Galois Theories

Galois Theories......Page 1
Contents......Page 4
Preface......Page 6
1.1 Algebraic extensions......Page 13
1.2 Separable extensions......Page 16
1.3 Normal extensions......Page 18
1.4 Galois extensions......Page 20
2.1 Algebras on a field......Page 27
2.2 Extension of scalars......Page 32
2.3 Split algebras......Page 35
2.4 The Galois equivalence......Page 39
3.1 Finitary Galois subextensions......Page 48
3.2 Infinitary Galois groups......Page 51
3.3 Classical infinitary Galois theory......Page 56
3.4 Profinite topological spaces......Page 59
3.5 Infinitary extension of the Galois theory of Grothendieck......Page 68
4.1 Stone duality......Page 77
4.2 Pierce representation of a commutative ring......Page 84
4.3 The adjoint of the \'spectrum\' functor......Page 92
4.4 Descent morphisms......Page 103
4.5 Morphisms of Galois descent......Page 110
4.6 Internal presheaves......Page 114
4.7 The Galois theorem for rings......Page 118
5. Categorical Galois theorem and factorization systems......Page 128
5.1 The abstract categorical Galois theorem......Page 129
5.2 Central extensions of groups......Page 139
5.3 Factorization systems......Page 156
5.4 Reflective factorization systems......Page 161
5.5 Semi-exact rllfiections......Page 168
5.6 Connected components of a space......Page 180
5.7 Connected components of a compact Hausdorff space......Page 182
5.8 The monotone-light factorization......Page 189
6.1 Categories of abstract families......Page 198
6.2 Some limits in Fam(A)......Page 201
6.3 Involving extensivity......Page 205
6.4 Local connectedness and etale maps......Page 209
6.5 Localization and covering morphisms......Page 213
6.6 Classification of coverings......Page 219
6.7 The Chevalley fundrunental group......Page 224
6.8 Path and simply connected spaces......Page 228
7.1 Internal presheaves on an internal groupoid......Page 237
7.2 Internal precategories and their presheaves......Page 253
7.3 A factorization system for functors......Page 258
7.4 Generalized descent theory......Page 263
7.5 Generalized Galois theory......Page 270
7.6 Classical Galois theories......Page 273
7.7 Grothendieck toposes......Page 278
7.8 Geometric morphisms......Page 286
7.9 Two dimensional category theory......Page 299
7.10 The Joyal-Tierney theorem......Page 306
A.1 Separable algebras......Page 316
A.2 Back to the classical Galois theory......Page 322
A.3 Exhibiting some links......Page 328
A.4 A short summary of further results and developments......Page 340
Bibliography......Page 343
Index of symbols......Page 348
General index......Page 350