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Book Categories |
Preface | ||
1 | Classical Galois theory | 1 |
1.1 | Algebraic extensions | 1 |
1.2 | Separable extensions | 4 |
1.3 | Normal extensions | 6 |
1.4 | Galois extensions | 8 |
2 | Galois theory of Grothendieck | 15 |
2.1 | Algebras on a field | 15 |
2.2 | Extension of scalars | 20 |
2.3 | Split algebras | 23 |
2.4 | The Galois equivalence | 27 |
3 | Infinitary Galois theory | 36 |
3.1 | Finitary Galois subextensions | 36 |
3.2 | Infinitary Galois groups | 39 |
3.3 | Classical infinitary Galois theory | 44 |
3.4 | Profinite topological spaces | 47 |
3.5 | Infinitary extension of the Galois theory of Grothendieck | 56 |
4 | Categorical Galois theory of commutative rings | 65 |
4.1 | Stone duality | 65 |
4.2 | Pierce representation of a commutative ring | 72 |
4.3 | The adjoint of the 'spectrum' functor | 80 |
4.4 | Descent morphisms | 91 |
4.5 | Morphisms of Galois descent | 98 |
4.6 | Internal presheaves | 102 |
4.7 | The Galois theorem for rings | 106 |
5 | Categorical Galois theorem and factorization systems | 116 |
5.1 | The abstract categorical Galois theorem | 117 |
5.2 | Central extensions of groups | 127 |
5.3 | Factorization systems | 144 |
5.4 | Reflective factorization systems | 149 |
5.5 | Semi-exact reflections | 156 |
5.6 | Connected components of a space | 168 |
5.7 | Connected components of a compact Hausdorff space | 170 |
5.8 | The monotone-light factorization | 177 |
6 | Covering maps | 186 |
6.1 | Categories of abstract families | 186 |
6.2 | Some limits in Fam (A) | 189 |
6.3 | Involving extensivity | 193 |
6.4 | Local connectedness and etale maps | 197 |
6.5 | Localization and covering morphisms | 201 |
6.6 | Classification of coverings | 207 |
6.7 | The Chevalley fundamental group | 212 |
6.8 | Path and simply connected spaces | 216 |
7 | Non-galoisian Galois theory | 225 |
7.1 | Internal presheaves on an internal groupoid | 225 |
7.2 | Internal precategories and their presheaves | 241 |
7.3 | A factorization system for functors | 246 |
7.4 | Generalized descent theory | 251 |
7.5 | Generalized Galois theory | 258 |
7.6 | Classical Galois theories | 261 |
7.7 | Grothendieck toposes | 266 |
7.8 | Geometric morphisms | 274 |
7.9 | Two dimensional category theory | 287 |
7.10 | The Joyal-Tierney theorem | 294 |
App: Final remarks | 304 | |
Bibliography | 331 | |
Index of symbols | 336 | |
General index | 338 |
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