Categorical structure of closure operators with applications to topology

Cover......Page 1
Title Page......Page 3
Copyright Page......Page 4
Dedication......Page 5
Table of Contents......Page 7
Preface......Page 11
Introduction......Page 13
1.1 .M-subobjects......Page 19
1.3 Review of pairs of adjoint maps......Page 21
1.4 Adjointness of image and inverse image......Page 23
1.5 The right M-factorization of a morphism......Page 24
1.6 Constructing images from right M-factorizations......Page 25
1.7 Stability properties of M-subobjects......Page 27
1.8 M-subobjects of M-subobjects......Page 30
1.9 When the subobjects form a large-complete lattice......Page 32
1.10 The right M-factorization of a sink......Page 34
1.11 The last word on least and last subobjects......Page 37
Exercises......Page 38
Notes......Page 41
2.1 The categorical setting......Page 42
2.2 The local definition of closure operator......Page 43
2.3 Closed and dense subobjects......Page 44
2.4 Idempotent and weakly hereditary closure operators......Page 45
2.5 Minimal and hereditary closure operators......Page 48
2.6 Grounded and additive closure operators......Page 52
2.7 Productive closure operators......Page 54
2.8 Restriction of closure operators to full subcategories......Page 56
Exercises......Page 57
Notes......Page 61
3.1 Kuratowski closure operator, Cech closure operator......Page 62
3.2 Filter convergence spaces and Katetov closure operator......Page 63
3.3 Sequential closure, b-closure, 0-closure, t-closure......Page 65
3.4 Preradicals of R-modules and Abelian groups......Page 69
3.5 Groups, rings and fields......Page 73
3.6 Graphs and partially ordered sets......Page 74
3.7 Directed-complete posets and Scott closure......Page 79
Exercises......Page 82
Notes......Page 89
4.1 The lattice structure of all closure operators......Page 90
4.2 Composition of closure operators......Page 91
4.3 Cocomposition of closure operators......Page 93
4.4 Closedness and density for (co)composites......Page 96
4.5 Properties stable under meet or join......Page 97
4.6 Idempotent hull and weakly hereditary core......Page 99
4.7 Indiscrete operator, proper closure operators......Page 103
4.8 Additive core......Page 106
4.9 Fully additive core......Page 108
4.10 Minimal core and hereditary hull......Page 112
4.11 Productivity of idempotent closure operators......Page 116
Exercises......Page 120
Notes......Page 126
5.1 Pointed endofunctors and prereflections......Page 127
5.2 Closure operators are prereflections......Page 130
5.3 Factorization systems......Page 134
5.4 Recognizing classes of C-dense and C-closed subobjects......Page 138
5.5 Closure operators versus M-preradicals......Page 143
5.6 M-preradicals versus E-prereflections......Page 148
5.7 (C, D)-continuous functors......Page 153
5.8 Lifting closure operators along M-fibrations......Page 158
5.9 Application to topological groups......Page 162
5.10 Closure operators and CS-valued functors......Page 165
5.11 Closure-structured categories, uniform spaces......Page 169
5.12 Pointed modifications of closure operators......Page 172
5.13 Closure operators and adjoint functors......Page 176
5.14 External closure operators......Page 182
Exercises......Page 186
Notes......Page 194
6.1 A-epimorphisms and A-regular monomorphisms......Page 195
6.2 A-epi closure and A-regular closure......Page 198
6.3 Computing the A-regular closure for reflective A......Page 202
6.4 The magic cube......Page 205
6.5 Frolik's Lemma......Page 210
6.6 The strong modification of a closure operator......Page 213
6.7 Regular closure in pointed and additive......Page 216
6.8 Clementino's Theorem......Page 221
6.9 Regular closure for topological spaces......Page 227
6.10 Pointed topological spaces......Page 231
Exercises......Page 237
Notes......Page 242
7.1 The Salbany correspondence......Page 243
7.2 Two diagonal theorems......Page 246
7.3 Essentially equivalent closure operators......Page 249
7.4 Regular hull and essentially strong closure operators......Page 250
7.5 Characterization of additive regular closure operators......Page 253
7.6 The Pumplun-Rohrl correspondence......Page 256
7.7 The maximal epi-preserving extension......Page 260
7.8 Nabla subcategories......Page 265
7.9 Companions of A(C) in topological categories......Page 268
Exercises......Page 271
Notes......Page 276
8.1 Categorical preliminaries......Page 277
8.2 Reflectivity and cowellpoweredness......Page 280
8.3 Epimorphisms in subcategories of Top - a first summary......Page 283
8.4 Projective closure operators and the categories Haus(P)......Page 285
8.5 Cowellpowered subcategories of Top......Page 291
8.6 Non-cowellpowered subcategories of Top......Page 293
8.7 Quasi-uniform spaces......Page 298
8.8 Topological groups......Page 302
8.9 Epimorphisms and cowellpoweredness in algebra......Page 305
8.10 The Frobenius closure operator of fields......Page 310
Exercises......Page 313
Notes......Page 322
9.1 Hereditariness revisited......Page 323
9.2 Initial and open morphisms......Page 325
9.3 Modal closure operators......Page 329
9.4 Barr's reflector......Page 332
9.5 Total density......Page 338
Exercises......Page 344
Notes......Page 347
Bibliography......Page 349
Index of Definitions......Page 363
Notation Index......Page 369
Tables of Results......Page 375