CATEGORIES: HISTORY, BASICS AND
DEVELOPMENTS
By
Theofanis Alexoudas
SUBMITTED IN PARTIAL FULFILLMENT OF THE
REQUIREMENTS FOR THE DEGREE OF
MASTER OF SCIENCE
AT
QUEEN MARY
UNIVERSITY OF LONDON
10 SEPTEMBER
c� Copyright by Theofanis Alexoudas, 2007
QUEEN MARY
DEPARTMENT OF
MATHEMATICAL SCIENCES
The undersigned hereby certify that they have read and
recommend to the Faculty of Graduate Studies for acceptance a
thesis entitled “Categories: History, Basics and Developments”
by Theofanis Alexoudas in partial fulfillment of the requirements for
the degree of Master of Science.
Dated: 10 September
Supervisor:P. J. Cameron
Readers:
ii
Table of Contents
Table of Contents v
Abstract vii
Acknowledgements viii
Preface ix
1 Categories, Functors and Natural Transformations 11.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Basic Notions of Category Theory . . . . . . . . . . . . . . . . . . . . 11.3 Examples of Categories . . . . . . . . . . . . . . . . . . . . . . . . . . 31.4 Functors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51.5 Initial and Terminal Objects . . . . . . . . . . . . . . . . . . . . . . . 91.6 Products and Coproducts of Categories . . . . . . . . . . . . . . . . . 101.7 Examples of Products and Coproducts . . . . . . . . . . . . . . . . . 131.8 Hom-Sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171.9 Natural Transformations . . . . . . . . . . . . . . . . . . . . . . . . . 21
2 Category Theory in Algebraic Topology 262.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 262.2 Homology and Cohomology Groups as Functors . . . . . . . . . . . . 272.3 Duality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 322.4 Equivalences of Functors . . . . . . . . . . . . . . . . . . . . . . . . . 342.5 Cech homology groups as Functors . . . . . . . . . . . . . . . . . . . 37
3 Further Developments 413.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 413.2 Adjoint Functors in One Variable . . . . . . . . . . . . . . . . . . . . 42
v
3.3 Adjoint Functors in Several Variables . . . . . . . . . . . . . . . . . . 453.4 Applications of Adjoint Functors to C.S.S. Complexes . . . . . . . . . 50
Bibliography 58
vi
Abstract
This project was an opportunity for the author to be introduced to the basic notions of
Category Theory and their applications to Algebraic Topology during the early stages
of the development of the theory. In terms of the basics, the notions of Category,
Functor and Natural Transformation are introduced. Within this context, the process
of formulation of various Homology and Cohomology groups in terms of Functors are
examined. The notion of Adjoint Functors are also developed to some extent. Finally,
the applications of Adjoint Functors to Complexes are examined.
vii
Acknowledgements
It is inevitable, in such an emotional moment, that there will be a great many to
whom I owe my thanks and my gratitude. What I mean is that there are a lot of
people who helped me in various ways in order to be in this position today.
First of all, I would like to thank my parents for supporting me from the first
moment I took the decision to continue my studies at Queen Mary College in the
University of London. Without their support and encouragement, I would never be
able to be in this position today.
I would also like to specially thank my supervisor, Prof. Peter J. Cameron for
giving me the opportunity to do my MSc Thesis under his supervision. I would like to
thank him for being such a great teacher, being patient, generous, having extensively
read my drafts, pointing out a lot of mistakes and guiding me every time I had
problems. Also I would like to thank him for the time he spent, trying to satisfy my
infinite curiosity.
Furthermore, I owe my gratitude to my course director, Prof. Leonard H. Soicher
for supporting me every time I had problems this year. Without his support and
encouragement I would never be able to be in this position today.
Moreover, I would like to thank Dimitrios Manolopoulos and Irene Galstian for
showing me how to use Latex and Johanna Ramo for her invaluable support this year.
Last but not least, I would like to express my gratitude from the bottom of my
heart to Tina Pavlou and Dr. Wilfried K. Kohler who both showed me the road in
order to come back to life.
Queen Mary, University of London Theofanis Alexoudas
September 10, 2007
viii
Preface
The basic concepts of what later became called Category Theory were introduced
in 1945 by Samuel Eilenberg and Saunders Mac Lane. During the 1950s and 1960s,
Category Theory became an important conceptual framework in many areas of math-
ematical research, especially in algebraic topology and algebraic geometry. Later,
connections to questions in mathematical logic emerged. The theory was subject to
some discussions by set theorists and philosophers of science, since on the one hand
some di�culties in its set-theoretic presentation arose, while on the other hand it
became interpreted itself as a suitable foundation of mathematics.
In particular, it was only after some time of development that a corpus of concepts,
methods and results deserving the name “theory” was arrived at. For example, the
introduction of the concept of adjoint functor which is due to Kan was important,
since it brought about nontrivial questions to be answered inside the theory. The
characterization of certain constructions in diagram language had a similar e↵ect
since thus a carrying out of these constructions in general categories became possible,
and this led to the question of the “existence” of these constructions in given cate-
gories. Hence, Category Theory arrived at its own problems, for example problems
of classification, problems to find existence criteria for objects with certain properties
etc.
This project concentrates on the introduction of the basic notions of Category
Theory and their applications to Algebraic Topology during the early stages of the
development of the theory.
In chapter 1, the reader is introduced to the concepts of Category, Functor and
Natural Transformation. This is achieved by giving various examples which arose in
the first place from algebra. Also the concept of products is discussed in terms of
categorical language.
Chapter 2 presents a coherent summary of the last section of Eilenberg-Mac Lane’s
ix
x
article [7] where the first applications of Category Theory to Algebraic Topology
emerged. In Section 2.2, the process of formulation of homology and cohomology
groups in terms of functors is discussed. In Section 2.3, various duality properties
of such functors are presented. In the last two section, it is shown that the basic
constructions of group extensions may be regarded as functors and that the Cech
homology theory may be treated in terms of functors.
In chapter 3, the reader is introduced to the concept of adjoint functor and its
applications to functors involving complexes. In Section 3.2 the notion of adjoint
functor in one variable is developed. Section 3.3 deals with the concept of adjoint
functors in several variables. Finally, the applications of adjoint functors to complexes
are discussed.
