2.8.8 The p-exact category J......Page 146
3.1.1 Involutive categories......Page 148
3.1.2 RO-categories......Page 149
3.1.3 The transfer of projections......Page 150
3.1.6 Restrictions......Page 151
3.1.8 Null morphisms......Page 152
3.2.2 RO-squares......Page 153
3.2.4 The vertical composition......Page 154
3.2.7 Lemma......Page 155
3.3.1 Epi-mono factorisations......Page 156
3.3.3 Proposition (Split projections)......Page 157
3.3.5 Properties of the projection-completion......Page 158
3.3.7 Theorem (The biuniversal property)......Page 159
3.3.8 Remarks......Page 160
3.3.9 Lemma (Factorising a RO-square)......Page 161
3.4.2 Definition......Page 162
3.4.5 Proposition (The orders of projections)......Page 163
3.4.6 Theorem (Bicommutative squares)......Page 164
3.5 RE-categories, II......Page 166
3.5.3 The denominator of a projection......Page 167
3.5.4 Proposition......Page 168
3.5.6 Proposition and Definition (Special null morphisms)......Page 169
3.6.1 Definition......Page 170
3.6.4 Corollary......Page 171
3.6.6 The RE-subcategory spanned by a subgraph......Page 172
3.6.8 Local properties......Page 173
3.7.3 Comma objects......Page 174
3.7.4 Theorem (Strict completeness)......Page 175
4.1.1 Theorem (RE-categories and relations)......Page 176
4.1.2 Corollary (Further properties of relations)......Page 180
4.1.4 Theorem......Page 181
4.1.5 Componentwise p-exact categories......Page 182
4.1.7 *Pseudo-completeness of EX......Page 183
4.2.2 Lemma (Characterisations of the modular relations)......Page 184
4.2.4 A new double category of modular lattices......Page 186
4.2.5 The transfer functor of a RE-category......Page 187
4.2.6 RE-functors and restrictions......Page 188
4.3.1 Proposition (The modular operation for projections)......Page 189
4.3.3 Proposition and Definition (Lattices of projections)......Page 191
4.3.5 Proposition......Page 192
4.3.7 Theorem (Canonical isomorphisms and regular induction)......Page 193
4.4.1 Theorem and Definition (Distributive RE-categories)......Page 194
4.4.2 Domination in distributive RE-categories......Page 195
4.4.5 Theorem (The distributive expansion of a RE-category)......Page 196
4.4.6 The orthodox expansion......Page 197
4.5.1 The idempotent case......Page 198
4.5.2 Theorem......Page 199
4.5.4 Theorem......Page 201
4.5.5 Theorem (Pre-idempotent p-exact categories)......Page 203
4.6.1 A category of relations......Page 204
4.6.2 Sets and partial identities......Page 205
4.6.3 Semitopological spaces and partial identities......Page 206
4.6.4 Relations and idempotence......Page 207
4.6.6 Projections and subquotients......Page 208
4.6.7 Universal embeddings......Page 209
4.7.1 Distributive joins in semilattices......Page 210
4.7.3 Lemma (Homomorphisms and partitions)......Page 211
4.7.4 Distributive joins in inverse categories......Page 212
4.7.5 Theorem (Distributive joins and exact sequences)......Page 213
4.7.8 Examples......Page 215
5.1.1 Morphism of graphs......Page 217
5.1.3 Properties of morphisms of graphs......Page 218
5.1.4 Lemma......Page 219
5.1.6 RE-morphisms......Page 220
5.2.1 Definition (RE-theory)......Page 221
5.2.3 Theorem (Existence and naturality of the universal model)......Page 222
5.2.4 Theorem (Modular naturality)......Page 224
5.2.6 Remarks......Page 226
5.3.1 Definition (Equivalent RE-theories)......Page 227
5.3.4 Remarks......Page 228
5.4.1 Definition (The universal projective model)......Page 229
5.4.3 Detection properties......Page 230
5.4.5 Theorem......Page 231
5.4.6 Theorem (A criterion for universal projective models)......Page 232
5.4.7 Remark......Page 233
5.5.1 Theorem (The running knot theorem for idempotent categories, I)......Page 234
5.5.3 Lemma......Page 235
5.5.4 Theorem (Criterion I for idempotent theories)......Page 236
5.5.6 Universality for distributive and idempotent theories......Page 237
5.5.7 Plane orderings......Page 238
5.5.8 Theorem (The running knot theorem for idempotent categories, II)......Page 239
5.6.1 Definition (EX-theory)......Page 242
5.6.3 Associated theories......Page 243
5.6.4 Theorem (Existence of biuniversal models)......Page 244
5.6.6 The global representation functor......Page 245
5.7 Models in semitopological spaces......Page 246
5.7.1 Auxiliary subcategories......Page 247
5.7.2 Theorem......Page 248
5.7.4 Theorem (Union rule)......Page 250
5.7.5 Lemma......Page 251
5.7.6 Theorem (Deletion rule)......Page 252
5.8.2 Models in crossword spaces......Page 253
