Abelian Categories: An Introduction to the Theory of by Peter Freyd

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.

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