Chapter 1
Categories, Functors and NaturalTransformations
1.1 Introduction
Category Theory begins with the observation that the collection of all mathematical
structures of a given type, together with all the maps between them, is itself an in-
stance of a nontrivial structure which can be studied in its own right [7]. In keeping
with this idea, the real objects of study are not so much categories themselves as
the maps between them—functors, natural transformations and adjunctions. Cate-
gory Theory has had great success in the unification of ideas from di↵erent ares of
mathematics. It has now become an indispensable tool for anyone doing research in
topology, algebra, logic and theoretical computer science. In this chapter we introduce
the basic notions of Category Theory based on [1] and [14].
1.2 Basic Notions of Category Theory
Definition 1.2.1. A category consists of the following data:
1
2
1. Objects: A, B, C, ...
2. Arrows: f,g,h,...
3. For each arrow f there are given objects:
dom(f ), cod(f ), called the domain and codomain of f. We write:
f : A ! B
to indicate that A=dom(f ) and B=cod(f ).
4. Given arrows
f : A ! B
and
g : B ! C
that is with cod(f )=dom(g), there is a given arrow:
g � f : A ! C
called the composite of f and g.
5. For each object A there is given an arrow:
1A : A ! A
called the identity arrow of A.
These data are required to satisfy the following laws:
1. Associativity:
h � (g � f) = (h � g) � f
whenever both sides are defined, i.e cod(f) = dom(g) and cod(g) = dom(h).
3
2. Unit:
f � 1A = f = 1B � f
for all f : A ! B.
1.3 Examples of Categories
Let consider a few examples of categories:
1. 0 is the empty category (no objects, no arrows).
2. The category 1 looks like ⇤, it has one object and its identity arrow.
3. The category 2 looks like this:
⇤ �! ?
It has two objects, their required identity arrows, and exactly one arrow between the
objects.
4. The category 3 looks like this:
⇤
•
g
��???
????
????
???⇤ ?
f // ?
•
h
������
����
����
��
It has three objects, their required identity arrows, exactly one arrow from the first
to the second object f : ⇤ �! ? , exactly one arrow from the second to the third
object h : ? �! •, and exactly one arrow from the first to the third object g : ⇤ �! •
(which is therefore the composite to the other two, g = h � f).
5. The category Sets of sets and functions. There is also the category Setsfin of
all finite sets and functions between them. There are many categories like this, given
4
by restricting the sets that are to be the objects and the functions that are to be
the arrows. For example, take finite sets as objects and injective functions as arrows.
Since the composition of injective functions is injective and identity functions are by
definition injective, this also gives a category.
6. Another kind of example one often sees in mathematics is categories of struc-
tured sets , that is, sets with some further “structure” and functions which “preserve”
it. Familiar examples of this kind are:
- Groups and group homomorphisms,
- Vector Spaces and linear mappings,
- Graphs and graph homomorphisms,
- The set R of real numbers and continuous functions R �! R,
- Open subsets U ✓ R and continuous functions f : U �! V ✓ R defined on them,
- Topological Spaces and continuous mappings,
- Di↵erential Manifolds and smooth mappings.
7. Monoids. A monoid is a category with just one object. Each monoid is thus
determined by the set of all its arrows, by the identity arrow, and by the rule for
the composition of arrows. Since any two arrows in a monoid have a composite, a
monoid can be described as a set M with a binary operation M⇥M �! M which is
associative and has an identity. Thus a monoid is exactly a semigroup with identity
element. For any category C and any object ↵ 2 C, the set hom(↵,↵) of all arrows
↵ �! ↵ is a monoid.
5
1.4 Functors
Like many mathematical “abstract” objects (such as groups, rings, topological spaces
etc.) are equipped with mappings between them (homomorphisms in the case of
groups and rings, and continuous functions in the case of topological spaces), the
notion of a mapping between categories is coming in the natural way.
Definition 1.4.1. A functor
F : C �! D
between categories C and D is a mapping of objects to objects and arrows to arrows,
in such a way that:
1.F (f : A ! B) = F (f) : F (A) ! F (B),
2.F (g � f) = F (g) � F (f),
3.F (1A) = 1F (A).
Now, one can check that functors compose in the expected way, and that every
category C has an identity functor 1C : C ! C. So we have another example of a
category, namely C, the category of all categories and functors.
Functors were first explicitly recognized in algebraic topology, where they arise
naturally when geometric properties are described by means of algebraic invariants.
For example, singular homology in a given dimension n (n a natural number) assigns
to each topological space X an abelian group Hn(X), the n-th homology group of X,
and also to each continuous map f : X ! Y of spaces a corresponding homomorphism
Hn(f) : Hn(X) ! Hn(Y) of groups, and in this way Hn becomes a functor between the
category of topological spaces and the category of abelian groups.
6
Also, functors arise naturally in algebra. For any commutative ring K with iden-
tity, the set of all non-singular n⇥ n matrices with entries in K is the usual general
linear group GLn(K). Moreover, each homomorphism f : K ! K0 of rings produces
in the evident way a homomorphism GLnf : GLn(K) ! GLn(K0) of groups. These
data define for each natural number n a functor between the category of commutative
rings and the category of finite groups.
For any group G the set of all products of commutators x yx�1y�1(x, y 2 G) is
clearly a normal subgroup, written [G, G], of G called the commutator subgroup
of G (For more on the theory of Finite Groups see [19]). Since any homomorphism
f : G ! H of groups carries commutators to commutators, the assignment G 7! [G, G]
defines a functor F : Grp ! Grp (the category of groups), while G 7! G/[G, G] defines
a functor G : Grp ! Ab, the functor commutator between the category of groups and
the category of abelian groups. However, the centre Z(G) of G does not naturally
define a functor F : Grp ! Grp, because a homomorphism f : G ! H may carry an
element in the centre of G to one not in the centre of H.
