By Peter Freyd

CONTENTS

========

Contents

Introduction

Exercises on Extremal Categories

Exercises on normal Categories

CHAPTER 1. FUNDAMENTALS

1.1. Contravariant Functors and twin Categories

1.2. Notation

1.3. the traditional Functors

1.4. specified 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. targeted Sequences

2.3. The Additive constitution for Abelian Categories

2.4. attractiveness of Direct Sum Systems

2.5. The Pullback and Pushout Theorems

2.6. Classical Lemmas

Exercises

CHAPTER three. particular FUNCTORS AND SUBCATEGORIES

3.1. Additivity and Exactness

3.2. Embeddings

3.3. precise Objects

3.4. Subcategories

3.5. detailed Contravariant Functors

3.6. Bifunctors

Exercises

CHAPTER four. METATHEOREMS

4.1. Very Abelian Categories

4.2. First Metatheorem

4.3. absolutely 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 kinds of completely natural gadgets and Left-Exact Functors

Exercises

APPENDIX

BIBLIOGRAPHY

INDEX

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

**Sample text**

Dually, a category is right-complete if every pair of maps has a difference cokernel and every indexed set of maps a sum. If a category is both left- and right-complete it is complete. 9. ZERO OBJECTS, KERNELS, AND COKERNELS A zero object is an object with precisely one map to and from each object. We reserve the symbol 0 for a zero object. Hence the sets (O,A) and (A,O) have one object each, for all A. The category of sets does not have a zero object; the category of groups does: namely, the group with one element.

Let A1 - A and A2 ---+ A be monomorphisms, A ---+Fa cokernel of A 1 --. A and A12 - A2 a kernel of A2 ---+ A- F. 131) commutes. ) Let X - A1 and X ---+ A 2 be any pair of maps such that commutes. We shall show that there is a unique X---+ A 12 such that (when X "is a subobject" we will have proved containment in Al2). The map X- A12 exists since X---+ A2 - F = X---+ A 1 ---+ F = 0 and A12 - A 2 = Ker(A 2 - F). Thus there is a unique map X ---+ A12 such that X - A 12 ---+ A 2 = X ---+ A2 • The other equation follows from X - A12 ---+ A1 ---+ A = X - A 2 - A = X- A1 ---+A and the fact that A1 ---+A is a monomorphism.

Hence the sets (O,A) and (A,O) have one object each, for all A. The category of sets does not have a zero object; the category of groups does: namely, the group with one element. , B. ) The kernel of A ~ B is defined to be the difference kernel of A ~ B and A 4 B. Hence if K ---+ A is a kernel of A ~ B then 0 )C K 1. K ---+ A ---'-+ B = K ---'-+ B K2. For all X ---+A such that X t~ A~B commutes 27 FUNDAMENTALS there is a unique X--. K such that X /t K~A commutes. The usual notation for kernel of x is Ker(x).