basic algebra ii - nathan jacobson

This can always be arranged for a given class of sets horn A, B by replacing the given set horn A,B by the set of ordered triples A,B,f where fe horn A,B.. Ring, the category of associat

Library of Congress Cataloging-in-Publication Data

(Revised for vol 2)

ISBN 0-7167-1933-9 (v 2)

Copyright © 1989 by W H Freeman and Company

No part of this book may be reproduced by any mechanical, photographic, or electronic process, or in the form of a phonographic recording, nor may it be stored in a retrieval system, transmitted, or otherwise copied for public or private use, without written permission from the publisher

Printed in the United States of America

1 2 3 4 5 6 7 8 9 0 VB 7 6 5 4 3 2 1 0 8 9

0.2 Arithmetic of cardinal numbers 3

0.3 Ordinal and cardinal numbers 4

0.4 Sets and classes 6

References 7

1 C A T E G O R I E S 8

1.1 Definition and examples of categories 9

1.2 Some basic categorical concepts 15

1.3 Functors and natural transformations 18

1.4 Equivalence of categories 26

1.5 Products and coproducts 32

1.6 The horn functors Representable functors 37

2.2 Subalgebras and products 58

2.3 Homomorphisms and congruences 60

2.4 The lattice of congruences Subdirect products 66

2.5 Direct and inverse limits 70

2.6 Ultraproducts 75

2.7 Free Q-algebras 78

2.8 Varieties 81

2.9 Free products of groups 87

2.10 Internal characterization of varieties 91

References 93

3 M O D U L E S 9 4

3.1 The categories R-mod and mod-R 95

3.2 Artinian and Noetherian modules 100

3.3 Schreier refinement theorem Jordan-Holder theorem 104

3.4 The Krull-Schmidt theorem 110

3.5 Completely reducible modules 117

3.6 Abstract dependence relations Invariance of dimensionality 122 3.7 Tensor products of modules 125

3.13 The Wedderburn-Artin theorem for simple rings 171

3.14 Generators and progenerators 173

3.15 Equivalence of categories of modules 177

References 183

4 B A S I C S T R U C T U R E T H E O R Y OF R I N G S 1 8 4

4.1 Primitivity and semi-primitivity 185

4.2 The radical of a ring 192

4.3 Density theorems 197

4.4 Artinian rings 202

4.5 Structure theory of algebras 210

4.6 Finite dimensional central simple algebras 215

4.7 The Brauer group 226

Contents vii

5.5 Characters Orthogonality relations 269

5.6 Direct products of groups Characters of abelian groups 279

5.7 Some arithmetical considerations 282

5.8 Burnside's p a

q b

theorem 284 5.9 Induced modules 286

5.10 Properties of induction Frobenius reciprocity theorem 292

5.11 Further results on induced modules 299

5.12 Brauer's theorem on induced characters 305

5.13 Brauer's theorem on splitting fields 313

5.14 The Schur index 314

5.15 Frobenius groups 317

References 325

E L E M E N T S OF H O M O L O G I C A L A L G E B R A

W I T H A P P L I C A T I O N S 3 2 6

6.1 Additive and abelian categories 327

6.2 Complexes and homology 331

6.3 Long exact homology sequence 334

7.7 Integrally closed domains 412

7.8 Rank of projective modules 414

7.9 Projective class group 419

7.10 Noetherian rings 420

7.11 Commutative artinian rings 425

7.12 Affine algebraic varieties The Hilbert Nullstellensatz 427

7.13 Primary decompositions 433

7.14 Artin-Rees lemma Krull intersection theorem 440

7.15 Hilbert's polynomial for a graded module 443

7.16 The characteristic polynomial of a noetherian local ring 448

7.17 Krull dimension 450

7.18 J-adic topologies and completions 455

References 462

8 FIELD T H E O R Y 4 6 3

8.1 Algebraic closure of a field 464

8.2 The Jacobson-Bourbaki correspondence 468

8.3 Finite Galois theory 471

8.4 Crossed products and the Brauer group 475

8.5 Cyclic algebras 484

8.6 Infinite Galois theory 486

8.7 Separability and normality 489

8.8 Separable splitting fields 495

9.2 The approximation theorem 562

9.3 Absolute values on Q and F(x) 564

9.4 Completion of a field 566

9.5 Finite dimensional extensions of complete fields

The archimedean case 569

9.6 Valuations 573

9.7 Valuation rings and places 577

9.8 Extension of homomorphisms and valuations 580

9.9 Determination of the absolute values of a finite

dimensional extension field 585

9.10 Ramification index and residue degree Discrete valuations 588

9.11 Hensel's lemma 592

9.12 Local fields 595

Contents ix

9.13 Totally disconnected locally compact division rings 599

9.14 The Brauer group of a local field 608

9.15 Quadratic forms over local fields 611

References 618

10 D E D E K I N D D O M A I N S 6 1 9

10.1 Fractional ideals Dedekind domains 620

10.2 Characterizations of Dedekind domains 625

10.3 Integral extensions of Dedekind domains 631

10.4 Connections with valuation theory 634

10.5 Ramified primes and the discriminant 639

10.6 Finitely generated modules over a Dedekind domain 643

References 649

11 F O R M A L L Y R E A L FIELDS 6 5 0

11.1 Formally real fields 651

11.2 Real closures 655

11.3 Totally positive elements 657

11.4 Hilbert's seventeenth problem 660

11.5 Pfister theory of quadratic forms 663

11.6 Sums of squares in R(x u , x n ), R a real closed field 669

11.7 Artin-Schreier characterization of real closed fields 674

References 677

I N D E X 6 7 9

of Basic Algebra I

I N T R O D U C T I O N : C O N C E P T S F R O M SET T H E O R Y

T H E INTEGERS 1

0.1 The power set of a set 3

0.2 The Cartesian product set Maps 4

0.3 Equivalence relations Factoring a map through an equivalence relation 10 0.4 The natural numbers 15

0.5 The number system Z of integers 19

0.6 Some basic arithmetic facts about Z 22

0.7 A word on cardinal numbers 24

1 M O N O I D S A N D G R O U P S 2 6

1.1 Monoids of transformations and abstract monoids 28

1.2 Groups of transformations and abstract groups 31

1.3 Isomorphism Cayley's theorem 36

1.4 Generalized associativity Commutativity 39

1.5 Submonoids and subgroups generated by a subset Cyclic groups 42

1.6 Cycle decomposition of permutations 48

1.7 Orbits Cosets of a subgroup 51

1.8 Congruences Quotient monoids and groups 54

Contents xi

2.6 Ideals and quotient rings for Z 103

2.7 Homomorphisms of rings Basic theorems 106

2.14 Factorial monoids and rings 140

2.15 Principal ideal domains and Euclidean domains 147

2.16 Polynomial extensions of factorial domains 151

2.17 "Rings" (rings without unit) 155

3 M O D U L E S OVER A P R I N C I P A L I D E A L D O M A I N 1 5 7

3.1 Ring of endomorphisms of an abelian group 158

3.2 Left and right modules 163

3.3 Fundamental concepts and results 166

3.4 Free modules and matrices 170

3.5 Direct sums of modules 175

3.6 Finitely generated modules over a p.i.d Preliminary results 179

3.7 Equivalence of matrices with entries in a p.i.d 181

3.8 Structure theorem for finitely generated modules over a p.i.d 187

3.9 Torsion modules' primary components' invariance theorem 189

3.10 Applications to abelian groups and to linear transformations 194

3.11 The ring of endomorphisms of a finitely generated module over a p.i.d 204

4 G A L O I S T H E O R Y OF E Q U A T I O N S 2 1 0

4.1 Preliminary results1

some old1

some new 213 4.2 Construction with straight-edge and compass 216

4.3 Splitting field of a polynomial 224

4.4 Multiple roots 229

4.5 The Galois group The fundamental Galois pairing 234

4.6 Some results on finite groups 244

