# Book:Peter Freyd/Abelian Categories

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## Peter Freyd:

## Contents

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

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

### Subject Matter

### 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**

## Source work progress

- 1964: Peter Freyd:
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