Book:Peter Freyd/Abelian Categories

Subject Matter

 * Category Theory

Contents

 * Introduction
 * Exercises on Extremal Categories
 * Exercises on Typical Categories


 * CHAPTER 1. FUNDAMENTALS
 * 1.1. Contravariant Functors and Dual Categories
 * 1.2. Notation
 * 1.3. The Standard Functors
 * 1.4. Special Maps
 * 1.5. Subobjects and Quotient Objects
 * 1.6. Difference Kernels and Cokernels
 * 1.7. Products and Sums
 * 1.8. Complete Categories
 * 1.9. Zero Objects, Kernels, and Cokernels
 * Exercises
 * A. Epimorphisms need not be onto B. The automorphism class group C. The category of sets D. The category of small categories E. The category of abelian groups F. The category of groups G. Categories of topological spaces H. Conjugate maps I. Definition theory


 * CHAPTER 2. FUNDAMENTALS OF ABELIAN CATEGORIES
 * 2.1. Theorems for Abelian Categories
 * 2.2. Exact Sequences
 * 2.3. The Additive Structure for Abelian Categories
 * 2.4. Recognition of Direct Sum Systems
 * 2.5. The Pullback and Pushout Theorems
 * 2.6. Classical Lemmas
 * Exercises
 * A. Additive categories B. Idempotents C. Groups in categories


 * CHAPTER 3. SPECIAL FUNCTORS AND SUBCATEGORIES
 * 3.1. Additivity and Exactness
 * 3.2. Embeddings
 * 3.3. Special Objects
 * 3.4. Subcategories
 * 3.5. Special Contravariant Functors
 * 3.6. Bifunctors
 * Exercises
 * A. Equivalence of categories B. Roots C. Construction of roots D. Small complete categories are lattices E. The standard functors F. Reflections G. Adjoint functors H. Transformation adjoints I. The reflectivity of images of adjoint functors J. The adjoint functor theorem K. Some immediate applications of the adjoint functor theorem L. How to find solution sets M. The special adjoint functor theorem N. The special adjoint functor theorem at work O. Exercise for model theorists


 * CHAPTER 4. METATHEOREMS
 * 4.1. Very Abelian Categories
 * 4.2. First Metatheorem
 * 4.3. Fully Abelian Categories
 * 4.4. Mitchell's Theorem
 * Exercises
 * A. Abelian lattice theory B. Functor metatheory C. Correspondences in categories D. A specialized embedding theorem E. Small projectives F. Categories representable as categories of modules G. Compact abelian groups H. Fully is more than very I. Unembeddable categories


 * CHAPTER 5. FUNCTOR CATEGORIES
 * 5.1 Abelianness
 * 5.2 Grothendieck Categories
 * 5.3 The Representation Functor
 * Exercises
 * A. Duals of functor categories B. Co-Grothendieck categories C. Categories of modules D. Projectives and injectives in functor categories E. Grothendieck categories F. Left-completeness almost implies completeness G. Small projectives in functor categories H. Categories representable as functor categories I. Tensor products of additive functors


 * CHAPTER 6. INJECTIVE ENVELOPES
 * 6.1. Extensions
 * 6.2. Envelopes
 * Exercises
 * A. A very large Grothendieck category B. Divisible groups C. Modules over principal ideal domains D. Injectives over acc rings E. Semisimple rings and the Wedderburn theorems F. Noetherian ideal theory


 * CHAPTER 7. EMBEDDING THEOREMS
 * 7.1. First Embedding
 * 7.2. An Abstraction
 * 7.3. The Abelianness of the Categories of Absolutely Pure Objects and Left-Exact Functors
 * Exercises
 * A. Effaceable and torsion functors B. Effaceable functors and injective objects C. $$0$$th right-derived functors D. Absolutely pure objects E. Computations of $$0$$th right-derived functors F. Sheaf theory G. Relative homological algebra


 * APPENDIX


 * BIBLIOGRAPHY


 * INDEX