4.7 Galois' criterion for solvability by radicals 251

4.8 The Galois group as permutation group of the roots 256

4.9 The general equation of the nth degree 262

4.10 Equations with rational coefficients and symmetric group as

Galois group 267

4.11 Constructible regular rc-gons 271

4.12 Transcendence of e and n The Lindemann-Weierstrass theorem 277

4.13 Finite fields 287

4.14 Special bases for finite dimensional extension fields 290

4.15 Traces and norms 296

4.16 Mod p reduction 301

5 R E A L P O L Y N O M I A L E Q U A T I O N S A N D I N E Q U A L I T I E S 3 0 6

5.1 Ordered fields Real closed fields 307

5.2 Sturm's theorem 311

5.3 Formalized Euclidean algorithm and Sturm's theorem 316

5.4 Elimination procedures Resultants 322

5.5 Decision method for an algebraic curve 327

5.6 Generalized Sturm's theorem Tarski's principle 335

6 M E T R I C V E C T O R S P A C E S A N D T H E C L A S S I C A L G R O U P S 3 4 2

6.1 Linear functions and bilinear forms 343

6.2 Alternate forms 349

6.3 Quadratic forms and symmetric bilinear forms 354

6.4 Basic concepts of orthogonal geometry 361

6.5 Witt's cancellation theorem 367

6.6 The theorem of Cartan-Dieudonne 371

6.7 Structure of the linear group GL n (F) 375

6.8 Structure of orthogonal groups 382

6.9 Symplectic geometry The symplectic group 391

6.10 Orders of orthogonal and symplectic groups over a finite field 398

6.11 Postscript on hermitian forms and unitary geometry 401

7 A L G E B R A S OVER A FIELD 4 0 5

7.1 Definition and examples of associative algebras 406

7.2 Exterior algebras Application to determinants 411

7.3 Regular matrix representations of associative algebras Norms and traces 422 7.4 Change of base field Transitivity of trace and norm 426

7.5 Non-associative algebras Lie and Jordan algebras 430

7.6 Hurwitz' problem Composition algebras 438

7.7 Frobenius' and Wedderburn's theorems on associative division algebras 451

8 LATTICES A N D B O O L E A N A L G E B R A S 4 5 5

8.1 Partially ordered sets and lattices 456

8.2 Distributivity and modularity 461

8.3 The theorem of Jordan-Holder-Dedekind 466

8.4 The lattice of subspaces of a vector space

Fundamental theorem of projective geometry 468

8.5 Boolean algebras 474

8.6 The Mobius function of a partialy ordered set 480

Preface

The most extensive changes in this edition occur in the segment of the b o o k devoted to commutative algebra, especially in C h a p t e r 7, Commutative Ideal Theory: General Theory and Noetherian Rings; C h a p t e r 8, Field Theory; a n d Chapter 9, Valuation Theory In Chapter 7 we give an improved account of integral dependence, highlighting relations between a ring a n d its integral ex-tensions ("lying over," "going-up," and "going-down" theorems) Section 7.7, Integrally Closed Domains, is new, as are three sections in Chapter 8: 8.13, Transcendency Bases for Domains; 8.18, Tensor P r o d u c t s of Fields; and 8.19, Free Composites of Fields The latter two are taken from Volume III of our

Lectures in Abstract Algebra (D Van N o s t r a n d 1964; Springer-Verlag, 1980)

The most notable addition to Chapter 9 is Krasner's lemma, used to give an improved proof of a classical theorem of Kurschak's lemma (1913) We also give an improved proof of the theorem on extensions of absolute values to a finite dimensional extension of a field (Theorem 9.13) based on the concept of composite of a field considered in the new section 8.18

In Chapter 4, Basic Structure Theory of Rings, we give improved accounts

of the characterization of finite dimensional splitting fields of central simple algebras a n d of the fact that the Brauer classes of central simple algebras over

a given field constitute a set—a fact which is needed to define the Brauer group

Br(F) In the chapter on homological algebra (Chapter 6), we give an improved

proof of the existence of a projective resolution of a short exact sequence of modules

A n u m b e r of new exercises have been added and some defective ones have been deleted

Some of the changes we have m a d e were inspired by suggestions m a d e by our colleagues, Walter Feit, George Seligman, and Tsuneo Tamagawa They,

as well as Ronnie Lee, Sidney Porter (a former graduate student), and the Chinese translators of this book, Professors C a o Xi-hua and W a n g Jian-pan, pointed out some errors in the first edition which are now corrected W e are indeed grateful for their interest and their important inputs to the new edition

O u r heartfelt thanks are due also to F D Jacobson, for reading the proofs of this text and especially for updating the index

Preface to the First Edition

This volume is a text for a second course in algebra that presupposes an ductory course covering the type of material contained in the Introduction a n d

intro-the first three or four chapters of Basic Algebra I These chapters dealt with

the rudiments of set theory, group theory, rings, modules, especially modules over a principal ideal domain, and Galois theory focused on the classical p r o b -lems of solvability of equations by radicals a n d constructions with straight-edge and compass

Basic Algebra II contains a good deal m o r e material t h a n can be covered in

a year's course Selection of chapters as well as setting limits within chapters will be essential in designing a realistic p r o g r a m for a year We briefly indicate several alternatives for such a program: Chapter 1 with the addition of section 2.9 as a supplement to section 1.5, Chapters 3 and 4, Chapter 6 to section 6.11, Chapter 7 to section 7.13, sections 8.1-8.3, 8.6, 8.12, Chapter 9 to section 9.13

A slight modification of this p r o g r a m would be to trade off sections 4.6-4.8 for sections 5.1-5.5 a n d 5.9 F o r students who have h a d no Galois theory it will be desirable to supplement section 8.3 with some of the material of C h a p -

ter 4 of Basic Algebra I If an i m p o r t a n t objective of a course in algebra is an

understanding of the foundations of algebraic structures and the relation

be-tween algebra and mathematical logic, then all of Chapter 2 should be included

in the course This, of course, will necessitate thinning down other parts, e.g., homological algebra There are m a n y other possibilities for a one-year course based on this text

The material in each chapter is treated to a depth that permits the use of the text also for specialized courses F o r example, Chapters 3, 4, a n d 5 could con-stitute a one-semester course on representation theory of finite groups, and Chapter 7 and parts of Chapters 8, 9, and 10 could be used for a one-semester course in commutative algebras Chapters 1, 3, and 6 could be used for an introductory course in homological algebra

Chapter 11 on real fields is somewhat isolated from the remainder of the

book However, it constitutes a direct extension of Chapter 5 of Basic Algebra

I and includes a solution of Hilbert's problem on positive semi-definite rational functions, based on a theorem of Tarski's t h a t was proved in Chapter 5 of the first volume Chapter 11 also includes Pfister's beautiful theory of quadratic forms that gives an answer to the question of the minimum n u m b e r of squares

required to express a sum of squares of rational functions of n real variables

(see section 11.5)

