An introduction to algebraic topology

Joseph J. Rotman

An Introduction
to Algebraic
Topology

Springer


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Graduate Texts in Mathematics
1

2

3

4
5

6
7

8
9
10
11

12
13

14

TAKEUTI/ZAPJNO. Introduction to

33

Axiomatic Set Theory. 2nd ed
OxroBv. Measure and Category. 2nd ed.
SCiMErER. Topological Vector Spaces.
HILTON/STAMMEACH. A Course in

34

1-lomological Algebra, 2nd ed.
MAC LANE. Categories for the Working
Mathematician 2nd ed
HuoirasfPu'wt. Projective Planes.
A Course in Arithmetic.
Axiomatic Set Theory.
HUMPHREYS. Introduction to Lie Algebras
and Representation Theory.
A Course in Simple Homotopy
Theory.
CONWAY. Functions of One Complex
Variable I. 2nd ed.
BRALs. Advanced Mathematical Analysis.
Rings and Categories
of Modules. 2nd ed.

18

Measure Theory.
HAI.MOS. A HiThert Space Problem Book.
2nd ed.
HUSEM0LLER. Fibre Bundles. 3rd ed.
Linear Algebraic Groups.
BARNES/MACK. An Algebraic Introduction

to Mathematical Logic.
GREUB. Linear Algebra. 4th ed.
24 HoLMES. Geometric Functional Analysis
and Its Applications.
Real and Abstract
25
Analysis.
23

28

29

MANES. Algebraic Theories.
General Topology.
ZARISIU/SAMuEL. Commutative Algebra.
Vol.1.
Commutative Algebra.

Vol.!!.
30

31

JACOBSON. Lectures in Abstract Algebra
1. Basic Concepts.
JACoBsoN. Lectures in Abstract Algebra

II. Linear Algebra.
32

41

Markov Chains. 2nd ed.
APosroL Modular Functions and
Dinchlet Series in Number Theory.

LoRve. Probability Theory 1. 4th ed.
L0EvE. Probability Theory II. 4th ed.
Moiss. Geometric Topology in

ROSENBLATT. Random Processes. 2nd ed.

26

KEMENY/SNEUJKNAPP. Dcnumcrablc

45
46
47

17

27

40

44

Analysis and Operator Theory.
WwrrER. The Structure of Fields.

22

WERMER. Several Complex

Variables and Banach Algebras. 3rd ed.
36 KELLEY/NAMIOKA ci al. Linear
Topological Spaces.
Mathematical Logic.
37
38
Several Complex
Variables.
39 ARVESON. An Invitation to C'-Algcbras.

and Their Singularities.

16

21

ALEXAND

GOLUBrISKY/GUILLEMIN. Stable Mappings

BERBERIAN. Lectures in Functional

20

2nd ed.
35

2nd ed.
SERER. Linear Representations of Finite
Groups.
Rings of Continuous
Functions.
KENDIG. Elementary Algebraic Geometry.

15

19

Hutsct. Differential Topology.
Principles of Random Walk.

42
43

Dimensions 2 and 3.
SACIIS/Wu. General Relativity for

Mathematicians.
49 GRIJENBERG/WEIR. Linear Geometry.
2nd ed.
48

50
51

52
$3

EDWARDS. Fermat's Last Theorem.
KUNGENBERG A Course in Differential
Geometry.
HAR1SH0RNE. Algebraic Geometry.

A Course in Mathematical Logic.
54 GRAVER/WATKINS. Combinatorics with
Emphasis on the Theory of Graphs.
55 BROWN/PEARCY. Introduction to Operator
Theory 1: Elements of Functional
Analysis.
56

MAssey. Algebraic Topology: An
Introduction.

