Handbook of Categorical Algebra 3: Categories of Sheaves

Cover......Page A001.djvu
Title page......Page page0001.djvu
Contents......Page page0004.djvu
Preface to volume 3......Page page0007.djvu
Introduction to this handbook......Page page0010.djvu
Contents of the three volumes......Page page0012.djvu
1.1 The intuitionistic propositional calculus......Page page0013.djvu
1.2 Heyting algebras......Page page0017.djvu
1.3 Locales......Page page0025.djvu
1.4 Limits and colimits of locales......Page page0030.djvu
1.5 Nuclei......Page page0041.djvu
1.6 Open morphisms of locales......Page page0050.djvu
1.7 Etale morphisms of locales......Page page0059.djvu
1.8 The points of a locale......Page page0073.djvu
1.9 Sober spaces......Page page0082.djvu
1.10 Compactness conditions......Page page0085.djvu
1.11 Regularity conditions......Page page0094.djvu
1.12 Exercises......Page page0096.djvu
2 Sheaves......Page page0099.djvu
2.1 Sheaves on a locale......Page page0100.djvu
2.2 Closed subobjects......Page page0106.djvu
2.3 Some categorical properties of sheaves......Page page0114.djvu
2.4 Etale spaces......Page page0120.djvu
2.5 The stalks of a topological sheaf......Page page0125.djvu
2.6 Associated sheaves and étale morphisms......Page page0135.djvu
2.7 Systems of generators for a sheaf......Page page0150.djvu
2.8 The theory of Ω-sets......Page page0156.djvu
2.9 Complete Ω-sets......Page page0168.djvu
2.10 Some basic facts in ring theory......Page page0180.djvu
2.11 Sheaf representation of a ring......Page page0185.djvu
2.12 Change of base......Page page0195.djvu
2.13 Exercises......Page page0201.djvu
3.1 A categorical glance at sheaves......Page page0204.djvu
3.2 Grothendieck topologies......Page page0207.djvu
3.3 The associated sheaf functor theorem......Page page0217.djvu
3.4 Categorical properties of Grothendieck toposes......Page page0226.djvu
3.5 Localizations of Grothendieck toposes......Page page0234.djvu
3.6 Characterization of Grothendieck toposes......Page page0242.djvu
3.7 Exercises......Page page0255.djvu
4.1 The points of a topos......Page page0256.djvu
4.2 The classifying topos of a finite limit theory......Page page0262.djvu
4.3 The classifying topos of a geometric sketch......Page page0269.djvu
4.4 The classifying topos of a coherent theory......Page page0282.djvu
4.5 Diaconescu\'s theorem......Page page0292.djvu
4.6 Exercises......Page page0299.djvu
5 Elementary toposes......Page page0001_1.djvu
5.1 The notion of a topos......Page page0002_1.djvu
5.2 Examples of toposes......Page page0006_1.djvu
5.3 Monomorphisms in a topos......Page page0016_1.djvu
5.4 Some set theoretical notions in a topos......Page page0017_1.djvu
5.5 Partial morphisms......Page page0021_1.djvu
5.6 Injective objects......Page page0027_1.djvu
5.7 Finite colimits......Page page0028_1.djvu
5.8 The slice toposes......Page page0034_1.djvu
5.9 Exactness properties of toposes......Page page0041_1.djvu
5.10 Union of subobjects......Page page0047_1.djvu
5.11 Morphisms of toposes......Page page0051_1.djvu
5.12 Exercises......Page page0052_1.djvu
6 Internal logic of a topos......Page page0055_1.djvu
6.1 The language of a topos......Page page0057_1.djvu
6.2 Categorical foundations of the logic of toposes......Page page0061_1.djvu
6.3 The calculus of truth tables......Page page0067_1.djvu
6.4 The point about “ghost” variables ......Page page0072_1.djvu
6.5 Coherent theories......Page page0074_1.djvu
6.6 The Kripke-Joyal semantics......Page page0084_1.djvu
6.7 The intuitionistic propositional calculus in a topos......Page page0108_1.djvu
6.8 The intuitionistic predicate calculus in a topos......Page page0113_1.djvu
6.9 Intuitionistic set theory in a topos......Page page0122_1.djvu
6.10 The structure of a topos in its internal language......Page page0130_1.djvu
6.11 Locales in a topos......Page page0137_1.djvu
6.12 Exercises......Page page0143_1.djvu
7.1 The regular elements of Ω......Page page0145_1.djvu
7.2 Boolean toposes......Page page0153_1.djvu
7.3 De Morgan toposes......Page page0155_1.djvu
7.4 Decidable objects......Page page0157_1.djvu
7.5 The axiom of choice......Page page0160_1.djvu
7.6 Exercises......Page page0165_1.djvu
8.1 The natural number object......Page page0166_1.djvu
8.2 Infinite objects in a topos......Page page0178_1.djvu
8.3 Arithmetic in a topos......Page page0181_1.djvu
8.4 The trichotomy......Page page0186_1.djvu
8.5 Finite objects in a topos......Page page0191_1.djvu
8.6 Exercises......Page page0196_1.djvu
9.1 Topologies in a topos......Page page0199_1.djvu
9.2 Sheaves for a topology......Page page0205_1.djvu
9.3 The localizations of a topos......Page page0213_1.djvu
9.4 The double negation sheaves......Page page0223_1.djvu
9.5 Exercises......Page page0224_1.djvu
Bibliography......Page page0227_1.djvu
Index......Page page0230_1.djvu