Aside from its use as a text for a course, the b o o k is designed for independent reading by students possessing the b a c k g r o u n d indicated A great deal of ma-terial is included However, we believe that nearly all of this is of interest to mathematicians of diverse orientations and not just to specialists in algebra W e have kept in mind a general audience also in seeking to reduce to a minimum the technical terminology a n d in avoiding the creation of an overly elaborate machinery before presenting the interesting results Occasionally we have h a d

to pay a price for this in proofs that may appear a bit heavy to the specialist

M a n y exercises have been included in the text Some of these state interesting additional results, accompanied with sketches of proofs Relegation of these to the exercises was motivated simply by the desire to reduce the size of the text somewhat The reader would be well advised to work a substantial n u m b e r of the exercises

An extensive bibliography seemed inappropriate in a text of this type In its place we have listed at the end of each chapter one or two specialized texts in which the reader can find extensive bibliographies on the subject of the chapter Occasionally, we have included in our short list of references one or two papers

of historical importance N o n e of this has been done in a systematic or prehensive manner

com-Again it is a pleasure for me to acknowledge the assistance of m a n y friends in suggesting improvements of earlier versions of this text I should mention first the students whose perceptions detected flaws in the exposition a n d sometimes suggested better proofs that they h a d seen elsewhere Some of the students who

Preface to the First Edition xvii

have contributed in this way are M o n i c a Barattieri, Ying Cheng, Daniel Corro, William Ellis, Craig H u n e k e , and Kenneth M c K e n n a Valuable suggestions have been communicated to me by Professors Kevin M c C r i m m o n , James D Reid, Robert L Wilson, and Daniel Zelinsky I have received such suggestions also from my colleagues Professors Walter Feit, George Seligman, and Tsuneo Tamagawa The a r d u o u s task of proofreading was largely taken over by Ying Cheng, Professor Florence Jacobson, and James Reid Florence Jacobson assis-ted in compiling the index Finally we should mention the fine j o b of typing that was done by Joyce H a r r y and D o n n a Belli I am greatly indebted to all

of these individuals, and I take this opportunity to offer them my sincere thanks

Introduction

In the Introduction to Basic Algebra I (abbreviated t h r o u g h o u t as "BAI") we

gave an account of the set theoretic concepts that were needed for that volume

These included the power set 0>(S) of a set S, the Cartesian product S 1 x S 2 of

two sets S 1 and S 2 , m a p s ( = functions), a n d equivalence relations In the first

volume we generally gave preference to constructive arguments and avoided transfinite methods altogether

The results that are presented in this volume require more powerful tools, particularly for the proofs of certain existence theorems M a n y of these proofs will be based on a result, called Zorn's lemma, whose usefulness for proving such existence theorems was first noted by M a x Zorn We shall require also some results on the arithmetic of cardinal numbers All of this fits into the framework of the Z e r m e l o - F r a e n k e l axiomatization of set theory, including the axiom of choice (the so-called Z F C set theory) Two excellent texts that can be used to fill in the details omitted in our discussion are P R H a l m o s '

Naive Set Theory a n d the m o r e substantial Set Theory and the Continuum Hypothesis by P J Cohen

Classical mathematics deals almost exclusively with structures based on sets

O n the other hand, category theory—which will be introduced in Chapter 1 —

deals with collections of sets, such as all groups, that need to be treated differently from sets Such collections are called classes A brief indication of a suitable foundation for category theory is given in the last section of this Introduction

0.1 Z O R N ' S L E M M A

We shall now formulate a m a x i m u m principle of set theory called Zorn's

lemma We state this first for subsets of a given set We recall that a set C of subsets of a set S (that is, a subset of the power set £?{S)) is called a chain if C

is totally ordered by inclusion, that is, for any A,BeC either A a B or B cz A

A set T of subsets of S is called inductive if the union [JA a of any chain

C = {A^} cz T is a member of T W e can now state

Z O R N ' S L E M M A (First formulation) Let T be a non-vacuous set of subsets

of a set S Assume T is inductive Then T contains a maximal element, that is, there exists an MeT such that no AeT properly contains M

There is another formulation of Zorn's lemma in terms of partially ordered

sets (BAI, p 456) Let P, ^ be a partially ordered set We call P, ^ (totally or

linearly) ordered if for every a,beP either a ^ b or b ^ a We call P inductive if

every non-vacuous subset C of P that is (totally) ordered by ^ as defined in P has a least upper b o u n d in P, that is, there exists a u e P such that u ^ a for every aeC and ifv^a for every aeC then v ^ u Then we have

Z O R N ' S L E M M A (Second formulation) Let P,^ be a partially ordered set

that is inductive Then P contains maximal elements, that is, there exists meP such that no aeP satisfies m < a

It is easily seen that the two formulations of Zorn's lemma are equivalent, so there is n o h a r m in referring to either as "Zorn's lemma." It can be shown that Zorn's lemma is equivalent to the

A X I O M O F C H O I C E Let S be a set, ^(<S)* the set of non-vacuous subsets of

S Then there exists a map f (a "choice function") of ^%S)* into S such that f(A)eAfor every Ae0>(S)*

This is equivalent also to the following: If {A a } is a set of non-vacuous sets

A a , then the Cartesian p r o d u c t J][v4a # 0

T h e statement that the axiom of choice implies Zorn's lemma can be proved

0.2 Arithmetic of Cardinal Numbers

by an argument that was used by E Zermelo to prove that every set can be

well ordered (see H a l m o s , pp 62-65) A set S is well ordered by an order relation ^ if every non-vacuous subset of S has a least element The well-

ordering theorem is also equivalent to Zorn's lemma and to the axiom of choice We shall illustrate the use of Zorn's lemma in the next section

0.2 A R I T H M E T I C OF C A R D I N A L N U M B E R S

Following Halmos, we shall first state the main results on cardinal arithmetic

without defining cardinal numbers We say that the sets A and B have the

same cardinality and indicate this by \A\ = \B\ if there exists a bijective m a p of

A onto B W e write \A\ < \B\ if there is an injective m a p of A into B and

\A\ < \B\ if \A\ ^ \B\ and \A\ ^ \B\ Using these notations, the

Schroder-Bernstein theorem (BAI, p 25) can be stated as: \A\ ^ \B\ and

\B\ ^ \A\ if and only if \A\ = \B\ A set F is finite if \F\ = \N\ for some

N = { 0 , 1 , , n — 1 } and A is countably infinite if \A\ = \co\ for co = { 0 , 1 , 2 , }

It follows from the axiom of choice that if A is infinite ( = not finite), then

\co\ < \A\ W e also have Cantor's theorem t h a t for any A, \A\ < \&(A)\

W e write C = AO B for sets A,B,C if C = A uB a n d A nB = 0 It is

clear that if \A X | ^\A 2 \ a n d j i ? ^ ^ ^ ! , then \A 1 0B x \ ^\A 2 0B 2 \ Let

C = F Oco where F is finite, say, F = { x0, , xn_ i } where x t # Xj for i ^ j

Then the m a p of C into co such that x t ~> i, k^k + n for keco is bijective Hence |C| = |co| It follows from |co| ^ \A\ for any infinite A that if C = F 0 ^4, then |C| = |^4| F o r we can write A = DOB where \D\ = \co\ Then we have a bijective m a p o f f u D o n t o D a n d we can extend this by the identity on B to obtain a bijective m a p of C onto A