A functor which simply “forgets” some or all of the structures of an algebraic
object is called a forgetful functor. Thus the forgetful functor U : Grp ! Set (where
Set is the category of sets), assigns to each group G the set UG of its elements
(“forgetting” the multiplication and hence the group structure), and assigns to each
morphism f : G ! G0 of groups the same function f, regarded just as function between
sets. The forgetful functor U : Rng ! Ab (where Rng is the category of rings), assigns
to each ring R the additive abelian group of R and to each morphism R ! R0 of rings
the same function, regarded just as a morphism of addition.
Definition 1.4.2. An isomorphism T : C ! B of categories is a functor T from
7
C to B which is a bijection, both on objects and arrows. Equivalently, a functor
T : C ! B is an isomorphism if and only if there is a functor S : B ! C for which
both composites S �T and T � S are the identity functors 1C and 1B respectively. We
say that C is isomorphic to B, written C ⇠= B, if there exists an isomorphism between
them.
A functor T : C ! B is called full when to every pair (A,A0) of objects of C
and to every arrow g : T(A) ! T(A0) of B, there is an arrow f : A ! A0 of C with
g = T(f). Clearly, the composite of two full functors is a full functor.
A functor T : C ! B is called faithful when to every pair (A,A0) of objects of C
and to every pair f1, f2 : A ! A0 of parallel arrows of C, the equality T(f1) = T (f2) :
T(A) ! T(A0) implies f1 = f2. Again, the composite of two faithful functors is a
faithful functor.
As usual in mathematics, when we define an “abstract” mathematical object, it
is natural to look for its subobjects (subgroups in the case of groups, subrings in the
case of rings etc.).
Definition 1.4.3. A subcategory S of a category C is a collection of some of the
objects and some of the arrows of C, which includes with each arrow f both the
objects dom(f ) and cod(f ), with each object A its identity arrow 1A and with each
pair of composable arrows f ! g ! h their composite.
These conditions ensure that these collections of objects and arrows themselves
constitute a category S. Moreover, the inclusion map S ! C which sends each object
and arrow of S to itself in C is a functor, called the inclusion functor. This inclusion
functor is clearly faithful. We say that S is a full subcategory of C when the inclusion
functor S ! C is full. A full subcategory, given C, is thus determined by giving just
8
the set of its objects, since the arrows between any two of these objects A, A0 are all
morphisms A ! A0 in C.
Various properties of groups are reflected by properties of functors with values in
the category of groups. The simplest such case is the fact that subgroups can give rise
to the notions of “subfunctors”. The concept of “subfunctor” thus developed applies
with equal force to functors whose values are in the category of rings, spaces and so
on.
Definition 1.4.4. In the category D of all topological groups, a mapping �0 : G01 !
G02 is a submapping of a mapping � : G1 ! G2, written �0 �, whenever G01 G1,
G02 G2 and �0(g1) = �(g1) for all g1 2 G01.
Definition 1.4.5. Given functors T (A, B) and T 0(A, B) in two variables on the cat-
egories U and W with values in the category D (where U and W are any categories
and D is the category of topological groups), T 0 is called a subfunctor of T , written
T 0 T , if T 0(A, B) T (A, B) (where means a subgroup of) for each pair of objects
A 2 U , B 2 W and T 0(↵, �) T (↵, �) for each pair of mappings ↵ 2 U , � 2 W .
Clearly T 0 T and T T 0 implies T 0 = T . Furthermore this inclusion satisfies
the transitive law. If T 0 and T 00 are both subfunctors of the same functor T , then in
order to prove that T 0 T 00 it is su�cient to verify that T 0(A, B) T 00(A, B) for all
A and B.
The operation of forming a quotient group leads to an analogous operation of
taking the “quotient functor” of a functor T by a normal subfunctor T 0.
Definition 1.4.6. Let T be a functor covariant in the category U and contravariant
in the category W with values in the category D (where U and W are any categories
9
and D is the category of topological groups). A normal subfunctor T 0 is a subfunctor
T 0 T such that each T 0(A, B) is a normal subgroup of T (A, B), written T 0 � T .
Thus if T 0�T , the quotient functor Q = T/T 0 has an object function given as the
factor group
Q(A, B) = T (A, B)/T 0(A, B).
For homomorphisms ↵ : A1 ! A2 and � : B1 ! B2 the corresponding mapping
function Q(A, B) is defined for each coset xT 0(A, B) as
Q(↵, �)[xT 0(A1, B2)] = [T (↵, �)x]T 0(A2, B1).
One may easily verify that Q gives a uniquely defined homomorphism
Q(↵, �) : Q(A1, B2) ! Q(A2, B1).
As one might expect, the next obvious task is to formulate the isomorphism theorems
of group theory in terms of functors. This will not be discussed here (For a detailed
discussion see [7, pg.260–265]).
1.5 Initial and Terminal Objects
We now consider abstract characterizations of the empty set and the one-element sets
in the category Sets and structurally similar objects in general categories.
Definition 1.5.1. In any category C, an object A is called initial, if for any object
B there is a unique morphism
A ! B.
D is called terminal, if for any object B there is a unique morphism
B ! D.
10
Note that there is a kind of “duality” in these definitions. Precisely, a terminal
object in the category C is exactly an initial object in the category Cop (where Cop
is the opposite category, in vague terms the category with all arrows reversed).
Definition 1.5.2. An isomorphism f : A ! B of objects A and B in a category C
is just a bijective morphism from A to B.
Proposition 1.5.1. Initial (terminal) objects are unique up to isomorphism.
Proof. See [1, pg.28]
Examples
1. In the category Sets the empty set is initial and any singleton set is terminal.
Observe that Sets has just one initial object but many terminal objects.
2. In the category Grp, the one element group is both initial and terminal (sim-
ilarly for the category of vector spaces and linear transformations, as well as the
category of monoids and monoid homomorphisms).
1.6 Products and Coproducts of Categories
Next we are going to see the categorical definition of a product of two objects in
a category. This was first given by Mac Lane [13], and it is probably the earliest
example of category theory being used to define a fundamental mathematical notion.
Let us begin by considering products of sets. Given sets A and B the cartesian product
of A and B is the set of ordered pairs
A⇥ B = {(a, b)|a 2 A, b 2 B}.
11
Observe that there are two “coordinate projections”
A oo ⇡1A⇥B
⇡2 // B
with
⇡1(a, b) = a, ⇡2(a, b) = b.
