# Book:Barry Mitchell/Theory of Categories

## Barry Mitchell: Theory of Categories

Published $\text {1965}$, Academic Press.

### 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
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

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
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