Book:Barry Mitchell/Theory of Categories

Subject Matter

 * Category Theory

Contents

 * Preface (May, 1964)


 * I. Preliminaries
 * Introduction
 * 1. Definition
 * 2. The Nonobjective Approach
 * 3. Examples
 * 4. Duality
 * 5. Special Morphisms
 * 6. Equalizers
 * 7. Pullbacks, Pushouts
 * 8. Intersections
 * 9. Unions
 * 10. Images
 * 11. Inverse Images
 * 12. Zero Objects
 * 13. Kernels
 * 14. Normality
 * 15. Exact Categories
 * 16. The 9 Lemma
 * 17. Products
 * 18. Additive Categories
 * 19. Exact Additive Categories
 * 20. Abelian Categories
 * 21. The Category of Abelian Groups $\mathscr G$
 * Exercises


 * II. Diagrams and Functors
 * Introduction
 * 2. Limits
 * 3. Functors
 * 4. Preservation Properties of Functors
 * 5. Morphism Functors
 * 6. Limit Preserving Functors
 * 7. Faithful Functors
 * 8. Functors of Several Variables
 * 9. Natural Transformations
 * 10. Equivalence of Categories
 * 11. Functor Categories
 * 12. Diagrams as Functors
 * 13. Categories of Additive Functors; Modules
 * 14. Projectives, Injectives
 * 15. Generators
 * 16. Small Objects
 * Exercises


 * III. Complete Categories
 * Introduction
 * 1. $C_i$ Categories
 * 2. Injective Envelopes
 * 3. Existence of Injectives
 * Exercises


 * IV. Group Valued Functors
 * Introduction
 * 1. Metatheorems
 * 2. The Group Valued Imbedding Theorem
 * 3. An Imbedding for Big Categories
 * 4. Characterization of Categories of Modules
 * 5. Characterization of Functor Categories
 * Exercises


 * V. Adjoint Functors
 * Introduction
 * 1. Generalities
 * 2. Conjugate Transformations
 * 3. Existence of Adjoints
 * 4. Functor Categories
 * 5. Reflections
 * 6. Monosubcategories
 * 7. Projective Classes
 * Exercises


 * VI. Applications of Adjoint Functors
 * Introduction
 * 1. Application to Limits
 * 2. Module-Valued Adjoints
 * 3. The Tensor Product
 * 4. Functor Categories
 * 5. Derived Functors
 * 6. The Category of Kernel Preserving Functors
 * 7. The Full Imbedding Theorem
 * 8. Complexes
 * Exercises


 * VII. Extensions
 * Introduction
 * 1. $\operatorname{Ext}^1$
 * 2. The Exact Sequence (Special Case)
 * 3. $\operatorname{Ext}^n$
 * 4. The Relation $\sim$
 * 5. The Exact Sequence
 * 6. Global Dimension
 * 7. Appendix: Alternative Description of $\operatorname{Ext}$
 * Exercises


 * VIII. Satellites
 * Introduction
 * 1. Connected Sequences of Functors
 * 2. Existence of Satellites
 * 3. The Exact Sequence
 * 4. Satellites of Group Valued Functors
 * 5. Projective Sequences
 * 6. Several Variables
 * Exercises


 * IX. Global Dimension
 * Introduction
 * 1. Free Categories
 * 2. Polynomial Categories
 * 3. Grassmann Categories
 * 4. Graded Free Categories
 * 5. Graded Polynomial Categories
 * 6. Graded Grassmann Categories
 * 7. Finite Commutative Diagrams
 * 8. Homological Tic Tac Toe
 * 9. Normal Subsets
 * 10. Dimension for Finite Ordered Sets
 * Exercises


 * X. Sheaves
 * Introduction
 * 1. Preliminaries
 * 2. $\mathscr F$-Categories
 * 3. Associated Sheaves
 * 4. Direct Images of Sheaves
 * 5. Inverse Images of Sheaves
 * 6. Sheaves in Abelian Categories
 * 7. Injective Sheaves
 * 8. Induced Sheaves
 * Exercises


 * BIBLIOGRAPHY


 * SUBJECT INDEX