Indeed, given any element c 2 A⇥ B we have
c = (⇡1c, ⇡2c).
The situation is captured concisely in the following diagram:
A A⇥Boo⇡1
A⇥B B⇡2//
1
A
a
������
����
����
�1
A⇥B
(a,b)
✏✏
1
B
b
��???
????
????
??
Definition 1.6.1. In any category C, a product diagram for the objects A and B
consists of an object P and arrows
A oo p1P
p2 // B
satisfying the following Universal Mapping Property:
Given any diagram of the form
A oo x1 Xx2 // B
there exists a unique u : X ! P, making the diagram
A Poop1
P Bp2//
X
A
x1
������
����
����
�X
P
u
✏✏
X
B
x2
��???
????
????
??
commutative, that is, such that x1 = p1u and x2 = p2u.
12
Proposition 1.6.1. Products are unique up to isomorphism.
Proof. See [1, pg.35].
If the objects A and B have a product, we write
A oo p1A⇥B
p2 // B.
for one such product. Then given X, x1, x2 as in the definition, we write
(x1, x2) for u : X ! A⇥ B.
However, a pair of objects may have many di↵erent products in a category. For
example, given a product A ⇥ B, p1, p2, and any isomorphism h : A ⇥ B ! Q, the
diagram Q, p1 � h, p2 � h is also a product of A and B.
Now an arrow into a product
f : X ! A⇥ B
is “the same thing” as a pair of arrows
f1 : X ! A, f2 : X ! B.
So we can essentially forget about such arrows, in that they are uniquely determined
by pairs of arrows.
There is a “dual” version of the notion of products in a category, called the
“coproduct” of two objects. One can think of coproducts as products with the arrows
reversed.
Definition 1.6.2. In any category C, a coproduct diagram for the objects A and B
consists of an object Q and arrows
Aq1 // Q oo q2
B
13
satisfying the following Universal Mapping Property:
Given any diagram of the form
Az1 // Z oo z2
B
there exists a unique u : Q ! Z, making the diagram
A Qq1// Q Boo
q2
Z
A
??
z1
����
����
����
�Z
Q
OO
u
Z
B
__
z2
????
????
????
?
commutative, that is, such that z1 = q1u and z2 = q2u.
1.7 Examples of Products and Coproducts
1. One can show that the product of two topological spaces X and Y, as usually
defined, is a product in Top (the category of topological spaces and continuous func-
tions). Thus, suppose that X and Y are two topological spaces and consider their
product space X⇥ Y with its projections
X oo p1X ⇥ Y
p2 // Y .
Recall that X⇥Y is generated by basic open sets of the form U⇥V where U is open in
X and V is open in Y . Therefore every W 2 X⇥Y is a union of open sets. Firstly, it
is obvious that p1 is continuous, since p�11 U = U⇥Y. Secondly, given any continuous
mappings f1 : Z ! X and f2 : Z ! Y, let f : Z ! X ⇥ Y be the function f = (f1, f2).
Now one needs to show that f is continuous. Given any W =S
i(Ui ⇥ Vi) 2 X⇥ Y,
14
f�1(W) =S
i f�1(Ui ⇥ Vi), so it su�ces to show that f�1(U⇥ V) is open. But
f�1(U⇥ V) = f�1((U⇥ Y) \ (X⇥ V))
= f�1(U⇥ Y) \ f�1(X⇥ V)
= f�1 � p�11 (U) \ f�1 � p�1
2 (V)
= (f1)�1(U) \ (f2)
�1(U) \ (f2)�1(V)
(1.7.1)
where (f1)�1(U)) and (f2)
�1(V) are open, since f1 and f2 are continuous.
The following diagram concisely captures the situation at hand:
X X ⇥ Yp�11
// X ⇥ Y Yoop�12
Z
X
??
f�11
����
����
����
�Z
X ⇥ Y
OO
f�1
Z
Y
__
f�12
????
????
????
?
Next, consider the following example of a coproduct:
2. For abelian groups A and B, the free product A � B need not be abelian.
One could take a quotient of A�B to get a coproduct in the category Ab of abelian
groups, but there is a more convenient presentation, which we now consider.
Definition 1.7.1. Let X = A � B, choose a set disjoint from X with the same
cardinality: for notational reasons we shall denote this by X�1 = {x�1|x 2 X} where
x�1 is merely a symbol. By a word in X we mean a finite sequence of symbols from
X [X�1, written in the form
w = xe11 ...xer
r
where xi 2 X, ei = ±1, and r � 0 (see [17, pg.45]).
Since the words in the free product A � B must be forced to satisfy the further
commutativity conditions
(a1b1b2a2...) ⇠ (a1a2...b1b2...)
15
we can shu✏e all the a0s to the front, and the b0s to the back, of the words. Further-
more, we already have
(a1a2...b1b2...) ⇠ (a1 + a2 + ... + b1 + b2 + ...).
Thus, we in e↵ect have pairs of elements (a, b). So we take the product set as the
underlying set of the coproduct
|A + B| = |A⇥B|.
As inclusions, we use the homomorphisms
iA(a) = (a, 0B)
iB(b) = (0A, b).
Then given any homomorphisms Af // X oo g
B, we let (f, g) : A + B ! X be
defined by
(f, g)(a, b) = f(a) +X g(b).
Proposition 1.7.1. In the category Ab of abelian groups, there is a canonical iso-
morphism between the binary coproduct and product,
A + B ⇠= A⇥B
Proof. See [1, pg.53].
This in fact was first observed by Mac Lane [13], and it was shown to lead to
a binary operation on parallel arrows f, g : A ! B between abelian groups (and
related structures like modules and vector spaces). In fact, the group structure of a
particular abelian group A can be recovered from this operation on arrows into A.
16
More generally, the existence of such an addition operation on arrows can be used
as the basis of an abstract description of categories like Ab, called abelian categories,
which are suitable for axiomatic homology theory (see [13]).
A very interesting situation arises in categories in which products and coproducts
di↵er.
First, consider the category Set of sets and mappings. The product of a family
(Ci)i2I of sets for some indexed set I, is just the cartesian product
Y
i2I
Ci = {(xi)i2I |xi 2 Ci}
with the obvious projection maps pi((xi)i2I) = xi.