We can use the preceding result and Zorn's l e m m a to prove the main result

on "addition of cardinals," which can be stated a s : If A is infinite and

C = A u B where \B\ = \A\, then \C\ = \A\ This is clear if A is countable from

the decomposition co = { 0 , 2 , 4 , } u { 1 , 3 , 5 , } It is clear also that the result

is equivalent to \A x 2| = \A\ if 2 = {0,1}, since \A x 2| = \A 0 B| W e proceed

to prove that \A x 2| = | ^ | for infinite A Consider the set of pairs (XJ) where

X is an infinite subset of A a n d / i s a bijective m a p of X x 2 onto X This set is

not vacuous, since A contains countably infinite subsets X and for such an X

we have bijective m a p s of 1 x 2 onto X W e order the set {(XJ)} by

(XJ) < (X'J 1

) if X c Xr

and / is an extension o f / It is clear that {(XJ)}, ^

is an inductive partially ordered set Hence, by Zorn's lemma, we have a

maximal (Y,g) in {(XJ)} W e claim that A — Y is finite For, if ^4 — Y is

infinite, then this contains a countably infinite subset D, and gr can be extended

to a bijective m a p of (YOD) x 2 o n t o 7 0 D contrary to the maximality of

(Y,g) Thus A-Yis finite Then

{(X,f)} as before By Zorn's lemma, we have a maximal (Y,g) in this set The

result we wish to prove will follow if \ Y\ = \A\ Hence suppose \ Y\ < \A\ Then the relation A = Y 0(A— Y) a n d the result on addition imply that

\A\ = \A-Y| Hence \A-Y\ > \ Y\ Then A-Y contains a subset Z such that

| Z | = | Y| Let W = Y u Z, so = Y 0 Z a n d x W is the disjoint union of

the sets Y x Y, Y x Z, Z x Y, and Z x Z We have

Hence we can extend g to a bijective m a p of W x W onto FT This contradicts the maximality of (Y,g) Hence \ Y\ = \A\ and the proof is complete

We also have the stronger result that if A ^ 0 and B is infinite and

In axiomatic set theory no distinction is m a d e between sets and elements One

writes AeB for "the set A is a member of the set JB." This must be

5

distinguished from A cz B, which reads "A is a subset of Br (In the texts on set theory one finds A c= B for our A cz B and A cz B for our v4 ^ 5 ) O n e defines

A cz B to mean that CeA => C e £ O n e of the axioms of set theory is that

given an arbitrary set C of sets, there is a set that is the union of the sets belonging to C, that is, for any set C there exists a set U such that A e U if and only if there exists a 5 such that AeB and £ e C In particular, for any set A

we can form the successor A +

=Au{A} where {^4} is the set having the

single member A

The process of forming successors gives a way of defining the set co (= N) of

natural numbers W e define 0 = 0 , 1 = 0 +

D E F I N I T I O N 0.1 An ordinal is a set oc that is well ordered by e and is

a set of representatives for the similarity classes of well-ordered sets F o r we

have the following theorem: If S, ^ is well ordered, then there exists a unique ordinal a and a unique bijective order-preserving m a p of S onto a If a and are ordinals, either a = p, a < /?, or f$ < a An ordinal a is called a successor if

there exists an ordinal /? such that oc = P +

Otherwise, a is called a Zimft ordinal

Any non-vacuous set of ordinals has a least element

D E F I N I T I O N 0.2 A cardinal number is an ordinal oc such that if P is any

ordinal such that the cardinality \P\ = |a|, then oc < /?

A cardinal number is either finite or it is a limit ordinal O n the other hand,

not every limit ordinal is a cardinal F o r example co + co is not a cardinal The smallest infinite cardinal is co Cardinals are often denoted by the Hebrew

letter "aleph" K with a subscript In this n o t a t i o n one writes K for co

Since any set can be well ordered, there exists a uniquely determined

cardinal a such that |a| = |S| for any given set S We shall now call oc the

cardinal number or cardinality of S and indicate this by |5| = a The results that

we obtained in the previous section yield the following formulas for cardinalities

T h e primitive objects in this system are "classes" and "sets" or m o r e precisely class variables and set variables together with a relation e A characteristic feature of this system is that classes that are members of other classes are sets,

that is, we have the axiom: Y e X => Y is a set

Intuitively classes may be thought of as collections of sets that are defined

by certain properties A part of the G B system is concerned with operations that can be performed on classes, corresponding to combinations of properties

A typical example is the intersection of classes, which is expressed by the

following: F o r all X and Y there exists a Z such that ueZ if and only if ueX and UEY We have given here the intuitive meaning of the axiom:

V X V Y 3 Z V u {ueZoueX a n d ueY) where V is read "for all" a n d 3 is read

"there exists " Another example is that for every X there exists a Y such that

ueY if and only if u£X i\JXlYMu (ue Y <=> ~ueX), where is the

negation of •••) Other class formations correspond to unions, etc We refer to Cohen's book for a discussion of the Z F and the G B systems and their relations We note here only that it can be shown that any theorem of Z F is a theorem of G B and every theorem of G B that refers only to sets is a theorem

o f Z F

In the sequel we shall use classes in considering categories a n d in a few other places where we encounter classes and then show that they are sets by showing that they can be regarded as members of other classes The familiar algebraic structures (groups, rings, fields, modules, etc.) will always be understood to be

"small," that is, to be based on sets

0.4 Sets and Classes 7

REFERENCES

Paul R Halmos, Naive Set Theory, Springer, New York, 1960

Paul J Cohen, Set Theory and the Continuum Hypothesis, Addison-Wesley, Reading,

Mass., 1966

Categories

In this chapter and the next one on universal algebra we consider two unifying concepts that permit us to study simultaneously certain aspects of a large number of mathematical structures The concept we shall study in this chapter

is that of category, and the related notions of functor and natural transformation These were introduced in 1945 by Eilenberg and M a c L a n e to provide a precise meaning to the statement that certain isomorphisms are

"natural." A typical example is the natural isomorphism between a

finite-dimensional vector space V over a field and its double dual F * * , the space of linear functions on the space F * of linear functions on V T h e isomorphism of

V onto F * * is the linear m a p associating with any vector xeV the evaluation

function f^f(x) defined for a l l / e F * T o describe the "naturality" of this

isomorphism, Eilenberg and M a c L a n e had to consider simultaneously all finite-dimensional vector spaces, the linear transformations between them, the double duals of the spaces, and the corresponding linear transformations between them These considerations led to the concepts of categories and functors as preliminaries to defining natural transformation We shall discuss a generalization of this example in detail in section 1.3

The concept of a category is m a d e up of two ingredients: a class of objects

9

and a class of morphisms between them Usually the objects are sets and the morphisms are certain m a p s between them, e.g., topological spaces and continuous maps The definition places on an equal footing the objects and the morphisms T h e a d o p t i o n of the category point of view represents a shift in emphasis from the usual one in which objects are primary and morphisms secondary O n e thereby gains precision by making explicit at the outset the morphisms that are allowed between the objects collected to form a category The language and elementary results of category theory have now pervaded

a substantial part of mathematics Besides the everyday use of these concepts and results, we should note that categorical notions are fundamental in some

of the most striking new developments in mathematics O n e of these is the extension of algebraic geometry, which originated as the study of solutions in the field of complex numbers of systems of polynomial equations with complex coefficients, to the study of such equations over an arbitrary commutative ring The proper foundation of this study, due mainly to A Grothendieck, is based

on the categorical concept of a scheme Another deep application of category theory is K Morita's equivalence theory for modules, which gives a new insight into the classical W e d d e r b u r n - A r t i n structure theorem for simple rings and plays an important role in the extension of a substantial part of the structure theory of algebras over fields to algebras over commutative rings

