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

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

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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).

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