Handbook of Categorical Algebra 3 Categories of Sheaves

Title page
Preface to volume 3
Introduction to this handbook
1 Locales
1.1 The intuitionistic propositional calculus
1.2 Heyting algebras
1.3 Locales
1.4 Limits and colimits of locales
1.5 Nuclei
1.6 Open morphisms of locales
1.7 Etale morphisms of locales
1.8 The points of a locale
1.9 Sober spaces
1.10 Compactness conditions
1.11 Regularity conditions
1.12 Exercises
2 Sheaves
2.1 Sheaves on a locale
2.2 Closed subobjects
2.3 Some categorical properties of sheaves
2.4 Etale spaces
2.5 The stalks of a topological sheaf
2.6 Associated sheaves and étale morphisms
2.7 Systems of generators for a sheaf
2.8 The theory of O-sets
2.9 Complete O-sets
2.10 Some basic facts in ring theory
2.11 Sheaf representation of a ring
2.12 Change of base
2.13 Exercises
3 Grothendieck toposes
3.1 A categorical glance at sheaves
3.2 Grothendieck topologies
3.3 The associated sheaf functor theorem
3.4 Categorical properties of Grothendieck toposes
3.5 Localizations of Grothendieck toposes
3.6 Characterization of Grothendieck toposes
3.7 Exercises
4 The classifying topos
4.1 The points of a topos
4.2 The classifying topos of a finite limit theory
4.3 The classifying topos of a geometric sketch
4.4 The classifying topos of a coherent theory
4.5 Diaconescu's theorem
4.6 Exercises
5 Elementary toposes
5.1 The notion of a topos
5.2 Examples of toposes
5.3 Monomorphisms in a topos
5.4 Some set theoretical notions in a topos
5.5 Partial morphisms
5.6 Injective objects
5.7 Finite colimits
5.8 The slice toposes
5.9 Exactness properties of toposes
5.10 Union of subobjects
5.11 Morphisms of toposes
5.12 Exercises
6 Internal logic of a topos
6.1 The language of a topos
6.2 Categorical foundations of the logic of toposes
6.3 The calculus of truth tables
6.4 The point about "ghost" variables
6.5 Coherent theories
6.6 The Kripke-Joyal semantics
6.7 The intuitionistic propositional calculus in a topos
6.8 The intuitionistic predicate calculus in a topos
6.9 Intuitionistic set theory in a topos
6.10 The structure of a topos in its internai language
6.11 Locales in a topos
6.12 Exercises
7 The law of excluded middle
7.1 The regular elements of n
7.2 Boolean toposes
7.3 De Morgan toposes
7.4 Decidable objects
7.5 The axiom of choice
7.6 Exercises
8 The axiom of infinity
8.1 The natural number object
8.2 Infinite objects in a topos
8.3 Arithmetic in a topos
8.4 The trichotomy
8.5 Finite objects in a topos
8.6 Exercises
9 Sheaves in a topos
9.1 Topologies in a topos
9.2 Sheaves for a topology
9.3 The localizations of a topos
9.4 The double negation sheaves
9.5 Exercises
Bibliography
Index