57

CiiownujFox. Introduction to Knot

Theory.
KoBLnz. p—adic Numbers, p-adic
Analysis, and Zeta-Functions. 2nd ed.
59 LANG. Cyclotomic Fields.
60 ARNOLD. Mathematical Methods in
Classical Mechanics. 2nd ed.
58

JACOBSON Lectures in Abstract Algebra

Ill. Theory of Fields and Galois Theory.

continued after index


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Graduate Texts in Mathematics
S. Axier

Springer
New York

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Editorial Board
F.W. Gehring K.A. Ribet


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Joseph J. Rotman

An Introduction
to Algebraic Topology

With 92 Illustrations

4

Springer


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Joseph 3. Rotman
Department of Mathematics
University of Illinois

Urbana, IL 61801
USA

Editorial Board

Axler
Mathematics Department
San Francisco State
S.

University
San Francisco, CA 94132
USA

F.W. Gchring

KA. Ribet

Mathematics Department
East Hall
University of Michigan
Ann Arbor, MI 48109

Department of Mathematics
University of California
at Berkeley

USA

Berkeley, CA 94720-3840

USA

Mathematics Subject Classification (1991): 55-01
Library of Congress Cataloging-in-Publication Data
Rotman, Joseph I.,
An introduction to algebraic topology.
(Graduate texts in mathematics; 119)
Bibliography: p.
Includes index.
1. Algebraic topology. L Title. II. Series.
QA612.R69

1988

514'.2

87-37646

1988 by Springer-Verlag New York Inc.
AN rights reserved. This work may not be translated or copied in whole or in part without the
written permission of the publisher (Springer-Verlag. 175 Fifth Avenue, New York, NY 10010.
USA), except for brief excerpts in connection with reviews or scholarly analysis. Use in connection
with any form of information storage and retrieval, electronic adaptation, computer software, or
by similar or dissimilar methodology now known or hereafter developed is forbidden.

The use of general descriptive names, trade names, trademarks, etc. in this publication, even if
the former are not especially identified, is not to be taken as a sign that such names, as understood
by the Trade Marks and Merchandise Marks Act, may
be used freely by anyone.
Typeset by Asco Trade Typesetting Ltd., Hong Kong.

Printed and bound by R. R. Donneiley & Sons, Harnsonburg. Virginia.
Printed in the United States of America.

9 8 7 6 5 4 (Fourth corrected printing, 1998)
ISBN 0-387.96678-1 Springer-Verlag New York Berlin Heidelberg
ISBN 3-540.96678-1 Springer-Verlag Berlin Heidelberg New York SPIN 10680226


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To my wife Marganit
and my children Ella Rose and Daniel Adam
without whom this book would have
been completed two years earlier


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Preface

There is a canard that every textbook of algebraic topology either ends with
the definition of the Klein bottle or is a personal communication to J. H. C.
Whitehead. Of course, this is false, as a glance at the books of Hilton and
Wylie, Maunder, Munkres, and Schubert reveals. Still, the canard does reflect
some truth. Too often one finds too much generality and too little attention
to details.
There are two types of obstacle for the student learning algebraic topology.

The first is the formidable array of new techniques (e.g., most students know
very little homological algebra); the second obstacle is that the basic definitions have been so abstracted that their geometric or analytic origins have
been obscured. I have tried to overcome these barriers. In the first instance,
new definitions are introduced only when needed (e.g., homology with coeflicients and cohomology are deferred until after the Eilenberg—Steenrod axioms
have been verified for the three homology theories we treat—singular, simplicial, and cellular). Moreover, many exercises are given to help the reader
assimilate material. In the second instance, important definitions are often
accompanied by an informal discussion describing their origins (e.g., winding
numbers are discussed before computing ir1(S'), Green's theorem occurs
before defining homology, and differential forms appear before introducing
cohomology).
We assume that the reader has had a first course in point-set topology, but
we do discuss quotient spaces, path connectedness, and function spaces. We
assume that the reader is familiar with groups and rings, but we do discuss
free abelian groups, free groups, exact sequences, tensor products (always over
Z), categories, and functors.
I am an algebraist with an interest in topology. The basic outline of this
book corresponds to the syllabus of a first-year's course in algebraic topology


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Preface

designed by geometers and topologists at the University of illinois, Urbana;
other expert advice came (indirectly) from my teachers, E. H. Spanier and S.
Mac Lane, and from J. F. Adams's Algebraic Topology: A Student's Guide. This

latter book is strongly recommended to the reader who, having finished this
book, wants direction for further study.
I am indebted to the many authors of books on algebraic topology, with
a special bow to Spanier's now classic text. My colleagues in Urbana, especially Ph. Tondeur, H. Osborn, and R. L. Bishop, listened and explained.

