By Peter Freyd

CONTENTS

========

Contents

Introduction

Exercises on Extremal Categories

Exercises on regular Categories

CHAPTER 1. FUNDAMENTALS

1.1. Contravariant Functors and twin Categories

1.2. Notation

1.3. the traditional Functors

1.4. precise Maps

1.5. Subobjects and Quotient Objects

1.6. distinction Kernels and Cokernels

1.7. items and Sums

1.8. entire Categories

1.9. 0 gadgets, Kernels, and Cokernels

Exercises

CHAPTER 2. basics OF ABELIAN CATEGORIES

2.1. Theorems for Abelian Categories

2.2. certain Sequences

2.3. The Additive constitution for Abelian Categories

2.4. reputation of Direct Sum Systems

2.5. The Pullback and Pushout Theorems

2.6. Classical Lemmas

Exercises

CHAPTER three. specified FUNCTORS AND SUBCATEGORIES

3.1. Additivity and Exactness

3.2. Embeddings

3.3. distinct Objects

3.4. Subcategories

3.5. precise Contravariant Functors

3.6. Bifunctors

Exercises

CHAPTER four. METATHEOREMS

4.1. Very Abelian Categories

4.2. First Metatheorem

4.3. totally Abelian Categories

4.4. Mitchell's Theorem

Exercises

CHAPTER five. FUNCTOR CATEGORIES

5.1. Abelianness

5.2. Grothendieck Categories

5.3. The illustration Functor

Exercises

CHAPTER 6. INJECTIVE ENVELOPES

6.1. Extensions

6.2. Envelopes

Exercises

CHAPTER 7. EMBEDDING THEOREMS

7.1. First Embedding

7.2. An Abstraction

7.3. The Abelianness of the types of totally natural gadgets and Left-Exact Functors

Exercises

APPENDIX

BIBLIOGRAPHY

INDEX

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**Extra resources for Abelian Categories: An Introduction to the Theory of Functors**

**Example text**

22 is exact iff K---+ A is monomorphic. B. is exact iff B---+ F is epimorphic. B. is exact iff A ---+ B is an isomorphism. A-B B---+F---+0 0---+ A ---+ B---+ 0 A - B~ B 0 - A ---+ B ---+ C ---+ is exact iff A ---+ B is the zero map. 0 is exact iff A - B is a monomorphism and B ---+ C is a cokernel of A -B. 3. 31 for abelian categories "t The sequence 0 - A ........... A (~) + B-+ . B---+ 0 ts exact. ABELIAN CATEGORIES Proof: . 1c smce A is. To prove that A + B (~) ~ B is a cokernel of uh let <;) A+ B---+ X be a map such that A~ A + B---+ X= 0.

FUNDAMENTALS 25 Sums of the same objects are isomorphic; the notation "t A + B refers to "the" sum of A and B; the maps A---'-+ A +B and B ~ A + Bare "the" associated maps. x > A u B~ X~ A = X~ B and x B. )) X= A~ Gt) "' h X to be t e unique X and 2 A +B--=-+ X= B~ X. 8. COMPLETE CATEGORIES Given an indexed set of objects {A;} 1 in a category, its product is defined to be an object 11 iEIA; together with maps {IIiE/Ai ~ A;}{ 26 ABELIAN CATEGORIES such thatforanyfamily {X~ A;}Jthere is a unique X---+ TI 1 Ai such that X---+ TI 1 A;~ A;= X~ A;.

Every map has a kernel and a cokernel. A 3. A 3*. Every monomorphism is a kernel of a map. Every epimorphism is a cokernel of a map. " Most categories that arise in nature satisfy Axioms A 0 through A 2. Often Axiom A 0 is satisfied by using base points. Many categories satisfy one of A 3 or A 3*. Compact Hausdorfspaces 15 ABELIAN CATEGORIES 36 with base points satisfy A 3; all groups (abelian or not) satisfy A 3*. 1. THEOREMS FOR ABELIAN CATEGORIES Consider an object A. LetS be the family of subobjects of A, Q the family of quotient objects.