By Peter Freyd

CONTENTS

========

Contents

Introduction

Exercises on Extremal Categories

Exercises on usual Categories

CHAPTER 1. FUNDAMENTALS

1.1. Contravariant Functors and twin Categories

1.2. Notation

1.3. the normal Functors

1.4. particular Maps

1.5. Subobjects and Quotient Objects

1.6. distinction Kernels and Cokernels

1.7. items and Sums

1.8. whole Categories

1.9. 0 gadgets, Kernels, and Cokernels

Exercises

CHAPTER 2. basics OF ABELIAN CATEGORIES

2.1. Theorems for Abelian Categories

2.2. specific Sequences

2.3. The Additive constitution for Abelian Categories

2.4. acceptance 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. detailed Objects

3.4. Subcategories

3.5. distinct 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 kinds of totally natural items and Left-Exact Functors

Exercises

APPENDIX

BIBLIOGRAPHY

INDEX

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**Sample text**

B ~ B is clearly a kernel of B-+ 0. By the last theorem so is A -+B. (Already we have shown that A and B are isomorphic-they are both kernels of the same map. ) Hence there is a map B ~ A such that • a bl A -a+ B = B -I+ B. Dua JJ y we note that 0 -+ A IS B----'-+ kernel of A ~ B and that both A ~ B and A ~ A are cokernels of 0 -+A. Hence there is a map B ~ A such that A~ B ~ A = A ~ A. By the definition of isomorphism, A ~ B is such. I The intersection of two subobjects of A is defined to be their greatest lower bound in the family of subobjects of A.

DC2. For all B ~X such that A --4. B--+ X= A 4 B ~ X there is a unique F ~ X such that n~F 'x/ X commutes. A difference cokernel must be epimorphic and if one exists it determines a quotient object of difference cokernels called the difference cokernel, symbolized by Cok(x-y). 7. PRODUCTS AND SUMS Given a pair of objects A, B we say that an object P is a product of A and B if there exists maps P ~ A and P ~ B such that for every pair of maps X ~ A and X - B there is a 21 FUNDAMENTALS unique X ---+ P such that commutes.

18 for abelian categories A ~ /m(x) is epimorphic. Proof: If Cok(A ~ Im(x)) =f=. 0, then A - Im(x) factors through a proper subobject of lm(x), which contradicts the definition of Im(x). I The dual of image is coimage. The coimage of A ~ B is the smallest quotient object of A through which A ~ B factors. Notation: Coim(A ~B), Coim(x). 16* for abelian categories Coim(A ~ B) = CokKer(A ~ B). 17* for abelian categories A - B is monomorphic iffCoim(A ---+B) = A iff Ker(A ---+B) = o. I Let A - I' be a coimage of A ---+ Band consider A --.

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