M.-E. Hamstrom took a particular interest in this book; she read almost the
entire manuscript and made many wise comments and suggestions that have

improved the text; my warmest thanks to her. Finally, I thank Mrs. Dee
Wrather for a superb job of typing and Springer-Verlag for its patience.

Joseph J. Rotman

Addendum to Second Corrected Printing
Though I did read the original galleys carefully, there were many errors that

eluded me. I thank all who apprised me of mistakes in the first printing,
especially David Carlton, Monica Nicolau., Howard Osborn, Rick Rarick,
and Lewis Stiller.
November 1992

Joseph J. Rotman

Addendum to Fourth Corrected Printing

Even though many errors in the first printing were corrected in the second
printing, some were unnoticed by me. I thank Bernhard J. Eisner and Martin
Meier for apprising me of errors that persisted into the the second and third
printings. I have corrected these errors, and the book is surely more readable
because of their kind efforts.
April, 1998

Joseph Rotman

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To the Reader

Doing exercises is an essential part of learning mathematics, and the serious
reader of this book should attempt to solve all the exercises as they arise. An
asterisk indicates only that an exercise is cited elsewhere in the text, sometimes
in a proof (those exercises used in proofs, however, are always routine).
I have never found references of the form I .2.1.1 convenient (after all, one
decimal point suffices for the usual description of real numbers). Thus, Theorem
7.28 here means the 28th theorem in Chapter 7.


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Contents

Preface

To the Reader

vii
ix

CHAPTER 0

Introduction

Notation
Brouwer Fixed Point Theorem
Categories and Functors

2
6

CHAPTER 1

Some Basic Topological Notions
Homotopy
Convexity, Contractibility, and Cones
Paths and Path Connectedness

14
14
18

24

CHAPTER 2

Simplexes
Affine Spaces
AlTine Maps

31

31

38

CHAPTER 3

The Fundamental Group
The Fundamental Oroupoid
The Functor

39
39

44


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Contents
CHAPTER 4

Singular Homology
Holes and Green's Theorem
Free Abelian Groups
The Singular Complex and Homology Functors
Dimension Axiom and Compact Supports
The Homotopy Axiom
The Hurewicz Theorem

57

57
59

62

68
72
80

CHAPTER 5

Long Exact Sequences
The Category Comp
Exact Homology Sequences
Reduced Homology

86
86
93
102

CHAPTER 6

Excision and Applications
Excision and Mayer—Vietoris
Homology of Spheres and Some Applications
Barycentric Subdivision and the Proof of Excision
More Applications to Euclidean Space

106
106
109

ill

119

7

Simplicial Complexes
Definitions
Simplicial Approximation
Abstract Simplicial Complexes
Simplicial Homology
Comparison with Singular Homology
Calculations

Fundamental Groups of Polyhedra
The Seifert—van Kampen Theorem

131

131

136
140
142
147
155
164
173

CHAPTER 8

CW Complexes
Hausdorif Quotient Spaces
Attaching Cells
Homology and Attaching Cells
CW Complexes
Cellular Homology

180
180
184
189
196

212

CHAPTER 9

Natural Transformations
Definitions and Examples
Eilenberg—Steenrod Axioms

228
228
230


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Contents

xiii

Chain Equivalences
Acyclic Models

233
237

Lefschetz Fixed Point Theorem

247

Tensor Products

253
256
265

Universal Coefficients
Eilcnberg - Zilber Theorem and the Kunneth Formula
CHAPTER 10

Covering Spaces
Basic Properties
Covering Transformations

Existence
Orbit Spaces

272

273
284
295
306

CHAPTER 11

Homotopy Groups

312

Function Spaces
Group Objects and Cogroup Objects
Loop Space and Suspension
1-lomotopy Groups
Exact Sequences