The coproduct of a family (Ci)i2I of sets for some indexed set I, is its disjoint
union, i.e. the union of sets (Ci)i2I considered as disjoint sets. When various sets Ci
are not disjoint, one may replace them by an isomorphic disjoint set
C 0i = {(x, i)|x 2 Ci, i 2 I}
and then perform the usual union of these sets C 0i. Thus in short
[
i2I
Ci = {(x, i)|x 2 Ci, i 2 I}
with the obvious inclusion maps si(x) = (x, i).
Another very interesting example of a category in which products and coproducts
di↵er arises in the category Grp of groups and group homomorphisms. The product
of a family of groups in Grp is just their direct product with componentwise binary
operation. For example,Y
i2I
Gi = {(gi)i2I |gi 2 Gi}
17
is defined by
(gi)i2I + (hi)i2I = (gi + hi)i2I for gi, hi 2 Gi.
The coproduct of a family of groups in Grp is obtained as follows: First, consider
the disjoint union, say V , of the underlying sets of the groups Gi. Let W be the set
of words of V . Thus W is the set of all finite sequences of elements in V . Then,
introduce on W the equivalence relation generated by the following data:
1. The unit element of each group Gi, is equivalent to the empty sequence.
2. If a sequence contains two consecutive elements belonging to the same compo-
nent Gi, the original sequence is equivalent to the sequence obtained by replacing the
two elements by their composite in Gi.
Then writeS
i2I Gi for the quotient of W by this equivalence relation. Concatena-
tion on W induces an associative composition law onS
i2I Gi with the empty sequence
as a unit. This indeed gives a group structure: the inverse of a sequence is the se-
quence of inverses of its elements in reversed order. Each group Gi is mapped intoS
i2I Gi, with each element g 2 Gi going to the equivalence class [x] of the sequence
consisting of that single element.S
i2I Gi is easily seen to be a coproduct of the
groups Gi, which is indeed the well known “free product” of the groups Gi.
1.8 Hom-Sets
Definition 1.8.1. A category C is called locally small if for all objects X,Y in C,
the collection Homc(X,Y) = {f 2 C1|f : X ! Y} is a set (called the hom-set).
In this section, we assume that all categories are locally small. Note that any
18
arrow g : B ! B0 in C induces a function:
Hom(A, g) : Hom(A,B) ! Hom(A,B0)
(f : A ! B) 7! (g � f : A ! B ! B0).
Thus Hom(A, g) = g � f. One sometimes writes g⇤ instead of Hom(A, g), so
g⇤(f) = g � f.
Let us show that this determines a functor
Hom(A,�) : C ! Sets
called the (covariant) representable functor of A. We need to show that
Hom(A, 1X) = 1Hom(A,X)
and that
Hom(A, g � f) = Hom(A, g) � Hom(A, f).
Taking an argument x : A ! X, we clearly have
Hom(A, 1X)(x) = 1X � x
= x
= 1Hom(A,X)(x)
(1.8.1)
and
Hom(A, g � f)(x) = (g � f) � x
= g � (f � x)
= Hom(A, g)(Hom(A, f)(x))
(1.8.2)
19
Also, there is a “dual” version of the covariant functor, which is of the form
F : Cop ! D and is called the contravariant functor of a category C. Explicitly, such
a functor takes f : A ! B to F(f) : F(B) ! F(A) and F(g � f) = F(f) �F(g). A typical
example of a contravariant functor is a representable functor of the form
HomC(�,C) : Cop ! Sets
for any C 2 C (where C is any locally small category). Such a contravariant repre-
sentable functor takes f : X ! Y to
f⇤ : Hom(Y,C) ! Hom(X,C)
by
f ⇤(g : X ! C) = g � f.
Now let see how how one can use the notion of Hom-sets to give another definition
of product.
An object P with arrows p1 : P ! A and p2 : P ! B gives an element (p1, p2) of
the set
Hom(P,A)⇥ Hom(P, B)
and similarly for any set X in place of P. Now, given any arrow
x : X ! P
composing with p1 and p2 gives a pair of arrows x1 = p1 �x : X ! A and x2 = p2 �x :
X ! B, as indicated in the following diagram:
A Poop1
P Bp2//
X
A
x1
������
����
����
�X
P
x
✏✏
X
B
x2
��???
????
????
??
20
In this way, we have a function
@X = (Hom(X, p1), Hom(X, p2)) : Hom(X, P ) ! Hom(X, A)⇥Hom(X, B)
defined by
@X(x) = (x1, x2).
This function @X can be used to express concisely the condition of being a product
as follows.
Proposition 1.8.1. A diagram of the form
A oo p1P
p2 // B
is a product for A and B if and only if for every object X, the canonical function
@X(x) = (x1, x2) is an isomorphism
@X : Hom(X, P ) ⇠= Hom(X, A)⇥Hom(X, B)
Proof. By the Universal Mapping Property, for every element (x1, x2) 2 Hom(X, A)⇥
Hom(X,B), there is a unique x 2 Hom(X, P ) such that @X = (x1, x2), that is, @X is
bijective.
Definition 1.8.2. Let C,D be categories with binary products. A functor F : C ! D
is said to preserve binary products if it takes every product diagram
A oo p1A⇥B
p2 // B
in the category C, to a product diagram
FA oo Fp1F (A⇥B)
Fp2 // FB
21
in the category D. It follows that F preserves products just if
F (A⇥B) ⇠= FA⇥ FB
that is, if and only if the canonical “comparison arrow”
(Fp1, Fp2) : F (A⇥B) ! FA⇥ FB
is an isomorphism.
For example, the forgetful functor U : Mon ! Sets preserves binary products.
Corollary 1.8.2. For any object X in a category C with products, the covariant
functor
HomC(X,�) : C ! Sets
preserves products.
Proof. For any A, B 2 C, the foregoing proposition says that there is a canonical
isomorphism:
HomC(X, A⇥B) ⇠= HomC(X,A)⇥HomC(X, B)
1.9 Natural Transformations
Eilenberg and Mac Lane first observed, that the notion of “category” has been defined
in order to be able to define the notion of “functor” and the notion of “functor” has
been defined in order to be able to define the notion of “natural transformation”. One
22
can think of natural transformations as morphisms of functors. For fixed categories C
and D we can regard the functors C ! D as the objects of a new category, and the
arrows between these objects are what we are going to call natural transformations.
