Handbook of Categorical Algebra 3: Categories of Sheaves (Encyclopedia of Mathematics and its Applications, Series Number 52)

This third volume turns to topos theory and the idea of sheaves. The theory of locales is considered first, and Grothendieck toposes are introduced. Notions of sketchability and accessible categories are discussed, and an axiomatic generalization of the category of sheaves is given. Cover Title Contents Preface to volume Introduction to this handbook Contents of the three volumes 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 etale morphisms 2.7 Systems of generators for a sheaf 2.8 The theory of ê-sets 2.9 Complete ê-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 Infective 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 internal language 6.11 Locales in a topos 6.12 Exercises 7 The law of excluded middle 7.1 The regular elements of ? 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 The Handbook of Categorical Algebra is intended to give, in three volumes, a rather detailed account of what, ideally, everybody working in category theory should know, whatever the specific topic of research they have chosen. The book is planned also to serve as a reference book for both specialists in the field and all those using category theory as a tool. Volume 3 begins with the essential aspects of the theory of locales, proceeding to a study in chapter 2 of the sheaves on a locale and on a topological space, in their various equivalent presentations: functors, etale maps or W-sets. Next, this situation is generalized to the case of sheaves on a site and the corresponding notion of Grothendieck topos is introduced. Chapter 4 relates the theory of Grothendieck toposes with that of accessible categories and sketches, by proving the existence of a classifying topos for all coherent theories. 1. Basic Category Theory -- 2. Categories And Structures -- 3. Categories Of Sheaves. Francis Borceux. Includes Bibliographical References And Indexes.