A typical example of a category is the category of groups Here one considers "all" groups, and to avoid the paradoxes of set theory, the foundations need to be handled with greater care t h a n is required in studying group theory itself O n e way of avoiding the well-known difficulties is to adopt the G o d e l - B e r n a y s distinction between sets and classes We shall follow this approach, a brief indication of which was given in the Introduction

In this chapter we introduce the principal definitions of category t h e o r y functors, natural transformations, products, coproducts, universals, and adjoints—and we illustrate these with m a n y algebraic examples This provides

-a review of -a l-arge n u m b e r of -algebr-aic concepts We h-ave included some trivial examples in order to add a bit of seasoning to a discussion that might otherwise appear too bland

non-1.1 D E F I N I T I O N A N D E X A M P L E S OF C A T E G O R I E S

D E F I N I T I O N 1.1 A category C consists of

1 A class ob C of objects (usually denoted as A, B, C, etc.)

2 For each ordered pair of objects (A,B), a set hom c (A,B) (or simply hom(A,B) if C is clear) whose elements are called morphisms with

domain A and codomain B (or from A to B)

3 For each ordered triple of objects (A,B,C), a map (fg)^gf of the

product set horn (A,B) x horn (B, C) into horn (A, C)

It is assumed that the objects and morphisms satisfy the following conditions:

01 //(A,B) ^ (C,D), then horn (A,B) and horn (C,D) are disjoint

02 (Associativity) If fe horn (A, B), gehom(B, C), and hehom(C,D), then (hg)f = h(gf) (As usual, we simplify this to hgf)

03 (Unit) For every object A we have an element l A ehorn (A, A) such that fl A =f for every fe horn (A, B) and l A g = g for every gehom (B, A) (1 A is unigue.)

I f f e h o m (A,B), we write f:A->BorA^B (sometimes A y> B), and we call / an arrow from A to B N o t e that gf is defined if and only if the d o m a i n of g

coincides with the codomain o f / a n d g / h a s the same domain a s / a n d the same

codomain as g

The fact that gf = h can be indicated by saying that

is a commutative diagram The meaning of more complicated diagrams is the same as for maps of sets (BAI, pp 7-8) F o r example, the commutativity of

D

means that gf = kh, and the associativity condition (hg)f = h(gf) is expressed

by the commutativity of

D

1.1 Definition and Examples of Categories

The condition that 1 A is the unit in hom(,4,,4) can be expressed by the commutativity of

for all fe horn (A,B) a n d all gehom (B,A)

We remark that in defining a category it is not necessary at the outset that

the sets horn (A,B) and horn (C,D) be disjoint whenever (A,B) # (C, D) This can always be arranged for a given class of sets horn (A, B) by replacing the given set horn (A,B) by the set of ordered triples (A,B,f) where fe horn (A,B)

This will give us considerably greater flexibility in constructing examples of categories (see exercises 3 - 6 below)

We shall now give a long list of examples of categories

EXAMPLES

1 Set, the category of sets Here ob Set is the class of all sets If A and B are sets, horn (A, B) = B A

, the set of maps from A to B The product gf is the usual composite of

maps and 1 A is the identity map on the set A The validity of the axioms CI, C2, and

C3 is clear

2 Mon, the category of monoids, ob Mon is the class of monoids (BAI, p 28),

horn (M, N) for monoids M and N is the set of homomorphisms of M into N, gf is the composite of the homomorphisms g and/, and 1M is the identity map on M (which is a

homomorphism) The validity of the axioms is clear

3 Grp, the category of groups The definition is exactly like example 2, with groups

replacing monoids

4 Ab, the category of abelian groups, ob Ab is the class of abelian groups

Otherwise, everything is the same as in example 2

A category D is called a subcategory of the category C if o b D is a subclass

of o b C and for any , 4 , £ e o b D , hom B (A,B) cz h o mc( , 4 , £ ) It is required also (as part of the definition) that 14 for AeobJ} and the product of morphisms for D is the same as for C The subcategory D is called full if

hom (A,B) = h o m ( , 4 , £ ) for every A, Be J} It is clear that Grp and Ab are

full subcategories of Mon O n the other hand, since a monoid is not just a set

but a triple (M, p, 1) where p is an associative binary composition in M and 1

is the unit, the category Mon is not a subcategory of Set We shall give below

an example of a subcategory that is not full (example 10)

We continue our list of examples

5 Let M be a monoid Then M defines a category M by specifying that o b M = {A},

a set with a single element A, and defining horn (A, A) = M, 1 A the unit of M, and xy

for x, ye horn (A, A), the product of x and y as given in M It is clear that M is a category with a single object Conversely, let M be a category with a single object:

o b M = {A} Then M = horn {A, A) is a monoid It is clear from this that monoids can

be identified with categories whose object classes are single-element sets

A category is called small if o b C is a set Example 5 is a small category; 1-4

are not

An element fe horn (A,B) is called an isomorphism if there exists a

g e h o m (B,A) such that fg = 1 B and gf = 1 A It is clear that g is uniquely

determined by f so we can denote it a s / "1

This is also an isomorphism and

maps, and in Grp they are the usual isomorphisms ( = bijective h o m o

-morphisms)

6 Let G be a group and let this define a category G with a single object as in

example 5 The characteristic property of this type of category is that it has a single object and all arrows ( = morphisms) are isomorphisms

7 A groupoid is a small category in which morphisms are isomorphisms

8 A discrete category is a category in which horn (A,B) = 0 if A ^ B and

horn (A, A) = {1 A } Small discrete categories can be identified with their sets of objects

9 Ring, the category of (associative) rings (with unit for the multiplication composition) obRing is the class of rings and the morphisms are homomorphisms

(mapping 1 into 1)

10 Rng, the category of (associative) rings without unit (BAI, p 155), phisms as usual Ring is a subcategory of Rng but is not a full subcategory, since there

homomor-exist maps of rings with unit that preserve addition and multiplication but do not map

1 into 1 (Give an example.)

