Introduction . . . . . Chapter 1 Preliminaries §1.1. ~~:-filtered colimits §1.2. ~~:-flat functors CONTENTS Chapter 2 Accessible categories and functors 1 11 14 §2.1. ~~:-accessible categories . . . . . 17 §2.2. Small categories and accessibility 20 §2.3. Raising the index of accessibility 22 §2.4. Accessible functors . . . . . . 31 §2.5. An equivalent definition of accessibility 34 Chapter 3 Sketches and logic §3.1. Sketches . . . . . . . . . . . . . . . . 39 §3.2. Logic . . . . . . . . . . . . . . . . . 42 §3.3. The downward Lowenheim-Skolem theorem 53 §3.4. Examples . . . . . . . . . . . . . . . 57 Chapter 4 Sketching accessible categories §4.1. Preliminaries on 2-categories . . . . . . . . . . . . . . 67 §4.2. The canonical sketch associated with a ~~:-accessible category 81 §4.3. Small sketches for an accessible category . . . . . . . . 86 §4.4. Diagrams of accessible categories and diagrams of sketches 91 §4.5. Axiomatizing an accessible subcategory . . . . . . . 95 Chapter 5 Limits and Colimits of accessible categories §5.1. Colimits of sketches and Limits of accessible categories 97 §5.2. Some results concerning Grothendieck toposes 107 §5.3. Accessible fibrations . . . . . . . . . . . 113 §5.4. Lax Colimits of accessible categories . . . . . 122 §5.5. The powerful image of an accessible functor 138 Chapter 6 Limits and colimits in accessible categories §6.1. Completeness and cocompleteness in accessible categories 141 §6.2. Models of a sketch in an accessible category 146 §6.3. Detectability of colimits . . . . . 153 §6.4. Completing an accessible category . . . . 160 vii
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