Definition 1.9.1. Given two functors S, T : C ! B, a natural transformation ⌧ :
S ! T is a function which assigns to each object A of C an arrow ⌧A = ⌧A : SA ! TA
of B in such a way that every arrow f : A ! A0 in C yields a commutative diagram:
A
A0
f
✏✏SA0 TA0
⌧A0//
SA
SA0
Sf
✏✏
SA TA⌧A // TA
TA0
Tf
✏✏
we call ⌧A, ⌧B, ... the components of the natural transformation ⌧ .
When this holds, we also say that ⌧A : SA ! TA is natural in A. If we think
of the functor S as giving a picture in B of all the objects and arrows of C, then a
natural transformation ⌧ is a set of arrows mapping the picture S to the picture T.
Definition 1.9.2. A natural transformation ⌧ with every component ⌧A invertible
in B is called a natural equivalence or a natural isomorphism, written as ⌧ : S ⇠= T .
In this case, the inverses (⌧A)�1 in B are the components of a natural isomorphism
⌧�1 : T ! S.
Examples of Natural Transformations
1. The determinant is a natural transformation. To be explicit, let detKM be the
determinant of the n⇥n matrix M with entries in the commutative ring K, while K⇤
denotes the group of units (invertible elements) of K. Thus, M is non-singular when
detKM is a unit, and detK is a morphism GLnK ! K⇤ of groups, an arrow in Grp,
23
(where GLn is obviously the general linear group). Because the determinant is defined
by the same formula for all rings K, each morphism f : K ! K 0 of commutative rings
leads to a commutative diagram
GLnK0 K 0⇤
detK0//
GLnK
GLnK0
GLnf
✏✏
GLnK K⇤detK // K⇤
K 0⇤
f⇤
✏✏
This states that the transformation det : GLn ! (⇤) is natural between two functors
C, T : Rng ! Grp.
2. For each group G the projection pG : G ! G/[G, G] to the factor-commutator
group, defines the transformation p from the identity functor on Grp to the factor-
commutator functor Grp ! Ab ! Grp. Moreover, p is natural because each group
homomorphism f : G ! H defines the evident homomorphism f 0 for which the
following diagram commutes:
H H/[H, H]pH
//
G
H
f
✏✏
G G/[G, G]pG // G/[G, G]
H/[H, H]
f 0
✏✏
3. Consider the category V ect(R) of real vector spaces and linear transformations
f : V ! W . Every vector space V has a dual space
V ⇤ = V ect(V, R)
of linear transformations and every linear transformation f : V ! W gives rise to a
dual linear transformation f ⇤ : W ⇤ ! V ⇤ defined by pre-composition, f ⇤(A) = A � f
for A : W ! R. In brief (�) : V ect(�, R) : V ectop ! V ect is the contravariant
24
representable functor endowed with vector space structure. There is a canonical
linear transformation from each vector space to its dual
⌘V : V ! V ⇤⇤
x 7! (evx : V ⇤ ! R)
where evV (A) = A(x) for every A : V ! R. This map is the component of a natural
transformation ⌘ : 1V ect ! ⇤⇤, since the following diagram always commutes in Vect
W W ⇤⇤⌘W
//
V
W
f
✏✏
V V ⇤⇤⌘V // V ⇤⇤
W ⇤⇤
f⇤⇤
✏✏
Indeed, given any v 2 V and A : W ! R in W ⇤, we have
(f ⇤⇤ � ⌘V (v)(A) = f ⇤⇤(evv)(A)
= evv(f⇤(A)
= evv(A � f)
= (A � f)(v)
= A(fv)
= evfv(A)
= (⌘W � f)(v)(A).
(1.9.1)
Now, it is well-known fact in linear algebra that every finite dimensional vector space
V is isomorphic to its dual space V ⇠= V ⇤, just for reasons of dimension. However,
there is no “natural” way to choose such an isomorphism. On the other hand, the
natural transformation
⌘V : V ! V ⇤⇤
25
is a natural isomorphism when V is finite dimensional. Thus, the formal notion of
naturality captures the informal fact that V ⇠= V ⇤⇤, “naturally”, unlike V ⇠= V ⇤.
Chapter 2
Category Theory in Algebraic
Topology
2.1 Introduction
In the years before 1945, Eilenberg and Mac Lane had been working intensively on
some homological problems within which natural transformations appear quite often,
some of them explicitly on the form of commutative diagrams. Broadly speaking,
homological procedures provide techniques for associating a suitably defined group
to every given topological space, so that properties of the space may be drawn from
more easily deducible properties of the homology groups. These procedures also
involve associating a group homomorphism to every continuous function between the
given topological spaces ([5, pg.347]).
In order to “translate” topological problems in the context of the new language
which category theory provided, Eilenberg and Mac Lane developed in the last chapter
of [7] a new framework by formulating well known concepts from algebraic topology
26
27
in terms of categorical language. Firstly, they formulated the homology group as a
functor from the category of complexes to the category of discrete abelian groups and
the cohomology group as a functor from the category of complexes to the category
of topological abelian groups. Secondly, they proved that the cohomology group is
isomorphic, indeed “naturally” isomorphic, to the character group of the homology
group (as functors). Thirdly, they showed that the basic constructions of group
extensions may be regarded as functors. Finally, they presented a new treatment of
the Cech homology theory in terms of functors. The aim of this chapter is to present
a coherent summary of the above developments and to show their significance in later
developments in Algebraic Topology.
2.2 Homology and Cohomology Groups as Func-
tors
Definition 2.2.1. An abstract complex K is a collection
{Cq(K)}, q = 0,±1,±2, ...
of free abelian discrete groups, together with a collection of homomorphisms
@q : Cq(K) ! Cq�1(K)
called boundary homomorphisms, such that
@q@q+1 = 0.
By selecting for each of the free abelian groups Cq a fixed basis {�qi }, called q-
dimensional cells, we obtain an abstract complex. The boundary operator @ can be
28
written as a finite sum
@�q =X
�q�1
[�q : �q�1]�q�1.