11 R-mod, the category of left modules for a fixed ring R (We assume lx = x for x

in a left R-module M.) obR-mod is the class of left modules for R and the morphisms

are R-module homomorphisms Products are composites of maps If R = A is a

division ring (in particular, a field), then R-mod is the category of (left) vector spaces over A In a similar manner one defines mod-R as the category of right modules for the

ring JR

12 Let S be a pre-ordered set, that is, a set S equipped with a binary relation a ^ b such that a < a and a < b and £> < c imply a ^ c S defines a category S in which obS = S and for a,beS, horn (a,b) is vacuous or consists of a single element according

as a ^ or a ^ 5 If fe horn (a, fe) and g e horn (fe, c), then gf is the unique element in

horn (a, c) It is clear that the axioms for a category are satisfied Conversely, any small

category such that for any pair of objects A,B, horn (A,B) is either vacuous or a single

element is the category of a pre-ordered set as just defined

13 Top, the category of topological spaces The objects are topological spaces and

the morphisms are continuous maps The axioms are readily verified

We conclude this section by giving two general constructions of new categories from old ones T h e first of these is analogous to the construction of the opposite of a given ring (BAI, p 113) Suppose C is a category; then we define Co p

by o b Co p

= o b C ; for A,BeobCop

, h o mc o p( ^ , B ) = h o mc( 5 , A ) ; if

fehom cop (A,B) a n d gehom coP (B,C), then g-f (in Co p) = fg"(as given in C) 1 A

is as in C It is clear that this de'fines a category W e call this the dual category

of C Pictorially we have the following: If A ^ B in C , then A ^-B in Co p

by reversing all of the arrows

Next let C a n d D be categories T h e n we define the product category C x D

by the following recipe: o b C x D = o b C x o b D ; if A, B e o b C and

v 4 ' , F e o b D , then h o mC x D( ( y l , y l/

) , (B,B f )) = hom c {A,B) xhom D (A',B f

EXERCISES

1 Show that the following data define a category Ring*: obRing* is the class of

rings; if R and S are rings, then homR i n g*(R,S) is the set of homomorphisms and

anti-homomorphisms of R into S; gf for morphisms is the composite g following/ for the m a p s / a n d g; and 1^ is the identity map on R

2 By a ring with involution we mean a pair (RJ) where R is a ring (with unit) and j

is an involution in R; that is, if j :a-> a*, then (a + b)* = a* + b*, (ab)* = b*a*, 1* = 1, (a*)* = a (Give some examples.) By a homomorphism of a ring with involution (RJ) into a second one (S, k) we mean a map rj of R into S such that rj

is a homomorphism of R into S (sending 1 into 1) such that rj(ja) = k(r\a) for all

aeR Show that the following data define a category Rinv: obRinv is the class of

rings with involution; if (RJ) and (S,k) are rings with involution, then horn ((RJ), (S, k)) is the set of homomorphisms of (RJ) into (S,k); gf for morphisms is the composite of maps; and 1 {RJ) = 1 R

3 Let C be a category, A an object of C Let obC/A = IJxeobc horn (X,A) so

obC/A is the class of arrows in C ending at A If fehom(B,A) and

g e horn (C, A), define horn ( / g) to be the set of u :B -> C such that

5

is commutative Note that horn (f,g) and horn (/',#') may not be disjoint for

(fg)^(f\Q')- If ushorn (fg) and uehom(0,/i) for h:D^A, then

G horn ( / h) Use this information to define a product from horn (fg) and horn /z) to horn ( / /z) Define l f = 1 B f o r / : 5 A Show that these data and the

remark on page 11 can be used to define a category C/A called the category of

objects of C over A

4 Use Co p

to dualize exercise 3 This defines the category C A of objects of C below

A

5 Let C be a category, A l9 A 2 eobC Show that the following data define a

category C/{A 1 ,A 2 }' The objects are the triples (Bf 1 f 2 ) w h e r e /G h o mc( B , ^f) A

morphism h '-(B,f u f 2 ) (C,g u g 2 )-is a m o r p h i s m h :B -> C in C such that

A 1

is commutative Arrange to have the horn sets disjoint as before Define

l ( B j i , / 2 ) = 1B and the product of morphisms as in C Verify the axioms C2 and C3 for a category

6 Dualize exercise 5 by applying the construction to Co p

and interpreting in C The

resulting category is denoted as c\{A 1 ,A 2

}-7 (Alternative definition of a groupoid.) Let G be a groupoid as defined in example

7 above and let G= {J ABeohG hom(A,B) Then G is a set equipped with a

c o m p o s i t i o n t h a t is defined for some pairs of elements (fg),fgeG, such that

the following conditions hold:

(i) For any f eG there exist a uniquely determined pair (u,v), u,veG such that uf

and/u are defined and uf = f = fv These elements are called the left and right

units respectively of/

(ii) If u is a unit (left and right for some / e G), then u is its own left unit and

hence its own right unit

(iii) The product fg is defined if and only if the right unit of / coincides with the left unit of g

(iv) If fg and gh are defined, then (fg)h a n d / (gh) are defined and (fg)h =f(gh) (v) If / h a s left unit u and right unit v, then there exists an element g having right unit u and left unit v such that/g = u and gf = v

Show that, conversely, if G is a set equipped with a partial composition satisfying conditions (i)-(v), then G defines a groupoid category G in which ob G is the set

of units of G; for any objects u, v, horn (u, v) is the subset of G of elements having

u as left unit and v as right unit; the product composition of horn (u, v) x

horn (v, w) is that given in G

8 Let G be as in exercise 7 and let G* be the disjoint union of G and a set {0} Extend the composition in G to G* by the rules that 0a = 0 = aO for all a e G* and fg = 0 if / , g e G and /# is not defined in G Show that G* is a semigroup

(BAI, p 29)

1.2 S O M E BASIC CATEGORICAL C O N C E P T S

We have defined a morphism / in a category C to be an isomorphism if

f\A B and there exists a g :B -+ A such t h a t / # = 1 B and # / = 14 If/:v4 -> J5,

g :B A, and # / = 1^, t h e n / i s called a section of # and # is called a retraction

of / M o r e interesting t h a n these two concepts are the concepts of monic and

epic that are defined by cancellation properties: A m o r p h i s m / : A -» B is called

monic (epic) if it is left (right) cancellable in C ; that is, if g t and g 2 e horn (C, A) (horn (B V C)) for any C and fg x =fg 2 (g t f= g 2 f \ then g x = g 2

The following facts are immediate consequences of the definitions:

1 If A B and B A C and / and g are monic (epic), then gf is monic

(epic)

2 If A -4 B and B -4 C and gf is monic (epic), t h e n / i s monic (g is epic)

3 I f / h a s a section t h e n / i s epic, and iff has a retraction t h e n / i s monic

Iff is a m a p of a set ,4 into a set 5 , then it is readily seen t h a t / i s injective

(that i s , / ( a ) ¥^f(d) for a ^ a' in A) if and only if for any set C and m a p s g u g 2

of C into A, / # ! =fg 2 implies g 1 = g 2 (exercise 3, p 10 of BAI) Thus

fe h o mS e t( ^ , j B ) is monic if and only iff is injective Similarly, / is epic in Set if

and only i f / i s surjective (f(A) = B) Similar results hold in the categories

R-mod and Grp: We have

P R O P O S I T I O N 1.1 A morphism f in R-mod or in Grp is monic (epic) if and

only if the map of the underlying set is injective (surjective)

Proof Let / : A -» B in R-mod or Grp Iff is injective (surjective) as a m a p of

sets, then it is left (right) cancellable in Set It follows that / is monic (epic) in

R-mod or Grp N o w suppose the set m a p / is not injective Then C = k e r / ^ 0

in the case of R-mod and # 1 in the case of Grp Let g be the injection

homomorphism of C into A (denoted by C ^ A), so g(x) = x for every xeC Then fg is the h o m o m o r p h i s m of C into B, sending every xeC into the identity element of B Next let h be the homomorphism of C into A, sending every element of C into the identity element of A Then h / g since C ^ 0 (or

