Table Of Content•
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An Introduction to
the Theory of Functors
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PETERFREYD
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Harper's Series in
!VIodern !VIathematics
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ABELIAN
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CATEGORIES
An Introduction to the Theory of Functors
PETER FREYD
University of Pennsylvania
A HARPER INTERNATIONAL EDITION
jointly published by
HARPER & ROW, New York, Evanston & London
and JOHN WEATHER HILL, INC., Tokyo
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This Harper International Edition Is an unabridged, photo-offset reproductiou of tke American
edition. It is for sale in Japan by J 0 H N WEATHER HILL, IN C., 50 Ryudo-cho, Azabu,
Minato-kn, Tokyo; In the United Kiugdom and on the continent of Europe by HARPER & R0 W,
LTD., 89 Great Russell Street, London W.C. I; and in otker designated couotries by HARPER
& ROW, PUBLISHERS, INCORPORATED, 49 East 33rd Street, New York, N.Y. 10016
I Copyright © 1964 by Peter Freyd 1 All rights reserved; no part of this book may be used
or reproduced In any lbnner wkatsoaver witkout writteu permission except In tke case of brief
quotations embodied in critical articles and reviews. For information write Harper & Row at the
New York address given above 1 Library of Congress catalog card number 64-12785 I Printed
in Japan I First printing, September, 1966.
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CONTENTS
Introduction 1
Exercises on Extremal Categories 11
Exercises on Typical Categories 12
CHAPTER 1. FUNDAMENTALS 14
1.1. Contravariant Functors and Dual Categories 15
1.2. Notation 16
1.3. The Standard Functors 16
1.4. Special Maps 17
1.5. Subobjects and Quotient Objects 19
1.6. Difference Kernels and Cokernels 21
1.7. Products and Sums 22
1.8. Complete Categories 25
1.9. Zero Objects, Kernels, and Cokernels 26
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Exercises 27
A. Epimorphisms need not be onto B. The automorphism
class group C. The category of sets D. The category of
small categories E. The category of abelian groups F. The
category of groups G. Categories of topological spaces
H. Conjugate maps I. Definition theory
CHAPTER 2. FUNDAMENTALS OF ABELIAN
CATEGORIES 35
2.1. Theorems for Abelian Categories 36
2.2. Exact Sequences 44
2.3. The Additive Structure for Abelian Categories 45
2.4. Recognition of Direct Sum Systems SO
2.5. The Pullback and Pushout Theorems 51
2.6. Classical Lemmas 54
Exercises 60
A. Additive categories B. Idempotents C. Groups in
categories
CHAPTER 3. SPECIAL FUNCTORS
AND SUBCATEGORIES 64
3.1. Additivity and Exactness 64
3.2. Embeddings 66
3.3. Special Objects 67
3.4. Subcategories 70
3.5. Special Contravariant Functors 72
3.6. Bifunctors 72
Exercises 74
A. Equivalence of categories 8. Roots C. Construction
of roots D. Small complete categories are lattices E. The
standard functors F. Reflections G. Adjoint functors
H. Transformation adjoints I. The reflectivity of images
of adjoint functors J. The adjoint functor theorem
K. Some immediate applications of the adjoint functor
theorem L. How to find solution sets M. The special ad
joint functor theorem N. The special adjoint functor
theorem at work 0. Exercise for model theorists
CONTENTS -5
CHAPTER 4. METATHEOREMS 94
4.1. Very Abelian Categories 95
4.2. First Metatheorem 96
4.3. Fully Abelian Categories 97
4.4. Mitchell's Theorem 100
Exercises 103
A. Abelian lattice theory B. Functor meta theory C. Cor
respondences in categories D. A specialized embedding
theorem E. Small projectives F. Categories representable
as categories of modules G. Compact abelian groups
H. Fully is more than very I. Unembeddable categories
CHAPTER 5. FUNCTOR CATEGORIES 109
5.1. Abelianness 109
5.2. Grothendieck Categories 111
5.3. The Representation Functor 112
Exercises 115
A. Duals of functor categories B. Co-Grothendieck
categories C. Categories of modules D. Projectives and
injectives in functor categories E. Grothendieck categories
F. Left-completeness almost implies completeness G. Small
projectives in functor categories H. Categories rep
resentable as functor categories I. Tensor products of
additive functors
CHAPTER 6. INJECTIVE ENVELOPES 123
6.1. Extensions 123
6.2. Envelopes 126
Exercises 131
A. A very large Grothendieck category B. Divisible
groups C. Modules over principal ideal domains D. Injec-
tives over ace rings E. Semisimple rings and the Wedder-
burn theorems F. Noetherian ideal tht:ory
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CHAPTER 7. EMBEDDING THEOREMS 138
7.1. First Embedding 138
7.2. An Abstraction 141
7.3. The Abelianness of the Categories of Absolutely Pure
Objects and Left-Exact Functors 148
Exercises 150
A. Effaceable and torsion functors B. Effaceable functors
and injective objects C. Oth right-derived functors D. Ab
solutely pure objects E. Computations of Oth right-
derived functors F. Sheaf theory G. Relative homological
algebra
APPENDIX 155
BmLIOGRAPHY 161
INDEX 163
DEDICATION
To the National Science Foundation for paying me while I wrote
part of this book.
To Columbia University for paying Sonja Levine, who typed
the preliminary manuscript of the book.
To the University of Pennsylvania for paying me while I finished
the book.
To Harper & Row for paying John Leahy, who proved the
book.
To Pamela Freyd for typing the final manuscript and for many,
many other things none of which have anything to do with
pay.
P. J. F.
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