The integers [�q : �q�1] are called incidence numbers, and satisfy the following con-
ditions:
1. Given �q, [�q : �q�1] 6= 0 only for a finite number of (q � 1)� cells �q�1,
2. Given �q+1 and �q�1,P
�q [�q+1 : �q][�q : �q�1] = 0.
Definition 2.2.2. Given two abstract complexes K1 and K2, a chain transformation
k : K1 ! K2
is a collection k = {kq} of homomorphisms,
kq : Cq(K1) ! Cq(K2)
such that
kq�1@q = @qkq,
as the following diagram indicates
Cq(K2) Cq�1(K2)@q//
Cq(K1)
Cq(K2)
kq
✏✏
Cq(K1) Cq�1(K1)@q
// Cq�1(K1)
Cq�1(K2)
kq�1
✏✏
In this way one gets the category ⌦ whose objects are the abstract complexes and
whose mappings are the chain transformations with obvious definition of the compo-
sition of chain transformations. The consideration of simplicial complexes and sim-
plicial transformations leads to a category ⌦s. Every simplicial complex determines
uniquely an abstract complex and every simplicial transformation a chain transfor-
mation. This leads to a covariant functor on ⌦s to ⌦.
29
Definition 2.2.3. For every complex K in the category ⌦ and every group G in
the category D0a of discrete abelian groups, define the groups Cq(K,G) of the q-
dimensional chains of K over G as the tensor product (of abelian groups)
Cq(K, G) = G⌦ Cq(K),
that is, Cq(K,G) is the group with the symbols
gcq, g 2 G, cq 2 Cq(K)
as generators, and
(g1 + g2)cq = g1c
q + g2cq, g(cq
1 + cq2) = gcq
1 + gcq2
as relations.
For every chain transformation k : K1 ! K2 and for every homomorphism � :
G1 ! G2 define a homomorphism
Cq(k, �) : Cq(K1, G1) ! Cq(K2, G2)
by setting
Cq(k, �)(g1cq1) = �(g1)k
q(c1q)
for each generator g1cq1 of Cq(K1, G1).
These definitions of Cq(K, G) and Cq(k, �) yield a functor Cq covariant in ⌦ and
D0a with values in D0a. This functor is called q-chain functor.
Define a homomorphism
@q(K, G) : Cq(K, G) ! Cq�1(K, G)
30
by setting
@q(K, G)(gcq) = g@cq
for each generator g1cq1 of Cq(K1, G1). Thus the boundary operator becomes a natural
transformation of the functor Cq into the functor Cq�1
@q : Cq ! Cq�1.
The kernel of this transformation is denoted Zq and is called the q-cycle functor. Its
object function is the group Zq(K, G) of the q-dimensional cycles of the complex K
over G. The image of Cq under the transformation @q is a subfunctor Bq�1 = @q(Cq)
of Cq�1. Its object function is the group Bq�1(K, G) of the (q � 1)-dimensional
boundaries in K over G.
The fact that @q@q+1 = 0 implies that Bq(K, G) is a subgroup of Zq(K, G). Con-
sequently Bq is a subfunctor of Zq. The quotient functor
Hq = Zq/Bq
is called the qth homology functor. Its object function associates with each complex K
and with each discrete abelian coe�cient group G the qth homology group Hq(K, G)
of K over G. The functor Hq is covariant in ⌦ and D0a with values in D0a.
Now in order to define the cohomology groups as functors, consider the category
⌦ as before and the category Da of topological abelian groups. Given a complex K
in ⌦ and a group G in Da define the group Cq(K,G) of the q-dimensional cochains
of K over G as
Cq(K, G) = Hom(Cq(K), G).
Given a chain transformation k : K1 ! K2 and a homomorphism � : G1 ! G2, define
31
a homomorphism
Cq(k, �) : Cq(K2, G1) ! Cq(K1, G2)
by associating with each homomorphism f 2 Cq(K2, G1) the homomorphism f 0 =
Cq(k, �)f defined as follows:
f 0(cq1) = �[f(kqcq
1)], cq1 2 Cq(K1).
The definitions of Cq(K, G) and Cq(k, �) yield a functor Cq contravariant in ⌦,
covariant in Da with values in Da. This functor is called the qth cochain functor.
The coboundary homomorphism
�q(K, G) : Cq(K, G) ! Cq+1(K, G)
is defined by setting, for each cochain f 2 Cq(K, G),
(�qf)(cq+1) = f(@q+1cq+1).
This leads to a natural transformation of functors
�q : Cq ! Cq+1.
One may observe that in terms of the functor Hom, �q(K,G) = Hom(@q+1, eG).
The kernel of the transformation �q is denoted by Zq and is called the (q + 1)-
coboundary functor. Since @q@q+1 = 0, one may easily deduce that Bq is a subfunctor
of Zq. The quotient functor
Hq = Zq/Bq
is by definition the qth cohomology functor. Hq is contravariant in ⌦, covariant in
Da with values in Da. Its object function associates with each complex K and each
topological abelian group G the qth cohomology group Hq(K, G).
32
2.3 Duality
Let G be a discrete abelian group and CharG be its character group. Given a chain
cq 2 Cq(K, G) = G⌦ Cq(K)
where
cq =X
i
gicqi , gi 2 G, cq
i 2 Cq(K)
and given a cochain
f 2 Cq(K, CharG) = Hom(Cq(K), Char)
one may define the Kronecker index
KI = (f, cq) =X
i
(f(cqi ), gi).
Since f(cqi ) is an element of CharG, its application to gi gives an element of the group
P of reals reduced mod1. The continuity of KI(f, cq) as a function of f follows from
the definition of the topology in CharG and in Cq(K, CharG).
Theorem 2.3.1. The cohomology group Cq(K,CharG) is naturally isomorphic to
the character group of the homology group Cq(K, G).
Proof. Define an isomorphism
⌧ q(K, G) : Cq(K, CharG) � CharCq(K, G) (2.3.1)
by defining for each cochain f 2 Cq(K, CharG) a character
⌧ q(K, G)f : Cq(K, G) ! P
33
as follows:
(⌧ qf, cq) = KI(f, cq).