312

Fibrations
A Glimpse Ahead

355

314
323

334
344
368

CHAPTER 12

Cohomology
Differential Forms
Cohomology Groups
Universal Coefficients Theorems for Cohomology
Cohomology Rings
Computations and Applications

Bibliography

Notation
Index

373
373
377
383
390
402

419
423
425


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CHAPTER 0

Introduction

One expects algebraic topology to be a mixture of algebra and topology, and
that is exactly what it is. The fundamental idea is to convert problems about
topological spaces and continuous functions into problems about algebraic
objects (e.g., groups, rings, vector spaces) and their homomorphisms; the
method may succeed when the algebraic problem is easier than the original

one. Before giving the appropriate setting, we illustrate how the method
works.

Notation
Let us first introduce notation for some standard spaces that is used throughout the book.

Z = integers (positive, negative, and zero).
Q = rational numbers.
C = complex numbers.
I = [0, 1), the (closed) unit interval.
R = real numbers.
W is called real a-space or euclidean space (of course, W is the cartesian
product of n copies of R). Also, R2 is homeomorphic to C; in symbols, R2 C.
(when
W, then its norm is defined by
If x = (x1, ...,
=
n

I, then lxii = lxi, the absolute value of x). We regard R' as the subspace
of RA+I consisting of all (n + 1)-tuples having last coordinate zero.

= l}.


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0. Introduction

2

is

(of radius I

called the

5'

and

center the origin). Observe that

R2); note also that the 0-sphere S° consists of the
two points {l, —1) and hence is a discrete two-point space. We may regard
5' as the equator oIS'41:
the circle S1

5' =

= {(x1, ..., x.÷2)e

..., 0,

511+1:

= 0).

S"; the south pole is (0, 0, .. ., 0, — 1). The
is
the other endpoint of the diameter having
...,
one endpoint x; thus the antipode of x is —x = (—x1, ..., —x,,.1), for the

The north pole is (0, 0,
antipode

of x = (xj,

1) e

e 5'

distancefrom —xtoxis2.

D'= {xER':

1).

D' is called the n-disk (or a-bull). Observe that S't

the boundary of 17' in R'.

= {(x1,

x2, ...,

R'4': each

xj

C W;

0 and

indeed S'' is
= I).

Es' is called the standard n-simplex. Observe that A° is a point, Es' is a closed
interval, Es2 is a triangle (with interior), & is a (solid) tetrahedron, and so on.
It is obvious that Es1' 17', although the reader may not want to construct' a
homeomorphism until Exercise 211.
There is a standard homeomorphism from 51' — {north pole) to R', called
szereograpbk projection. Denote the north pole by N, and define a: 5' — (N)

-. R' to be the intersection of R' and the line joining x and N. Points on
the latter line have the form tx + (1 — t)N; hence they have coordinates
tx,, tx,.,.1 + (I — t)). The last coordinate is zero fort = (1

—

hence

a(x) = (tx1,

..

.,

It is now routine to check that a is indeed a borneo(1 —
morphism. Note that a(x) = x if and only if x lies on the equator

where t

Brouwer Fixed Point Theorem
Having established notation, we now sketch a proof of the Brouwer fixed point
theorem: if f:17' -, 17' is continuous, then there exists x e 17' with 1(x) = x.
1, this theorem has a simple proof. The disk D' is the closed interval
When n

[—1, 1]; let us look at the graph off inside the squareD' x D'.

'liii an exerczee that a compact convex subset of k' containing an interior point is homcomorphic to 1? (convexity is defined in Chapter 1k it follows that t?, D', and P are


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Brouwer Fixed Point Theorem

3

(I. I)

(—I.

b

(I. —I)

(—I, —I)

Theorem 0.1. Every continuous f: D' —.

has a fixed point.