^ 1), but fg = fh H e n c e / i s not monic

Next suppose we are in the category R-mod and / is not surjective The

image / ( A ) is a submodule of B and we can form the module C = B/f (A), which is ^ 0 since f(A) ^ B Let g be the canonical h o m o m o r p h i s m of B onto

C and h the h o m o m o r p h i s m of B into C, sending every element of B into 0

Then g # h but gf = hf H e n c e / i s not epic

Finally, suppose we are in the category Grp and / is not surjective The

foregoing argument will apply if C =f(A)<\ B (C is a normal subgroup of B) This will generally not be the case, although it will be so if [B :C] = 2 Hence

we assume [B:C]> 2 and we shall complete the proof by showing that in this case there exist distinct homomorphisms g and h of B into the group Sym B of permutations of B such that gf = hf We let g be the h o m o m o r p h i s m b ^> b L of

B into Sym B where b L is the left translation x bx in B W e shall take h to have the form kg where k is an inner automorphism of Sym B T h u s k has the form y^pyp' 1

where y eSymB and p is a fixed element of S y m T h e n

h = kg will have the form b ~»pb L p~ x

and we want this to be different from

g :b ->b L This requires that the permutation p does not c o m m u t e with every

b L Since the permutations commuting with all of the left translations are the

right translations (exercise 1, p 42 of BAI), we shall have h = kg / g if p is not

1.2 Some Basic Categorical Concepts

a right translation Since translations / 1 have no fixed points, our condition

will be satisfied if p is any permutation # 1 having a fixed point O n the other hand, to achieve the condition gf = hf we require that p commutes with every

c L , ceC T o construct a p satisfying all of our conditions, we choose a

permutation n of the set C\B of right cosets Cb, beB, that is not the identity and has a fixed point Since \C\B\ > 2, this can be done Let / be a set of representatives of the right cosets Cb Then every element of B can be written

in one and only one way as a product cu, c e C, u e / We now define the m a p p

by p(cu) = cu' where %(Cu) = Cu' Then peSymB, p 1, and p has fixed points since n # 1 and n has fixed points It is clear that p commutes with every d L , deC Hence p satisfies all of our requirements a n d / i s not epic •

W h a t can be said a b o u t monies and epics in the category Ring? In the first

place, the proof of Proposition 1.1 shows that injective homomorphisms are monic and surjective ones are epic The next step of the argument showing that monies are injective breaks down totally, since the kernel of a ring homomorphism is an ideal and this may not be a ring (with unit) Moreover, even if it were, the injection m a p of the kernel is most likely not a ring homomorphism We shall now give a different argument, which we shall later

generalize (see p 82), to show t h a t monies in Ring are injective

L e t / b e a h o m o m o r p h i s m of the ring A into the ring B that is not injective

F o r m the ring A © A of pairs (a 1 ,a 2 ), a t eA, with component-wise addition

and multiplication and unit 1 = (1,1) Let K be the subset of A © A of elements (a 1 ,a 2 ) such that f(a x ) =f(a 2 ) It is clear that K is a subring of

A © A and K ^ D = {(a, a)\a e A} Let g± be the m a p (a u a 2 ) a x and g 2 the

m a p (a 1 ,a 2 ) ^ a 2 from K to A These are ring homomorphisms a.ndfg 1 =fg 2 ,

by the definition of K O n the other hand, since K ^ D, we have a pair

(a 1 ,a 2 )eK with a x / a 2 Then g±{a u a 2 ) = a x a 2 = g 2 (a u a 2 ) Hence

Qi ^ 9i> which shows t h a t / i s not monic

N o w we can show by an example that epics in Ring need not be surjective

For this purpose we consider the injection h o m o m o r p h i s m of the ring Z of

integers into the field Q of rationals If g and h are homomorphisms of Q into

a ring R, then gf = hf if and only if the restrictions g\Z = h\Z Since a homomorphism of Q is determined by its restriction to Z, it follows that

gf = hf implies g = h T h u s / i s epic and o b v i o u s l y / i s not surjective

We have proved

P R O P O S I T I O N 1.2 A morphism in Ring is monic if and only if it is injective

However, there exist epics in Ring that are not surjective

The concept of a monic can be used to define subobjects of an object A of a category C We consider the class of monies ending in A

We introduce a preorder in the class of these monies by declaring that / ^ g if there exists a k such t h a t / = gk It follows that k is monic W e w r i t e / = g if

/^ g and # ^ / In this case the element k is an isomorphism T h e relation = is

an equivalence and its equivalence classes are called the subobjects of A

By duality we obtain the concept of a quotient object of A Here we consider the epics issuing from A and d e f i n e / ^ g if there exists a k such that / = kg We have an equivalence relation / = g defined by f = kg where k is an

isomorphism The equivalence classes determined by this relation are called

the quotient objects of A

If the reader will consider the special case in which C = Grp, he will

convince himself that the foregoing terminology of subobjects and quotient

objects of the object A is reasonable However, it should be observed that the

quotient objects defined in Ring constitute a larger class than those provided

by surjective homomorphisms

EXERCISES

1 Give an example in Top of a morphism that is monic and epic but does not have

a retraction

2 Let G be a finite group, H a subgroup Show that the number of permutations of

G that commute with every h L , heH (acting in G), is [G \H] \\H§- G:H

Let K be a ring and let U(R) denote the multiplicative group of units ( = invertible elements) of R The m a p R U(R) is a m a p of rings into groups,

that is, a m a p of o b R i n g into ob Grp Moreover, if f:R-+S is a

homomorphism of rings, then the restriction f\U(R) maps U(R) into U(S) and

so may be regarded as a m a p of U(R) into 1/(5) Evidently this is a group homomorphism It is clear also that if g :S -> T is a ring homomorphism, then

(gfW(R) = (g\U{S))(f\U(R)) Moreover, the restriction 1 R \U(R) is the

identity m a p on U{R)

The m a p R^U(R) of rings into groups and f^>f\U(R) of ring

homomorphisms into g r o u p h o m o m o r p h i s m s constitute a functor from Ring

to Grp in the sense of the following definition

D E F I N I T I O N 1.2 / / C and D are categories, a (covariant) functor F from C

to D consists of

1 .4 map ,4 ~> i v l 0/ob C into o b D

2 For euery pafr o/objects (A,B) of C, a map f ^ F(f) ofhom c (A,B) into

in C is mapped into a commutative triangle in D

A contravariant functor from C to D is a functor from Co p

to D M o r e

directly, this is a m a p F of ob C into ob D a n d for each pair (A, B) of objects in

C, a m a p F of horn (A, B) into horn (FB,FA) such that F(fg) = F(g)F(f) and

F(1 A ) = 1 FA A functor from B x C to D is called a bifunctor from B and C into

D We can also combine bifunctors with contravariant functors to obtain functors from Bo p

x C t o D and from Bo p

x Co p

to D The first is called a

bifunctor that is contravariant in B and covariant in C and the second is a bifunctor that is contravariant in B and C

EXAMPLES

1 Let D be a subcategory of the category C Then we have the injection functor of D

into C that maps every object of D into the same object of C and maps any morphism

in D into the same morphism in C The special case in which D = C is called the

identity functor lc

2 We obtain a functor from Grp to Set by mapping any group into the underlying

set of the group and mapping any group homomorphism into the corresponding set map The type of functor that discards some of the given structure is called a "forgetful" functor Two other examples of forgetful functors are given in the next example

