Book:Barry Mitchell/Theory of Categories
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Barry Mitchell: Theory of Categories
Published $\text {1965}$, Academic Press
Subject Matter
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