PROOF. Lctf(—l) = aandf(1) = b. Ifeitherf(— 1) = —l orf(I) = l,weare
done.Thcrefore,wemayassumethatf(—l) = a> —land thatf(l) b < I,

as drawn. ff6 is the graph of f and A is the graph of the identity function (of
course, A is the diagonal), then we must prove that G fl A # 0. The idea is to
use a connectedness argument to show that every path in D' x D' from a to
b must cross A. Since f is continuous, G =
f(x)): XE D') is connected [G
is the image of the continuous map D' —.
x D' given by
Define A = ((x, 1(x)): f(x)> x) and B = {(x,f(x)): 1(x) < x}. Note that a A
and b E B, so that A 0 and B 0. If G fl A = 0. then G is the disjoint
union
G = A U B.

Finally, it is easy to see that both A and B are open in 6, and this contradicts
the connectedness of G.

0

Unfortunately, no one knows how to adapt this elementary topological
argument when n> 1; some new idea must be introduced. There is a proof
using the simplicial approximation theorem (see [Hirsch]). There are proofs
by analysis (see [Dunford and Schwartz, pp. 467—470] or [Milnor (1978)]),
the basic idea is to approximate a continuous function f: I)' —. TY' by smooth
functions g: 17'
in such a way that I has a fixed point if all the g do; one
can then apply analytic techniques to smooth functions.
Here is a proof of the Brouwer fixed point theorem by algebraic topology.
We shall eventually prove that, for each n 0, there is a homology frnctor H.
with the following properties: for each topological space X there is an abelian
group
and for each continuous function f: X —, Y there is a homomorphism
such that:
HN(X) —

H.(g of) =
whenever the composite g o

o

HR(f)

(1)

I is defined;

is the identity function on HN(X),

(2)


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0. Introduction

4

where

is the identity function on X;

#0

for alln 1;

(3)

for all n 1.

(4)

Using these III's, we now prove the Brouwer theorem.
Definition. A subspace X of a topological space Y is a retract of Y if there is
a continuous map2 r: Y -. X with r(x) = x for all x E X; such a map r is called
a retraction.
Remarks. (1) Recall that a topological space X contained in a topological

space Y is a subopace of Y if a subset V of X is open in X if and only if

V = X fl U for some open subset U of V. Observe that this guarantees that
= Xfl u is open in X
the inclusion I:
Y is continuous, because
whenever U is open in V. This parallels group theory: a group H contained
in a group G is a subgroup of G if and only if the inclusion I: H c. G is a
homomorphism (this says that the group operations in H and in G coincide).

(2) One may rephrase the defmition of retract in terms of functions. If
i: X .. V is the inclusion, then a continuous map r: V —, X is a retraction if
and only if

roi=
(3) For abelian groups, one can prove that a subgroup H of G is a retract
of 6 if and only if H is a direct summand of G; that is, there is a subgroup K

Lemma 0.2. If n 0, then

is not a retract

of D".

there were a retraction r:
—. Sa; then there would be a
'commutative diagram" of topological spaces and continuous maps
PROOF. Suppose

Da+1

(here commutative means that r o = 1, the identity function on S). Applying
Ha gives a diagram of abelian groups and homomorphisms:

H,(1)

Ha(Sa).

2We use the words map and fluscilon interchangeably.


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Brouwer Fixed Point Theorem

5

the new diagram commutes:
By property (I) of the homology functor
o
by
(3),
it follows that Ha(I) = 0. But
Since
=
0,
= HR(l).
is the identity on H.(SN), by (2). This contradicts (4) because H1(S")

Note how homology functors
into an algebraic one.

0.

0

have converted a topological problem

We mention that Lemma 0.2 has an elementary proof when n = 0. It
is plain that a retraction r: Y -. X is suijective. In particular, a retraction
r:
—' S° would be a continuous map from [—.1, 1) onto the two-point set
{ ±1), and this contradicts the fact that a continuous image of a connected
set is connected.
Theorem 0.3 (Brouwer).

1ff: IY -.

is continuous, then f has a fixed point.