3 Associated with any ring (R, + , •, 0,1) we have the additive group (R, 4-, 0) and the multiplicative monoid (R,-, 1) A ring homomorphism is in particular a homomorphism

of the additive group and of the multiplicative monoid These observations lead in an

obvious manner to definitions of the forgetful functors from Ring to Ab and from Ring

to Mon

4 Let n be a positive integer For any ring ,R we can form the ring M n (R) of n x n

matrices with entries in R A ring homomorphism f:R-*S determines a phism (r tj ) ~» (/(r tj )) of M n (R) into M n {S) In this way we obtain a functor M n of Ring into Ring

homomor-5 Let n and JR be as in example 4 and let GL n (R) denote the group of units of M n (R),

that is, the group of n x n invertible matrices with entries in R The maps R ~> GL n (R),f

into (r tj ) ~> (/(r t j)) define a functor GL n from Ring to Grp

6 We define the power functor in Set by mapping any set A into its power set

0>(A) and any set map f:A-+B into the induced map f & of £?(A) into ^(B), which sends any subset A x of A into its image/(A x ) cz B (0 ~> 0 )

7 The abelianizing functor from Grp to Ab Here we map any group G into the

abelian group G/(G, G) where (G, G) is the commutator group of G (BAI, p 238) I f / i s

a homomorphism of G into a second group H,f maps (G, G) into (H,H) and so induces

a homomorphism / of G/(G, G) into H/(H,H) The map f^f completes the definition

of the abelianizing functor

8 Let Poset be the category of partially ordered sets Its objects are partially ordered sets (BAI, p 456) and the morphisms are order-preserving maps We obtain a functor

from R-mod to Poset by mapping any R-module M into L(M), the set of submodules

of M ordered by inclusion If f:M -> N is a module homomorphism, / determines an order-preserving map of L(M) into L(N) These maps define a functor

9 We define a projection functor of C x D into C by mapping any object (A, B) of

C x D into the object A of C and mapping (f g)ehorn ((A,B),(A',B')) into

fe horn (A, A')

10 We define the diagonal functor C - ^ C x C by mapping A ~» (A, A) and f\A-+B into (A,A)^(B,B)

11 Consider the categories mod and mod-R of left modules and right

modules respectively for the ring R We shall define a contravariant functor D from

R-mod to R-mod-R as follows If M is a left R-R-module, we consider the set

M* = hom^(M,i^) of homomorphisms of M into R regarded as left R-module in the

usual way Thus M* is the set of maps of M into R such that

f(x + y)=f(x)+f{y) f(rx) = rf(x)

for x,yeM,reR Iff geM* and seR, then/+gr defined by (f+g) (x) =f(x) + g(x) and

fs defined by (fs) (x) =f(x)s are in M* In this way M* becomes a right R-module and

we have the map M M* of ob R-mod into ob mod-R Now let L.M-+N be a

homomorphism of the R-module M into the R-module N We have the transposed map

It is clear that (1M)* = 1M It follows that

defines a contravariant functor, the duality functor D from R-mod to mod-R In a similar fashion one obtains the duality functor D from mod-R to R-mod

It is clear that a functor m a p s an isomorphism into an isomorphism: If we

have fg = 1 gf= l then F(f)F(g) = l a n d F(g)F(f) = \ Similarly,

sections are mapped into sections and retractions are mapped into retractions

by functors O n the other hand, it is easy to give examples to show that monies (epics) need not be m a p p e d into monies (epics) by functors (see exercise 3 below)

If F is a functor from C to D and G is a functor from D to E, we obtain a functor GF from C to E by defining (GF)A = G(FA) for ^ e o b C and

(GF)(f) = G(F(f)) for fehom c (A,B) I n a similar m a n n e r we can define

composites of functors one or both of which are contravariant Then FG is contravariant if one of F, G is contravariant and the other is covariant, and FG

is covariant if both F and G are contravariant Example 5 above can be described as the composite UM n where M n is the functor defined in example 4

and U is the functor from Ring to Grp defined at the beginning of the section

As we shall see in a moment, the double dual functor D 2

from R-mod to itself

is a particularly interesting covariant functor

A functor F is called faithful (full) if for every pair of objects A, B in C the

m a p / ~ > F(f) of hom c (A,B) into hom D (FA,FB) is injective (surjective) In the

foregoing list of examples, example 1 is faithful and is full if and only if D is a full subcategory of C; examples 2 and 3 are faithful but not full (why?); and

example 9 is full but not faithful

We shall define next the important concept of natural transformation between functors However, before we proceed to the definition, it will be illuminating to examine in detail the example mentioned briefly in the introduction to this chapter W e shall consider the more general situation of

modules Accordingly, we begin with the category R-mod for a ring R and the

double dual functor D 2

in this category This maps a left R-module M into

M * * = (M*)* and a h o m o m o r p h i s m L : M -+ N into L** = (L*)* : M * * -> iV**

If x e M , geN*, then L ^ e M * and (Ifg)(x) = g(Lx) If cpeM**, L**cpeN** and (L**<p) (g) = tp(Ug) W e now consider the m a p

1M(X) -f^f(*)

of M * into R This is contained in M** = hom^(M*, R) and the m a p

^M: x ^ ? /M( x ) is an K-homomorphism of M into M** N o w for any homomorphism L : M -> N, the diagram

(1) L

I

1.3 Functors and Natural Transformations

is commutative, because if x e M, then rj N (Lx) is the m a p g ~> g(Lx) of AT* into

R and for <p = n M {x)eM**, (L**(p)(g) = cp(L?g) Hence {L**n M (x)) {g) =

n M (x)(L*g) = (L*g)(x) = g(Lx)

We now introduce the following definition of "naturality."

D E F I N I T I O N 1.3 Let F and G be functors from C to D PFe de/me a natural transformation n from F to G to be a map that assigns to every object A in C a

morphism n A ehomjy(FA, GA) such that for any objects A,B of C and any

fehom c (A,B) the rectangle in

In the foregoing example we consider the identity functor lR. mo d and the

double dual functor D 2

on the category of left R-modules F o r each object M

of R-mod we can associate the morphism n M of M into M** The commutativity of (1) shows that n :M ~> n M is a natural transformation of the identity functor into the double dual functor

We can state a stronger result if we specialize to finite dimensional vector

spaces over a division ring A These form a subcategory of A-mod If V is a

finite dimensional vector space over A, we can choose a base {e 1 ,e 2 , • ,e n ) for

V over A Let ef be the linear function on V such that ef(ej) = 5 {j Then (e*,e*,"",£*) is a base for F * as right vector space over A—the dual (or complementary) base to the base (e 1 , -,e n ) This shows that F * has the same

dimensionality n as V Hence F * * has the same dimensionality as V Since any non-zero vector x can be taken as the element e 1 of the base (e x , e2, • • •, e n ), it is

clear that for any x ^ 0 in V there exists a # e F * such that g(x) ^ 0 It follows that for any the m a p n v (x) :f ~>f{x) is non-zero Hence n v :x n v (x) is

an injective linear m a p of V into F * * Since dim F * * = dim F , it follows that

77K is an isomorphism Thus, in this case, n is a natural isomorphism of the

identity functor on the category of finite dimensional vector spaces over A onto the double dual functor on this category

We shall encounter many examples of natural transformations as we proceed in our discussion F o r this reason it m a y be adequate to record at this point only two additional examples