5.8.3 Locally closed rectangles......Page 254
5.8.5 Lemma......Page 256
6 Homological theories and their universal models......Page 258
6.1.2 Constructing the universal model......Page 259
6.1.3 Theorem (The universal bifiltered object)......Page 260
6.1.5 Theorem (The Jordan-Holder theorem for RE-categories, a crossword-chasing proof)......Page 261
6.1.7 Corollary (The Jordan-Holder Theorem for modular lattices)......Page 263
6.2.2 The model of a relation......Page 264
6.2.3 Constructing the universal model......Page 265
6.2.4 Theorem (The universal n-sequence of relations)......Page 266
6.2.5 The sequence of proper morphisms......Page 268
6.2.6 Other related theories......Page 269
6.2.8 Lemma (A crossword-chasing proof of the Snake Lemma)......Page 271
6.3.1 The theory of the bounded filtered chain complex......Page 273
6.3.2 Models......Page 274
6.3.3 Constructing the universal model......Page 275
6.3.4 Theorem (The universal model)......Page 276
6.3.5 The spectral sequence......Page 278
6.3.6 The homology of the spectral sequence......Page 280
6.3.7 The representation......Page 281
6.3.8 The short exact sequence of complexes......Page 282
6.4.1 Degeneracy......Page 283
6.4.2 Proper transgressions......Page 285
6.4.3 Theorem (Wang exact sequence)......Page 286
6.4.4 Theorem (Gysin exact sequence)......Page 288
6.4.6 Proposition......Page 289
6.4.7 Frame theorems......Page 290
6.5 The real filtered chain complex......Page 291
6.5.2 Constructing the universal model......Page 292
6.5.3 Theorem (The universal model)......Page 293
6.5.4 Partial homologies......Page 294
6.6.1 The double complex and its spectral sequences......Page 295
6.6.2 Comments......Page 296
6.6.3 The theory of the double complex......Page 297
6.6.5 Constructing the universal model......Page 298
6.6.6 Theorem (The universal double complex)......Page 300
6.6.7 The spectral sequences......Page 302
6.6.8 Degeneracy......Page 304
6.7 Eilenberg’s exact system......Page 305
6.7.1 Definition (The exact system)......Page 306
6.7.3 Lemma (Exact squares in exact systems)......Page 307
6.7.4 The theory......Page 308
6.7.6 A crossword model......Page 309
6.7.8 Spectral systems......Page 311
6.7.9 Representing the associated spectral system......Page 312
6.8.1 The discrete exact system......Page 313
6.8.2 The associated spectral sequence......Page 314
6.8.3 Definition (The exact couple)......Page 315
6.8.5 A crossword model......Page 316
6.8.6 Theorem (The universal model)......Page 317
6.8.8 The RE-theory of the exact square......Page 318
6.8.9 Whitehead’s semiexact couples......Page 320
6.9.1 Lemma......Page 322
6.9.3 The theory of an automorphism......Page 323
6.9.4 The differential object......Page 324
6.9.5 The endorelation......Page 325
6.9.6 The finitely-filtered differential object......Page 326
6.9.7 The real filtered differential object......Page 327
6.9.8 The  -filtered object......Page 328
A1.1 Categories......Page 330
A1.2 Small and large categories......Page 331
A1.3 Isomorphisms, monomorphism, epimorphisms......Page 332
A1.4 Subcategories, quotients and products of categories......Page 333
A1.6 Functors......Page 334
A1.7 Forgetful and structural functor......Page 335
A1.8 Faithful and full functors......Page 336
A1.9 Natural transformations......Page 337
A2.1 Subobject and quotients......Page 338
A2.2 Products and equalisers......Page 339
A2.3 Sums and coequalisers......Page 340
A2.5 Equivalence of categories......Page 342
A2.6 Categories of functors and presheaves......Page 343
A2.7 A digression on mathematical structures and categories......Page 344
A3.1 Universal arrows......Page 345
A3.3 Particular cases, pullbacks and pushouts......Page 346
A3.4 Complete categories and the preservation of limits......Page 348
A4 Adjoint functors......Page 349
A4.1 Main definitions......Page 350
A4.3 Main properties of adjunctions......Page 351
A4.5 The Initial Object Theorem (P. Freyd)......Page 352
A5.1 Monoidal categories......Page 353
A5.3 Cartesian closed categories......Page 354
A5.5 Two-categories......Page 355
A5.6 Two-dimensional functors and universal arrows......Page 356
A5.7 Double categories......Page 357
B2 A canonical bifiltration......Page 360
B3 The bifiltration of the crossword model......Page 361
B5 Lemma (Domination in exact systems)......Page 362
B6 Theorem......Page 364
B7 Theorem......Page 365
B8 Theorem......Page 367
References......Page 370
Index......Page 376