Suppose thatf(x) x for all x 1Y'; the distinct points x and 1(x) thus
as the function
determine a line. Define g: D' — S"' (the boundary of

gtx)

assigning to x that point where the ray from f(x) to x intersects S"'. Obimplies g(x) = x. The proof that g is continuous is left as an
viously, x e
exercise in analytic geometry. We have contradicted the lemma.
0

There is an extension of this theorem to infinite-dimensional spaces due to
Schauder (which explains why there is a proof of the Brouwer fixed point
theorem in [Dunford and Schwartz]): if D is a compact convex subset of a
Banach space, then every continuous f: D —. I) has a fixed point. The proof
by a sequence of continuous functions each of
involves approximating I —
which is defined on a finite-dimensional subspace of D where Brouwer's
theorem applies.
EXERasES

0.1. Let H be a subgroup of an abelian group G. If there is a homomorphism r: G -. H
with r(x) — x for all x e H, then G — H kerr. (Hint: If ye G, then y = r(y) +
(y — r(y)).)


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0. Introduction

6

theorem for n = I using the proof of
Theorem 0.3 and the remark preceding it.

0.2. Give a proof of Brouwer's fixed point

= Z if I = 0, n, and that He(S) = 0 otherwise.
0.3. Assume, for n 1, that
Using the technique of the proof of Lemma 0.2, prove that the equator of the
n-sphere is not a retract.

0.4. If X is a topological space homeomorphic to 17, then every continuous f: X
has a fixed point.
0.5.

X

Letf,g: I —, I x I be continuous; let 1(0) = (a, 0) and f(1) = (b, 1), and let g(0) =
c) and g(1) = (1, d) for some a, b, c, d El. Show that f(s) = g(t) for some 5,
I E I; that is, the paths intersect. (Hint: Use Theorem 0.3 for a suitable map
I x I -. I x I.) (There is a proof in [Maehara]; this paper also shows how to

(0,

derive the Jordan curve theorem from the Brouwer theorem.)

0.6. (Perron). Let A = [au] be a real n x n matrix with au >

0

for every i, j. Prove

that A has a positive eigenvalue A; moreover, there is a corresponding eigenvector
=
> 0.(Hinz:Firstdeflne
x =(x1,x2
—,
and then define g:
by
R -. R by o(x1, x2, ..., x,) =
c is regarded as a column vector. Apply the

g(x) = Ax/a(Ax), where x e
Brouwer fixed point theorem after showing that g is a well defined continuous

function.)

Categories and Functors
Having illustrated the technique, let us now give the appropriate setting for
algebraic topology.
Definition. A category consists of three ingredients: a class of objects, obj
sets of morphisms Hom(A, B), one for every ordered pair A, B e obj composition Hom(A, B) x Hom(B, C) —, Hom(A, C), denoted by (f, g) '—' g o f, for
every A, B, C e obj 's', satisfying the following axioms:

(1) the family of Hom(A, B)'s is pairwise disjoint;
(ii) composition is associative when defined;
(iii) for each A E obj 'P1, there exists an identity 'A
'A

g

Hom(A, A) satisfying
= gforevery

of = fforeveryfe Hom(B, A),allB€obj 'ó',andg
Hom(A, C), all C e obj

Remarks. (1) The associativity axiom stated more precisely is: if f, g, h are
morphisms with either h o (g o f) or (h o g) o defined, then the other is
also defined and both composites are equal.
(2) We distinguish class from set: a set is a class that is small enough
to have a cardinal number. Thus, we may speak of the class of all topological

spaces, but we cannot say the set of all topological spaces. (The set theory we
accept has primitive undefined terms: class, element, and the membership
relation e. All the usual constructs (e.g., functions, subclasses, Boolean opera-

f


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Categories and Functors

7

tions, relations) are permissible except that the statement x E A is always false
whenever x is a class that is not a set.)

(3) The only restriction on Hom(A, B) is that it be a set. In particular,

Hom(A, B) = 0 is allowed, although axiom (iii) shows that Hom(A, A)
because it contains 14.
(4) Instead of writing f e Hom(A, B), we usually write f: A —. B.

0

= Sets. Here obj
all sets, Hom(A, B) = {all functions
B), and composition is the usual composition of functions.

