# Book:Peter Freyd/Abelian Categories

## Peter Freyd: Abelian Categories: An Introduction to the Theory of Functors

Published $\text {1964}$, Harper International.

### 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
3.2. 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

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