ExAMPLE 0.1.
A —,

This example needs some discussion. Our requirement, in the definition of
category, that Horn sets are pairwise disjoint is a reflection of our insistence
that a function f: A — B is given by its domain A, its target B, and its graph:
{all (a,f(a)): a A) c A x B. In particular, if A is a proper subset of B, we
distinguish the inclusion 1: A B from the identity 14 even though both

functions have the same domain and the same graph; i e Horn(A, B) and
14 e Hom(A, A), and so I $ 14. This distinction is essential. For example, in
0 when A = and B =
the proof of Lemma 0.2, H,,(i) = 0 and

Here are two obvious consequences of this distinction: (1) If B c B' and
f: A —. B and g: A —' B' are functions with the same graph (and visibly the
same domain), then g = iof, where I: B . B' is the inclusion. (2) One may
form the composite h o g only when target g = domain h. Others may allow

one to compose g: A .- B with h: C D when B c C; we insist that the only
composite defined here is h o i o g, where I: B -. C is the given inclusion.
Now that we have explained the fine points of the definition, we continue
our list of examples of categories.

= Top. Here obj = all topological spaces, Hom(A, B) =
(all continuous functions A —.B), and composition is usual composition.

EXAMPLE 0.2.

Definition.
and d be categories with obj c obj d. If A, B e obj "€,
let us denote the two possible Horn sets by Hom114A, B) and Homd(A, B).
Then is a subcategory of d if

B) c Hom,,(A, B) for all A, B e
obj and if composition in is the same as composition in d; that is, the
function Hom,(A, B) x Hom5.(B, C) —
C) is the restriction of the

corresponding composition with subscripts d.
EXAMPLE 0.2'. The category Top has many interesting subcategories. First, we
may restrict objects to be subspaces of euclidean spaces, or Hausdorif spaces,

or compact spaces, and so on. Second, we may restrict the maps to be differentiable or analytic (assuming that these make sense for the objects being
considered).
Ex.aj,tpii 0.3. = Groups Here obj = all groups, Hom(A, B) = {all homomorphisms A -, B), and composition is usual composition (Horn sets are so
called because of this example).


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0. Introduction

8

0.4.'e = Ab. Here obj'e = all abelian groups, and Hom(A, B) = (all
homomorphisms A -. B); Ab is a subcategory of Groups
0.5. 'e= Rings. Here obj 'e = all rings (always with a two-sided
identity element), Hom(A, B) = (all ring homomorphisms A —. B that preserve identity elements), and usual composition.
EXAMPLE 0.6.'e = Top2. Here obj 'econsists of all ordered pairs (X, A), where

X is a topological space and A is a subspace of X. A morphism f: (X, A)
(Y, B) is an ordered pair (f,f'), where f: X —+ Y is continuous and ft = If'
(where I and j are inclusions),

and composition is coordinatewise (usually one is less pedantic, and one says
that a morphism is a continuous map f: X -. Y with f(A) B). Top' is called
the category of pairs (of topological spaces).
Top,. Here obj 'econsists of all ordered pairs (X, x0), where
X is a topological space and x0 is a point of X. Top, is a subcategory of Top2
(subspaces here are always one-point subspaces), and it is called the category
of pointed spaces; x0 is called the basepoint of(X, x0), and morphisms are called
pointed maps (or basepoint preserving maps). The category Sets, of pointed
sets is defined similarly.

Of course, there arc many other examples of categories, and others arise
as we proceed.
EXERCISES
0.7.

Letf€Hom(A,B) bea morphism in
Iffhasaleft inverse g
(geHom(B.A) and gof= lÀ) and a right inverse It (h€Hom(B,A) and

foh= l,)theng=h.

0.8. (i) Let'e be a category and let A e obj'e. Prove that Hom(A, A) has a unique

identity lÀ.

(ii) If'e' is a subcategory of 'e, and if A E obj'e', then the identity of A in
A) is the identity lÀ in
A

set X is called quail-ordered (or pre-ordered) if X has a transitive and

reflexive relation . (Of course, such a set is partially ordered if, in addition,
is antisymmetric.) Prove that the following construction gives a category'e.
Define obj 'em X; lix, ye X and x y, define Hom(x. y) — 0; lix y, define

Hom(x, y) to be a set with exactly one element, denoted by i; if x
o
=

define composition by i

y z,


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Categories and Functors

9

*0.10. Let G be a monold, that is, a semigroup with I. Show that the following
Let obj have exactly one element, denoted
construction gives a category
by s; define Hom(., .) = G, and define composition G x G -. G as the given
multiplication in G. (This example shows that morphisms may not be functions.)

0.11. Show that one may regard Top as a subcategory of Top2 if one identifies
a space X with the pair (X, 0).

Definition. A diagram in a category is a directed graph whose vertices are
labeled by objects of and whose directed edges are labeled by morphisms

in
A commutative diagram in %' is a diagram in which, for each pair of
vertices, every two paths (composites) between them are equal as morphisms.

This terminology comes from the particular diagram
A

B

f f'

which commutes if g o = o g'. Of course, we have already encountered
commutative diagrams in the proof of Lemma 0.2.

show that the following construction gives a category .4'.
First, an object of 4' is a morphism of Next, 1ff. g E obj 9, say, f: A —. B
and g: C D, then a morphism in .4' is an ordered pair (Is, k) of morphisms in
such that the diagram

*0.12. Given a category

48

A

C

g

commutes. Define composition coordinatewise:

(h',k')o(h,k) =(h'oh,k' o k).
0.13.

Show that Top2 is a subcategory of a suitable morphism category (as constructed in Exercise 0.12). (Hint: Take %' = Top, and let .4' be the corresponding
morphism category; regard a pair (X, A) as an inclusion i: A -. X.)

The next simple construction is useful.
Definition. A congruence on a category is an equivalence relation
class U(Aa) Hom(A, B) of all morphisms in such that:

on the


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0. Introduction

10

(I) f e Hom(A, B) and f f' implies f' E Hom(A, B);
(ii) 1 1'. g — g', and the composite g o I exists imply that

g'of'.
Theorem 0.4. Let be a category with congruence —, and let [f) denote the
as follows:
equivalence class of a morphism f. Define
obj

(f = obj

B) =

[g] °
Then

is

{[f]: f E

B)};

If] = [g of].

a category.

PRooF. Property (i) in the definition of congruence shows that — partitions
each set Hom?(A, B), and this implies that Hom?.(A, B) is a set; moreover,
the family of these sets is pairwise disjoint. Property (ii) in the definition of
congruence shows that composition in is well defined, and it is routine to
see that composition in is associative and that [14) is the identity morphism

0

onA.
The category %" just constructed is called a quotient category of
usually denotes
B) by [A, B].

one

The most important quotient category for us is the homozopy category
described in Chapter 1. Here is a lesser example. Let groups and let f, f' Hom(G, H). Define f f' if there exists a e H with
f(x) = af'(x)a' for all x e G (one may say that f and f' are conjugate). It is
routine to check that
is an equivalence relation on each Hom(G, H). To
see that — is a congruence, assume that I
that g g', and that g of
exists. Thus f and f e Hom(G, H), g and g' e Hom(H, K), there is a H with
for all
f(x) =
for all x G, and there is b e K with g(y) =
ye H. It is easy to see that g(f(x)) = [g(a)b)g'(f'(x))[g(a)b]1 for all x E G,
that is, g o f g' o f'. Thus the quotient category is defined. If G and H are
groups, then (G, H) is the set of all "conjugacy classes" [f], where f: G —' H
is a homomorphism.
ExmtclsE
0.14. Let G be a group and let 'Pd'

the one-object category it defines (Exercise 0.10
= {.}, Hom(., .) = G, and
composition is the group operation. 1(8 is a normal subgroup oIG, define x — y
to mean xy' H. Show that is a congruence on and that [s, s) = G/H
in the corresponding quotient category.
be

applies because every group is a monoid): obj

Just as topological spaces are important because they carry continuous
functions, so categories are important because they carry functors.