The fact that ⌧ q(K, G) is an isomorphism is a direct consequence of the character
theory (see [18]). In (2.3.1) both sides should be interpreted as object functions of
functors, contravariant in both K and G, suitably compounded from the functors Cq,
Cq and CharG. In order to prove that (2.3.1) is natural, consider
k : K1 ! K2 in ⌦
and
� : G1 ! G2 in D.
One must prove that
⌧ q(K1, G1)Cq(k, Char�) = [CharCq(k, �)]⌧ q(K2, G2). (2.3.2)
If now
f 2 Cq(K2, G2), cq 2 Cq(K1, G1),
then the definition of ⌧ q shows that (2.3.2) is equivalent to the identity
KI(Cq(k, Char�)f, cq) = KI(f, Cq(k, �)cq). (2.3.3)
It will be su�cient to establish (2.3.3) in the case when cq is a generator of Cq(K1, G1),
cq = g1cq1, g1 2 G1, cq
1 2 Cq(K2).
Using the definition of the terms involved in (2.3.3) one have on the one hand
KI(Cq(k, Char�)f, g1cq1) = ([Cq(k, Char�)f ]cq,g1
1 )
= (Char�[f(kcq1)]g1)
= (f(kc21), �g1)
(2.3.4)
34
and on the other hand
KI(f, Cq(k, �)g1cq1) = KI(f, (�g1)(kcq
1)) = (f(kcq1), �g1).
2.4 Equivalences of Functors
Elenberg and Mac Lane in [6], expressed the cohomology groups of a complex, for
an arbitrary coe�cient group, in terms of the integral homology groups and the
coe�cient group itself. They claimed that a general form of the “Universal Coe�cient
Theorems” can be stated in terms of certain groups of group extensions by showing
that the basic constructions of group extensions may be regarded as functors.
Definition 2.4.1. Let G be a topological abelian group and H a discrete abelian
group. A factor set of H in G is a function f(h, k) which assigns to each pair (h, k)
of elements in H an element f(h, k) 2 G in such a wise that
f(h, k) = f(k, h)
and
f(h, k) + f(h + k, l) = f(h, k + l) + f(k, l) for all h, k, l 2 H.
With the natural addition and topology, the set of all factor sets f of H in G
constitute a topological abelian group, written Fact(G, H). If � : G1 ! G2 and
⌘ : H1 ! H2 are homomorphisms, one can define a corresponding mapping
Fact(�, ⌘) : Fact(G1, H2) ! Fact(G2, H1)
35
by setting
(Fact(�, ⌘)f)(h1, k1) = �f(⌘h1, ⌘k1)
for each factor set f in Fact(G1, H2). Thus it appears that Fact is a functor, covariant
in the category Da of topological abelian groups and contravariant in the category
D0a of discrete abelian groups.
Guven any g(h) with values in G, the combination
f(h, k) = g(h) + g(k)� g(h + k)
is always a factor set, the sets of this form are called transformation sets and the set
of all such sets is a subgroup, written Trans(G, H), of the group Fact(G, H). This
subgroup is the object function of a subfunctor. The corresponding quotient functor
Ext = Fact/Trans
is thus covariant in Da and contravariant in D0a with values in Da. Its object function
assigns to the group G and H the group Ext(G, H) of the so-called abelian group
extension of G by H.
Since Cq(K, G) = Hom(Cq(K), G) and since Cq(K, I) = I ⌦ Cq(K) = Cq(K),
where I is the additive group of integers, one gets
Cq(K, G) = Hom(Cq(K, I), G).
Therefore, one may define a subgroup
Aq(K, G) = AnnihZq(K, I)1
1AnnihZq(K, I) is the set of all characters x 2 CharZq(K, I) with (x, g) = 0 for each g 2Zq(K, I).
36
of Cq(K, G), consisting of all homomorphisms f such that f(zq) = 0 for zq 2 Zq(K, I).
Thus one gets a subfunctor Aq of Cq, and one may show that the coboundary functor
Bq is a subfunctor of Aq, which is a subfunctor of the cocycle functor Zq. Therefore,
the quotient functor
Qq = Aq/Bq
is a subfunctor of the cohomology functor Hq, and one may consider the quotient
functor Hq/Qq. The functors Qq and Hq/Qq have the following object functions:
Qq(K, G) = Aq(K, G)/Bq(K, G),
(Hq/QQ)(K, G) = Hq(K, G)/Qq(K, G) ⇠= Zq(K, G)/Aq(K, G).
Based on [6], Eilenberg and Mac Lane made the following assertions, interpreting
the isomorphisms as equivalences of functors.
1. Qq(K, G) is a direct factor of Hq(K,G).
2. Qq(K, G) ⇠= Ext(G, Hq+1(K, I)).
3. Hq(K, G)/Qq(K, G) ⇠= Hom(Hq(K, I), G).
The procedure of forming the topological abelian group Fact and its subgroup
Trans comes directly from the theory of group extensions as follows:
Let A be an abelian group with a normal subgroup isomorphic to a group G such
that A/G ⇠= H. Let f be a function which assigns to each pair (h, k) of elements
in H an element f(h, k) 2 G. Choose a set {t(h)|h 2 H} of coset representatives
for G in A. Then f(h) + t(k) lies in the coset with representatives t(h + k), so
f(h) + t(k) = f(h, k) + t(h + k) for some f(h, k) 2 G. It is straightforward to check
that f is a factor set.
Now suppose that {t0(h)|h 2 H} is another set of coset representatives, defining
a factor set f 0. Then, t0(h) = t(h) + g(h) for some g(h) 2 G and then one may easily
37
check that
f 0(h, k) = f(h, k) + g(h) + g(k)� g(h, k).
In other words, f and f 0 di↵er by a transformation set.
Conversely, given a factor set, one may define a group A whose elements are
ordered pairs (g, h) for g 2 G, h 2 H with addition
(g1, h1) + (g2, h2) = (g1 + g2 + f(h1, h2), h1 + h2).
This has a normal subgroup {(g, 0)|g 2 G} isomorphic to G, with A/G ⇠= H. There-
fore elements of Ext(H, G) define extensions of G by H, i.e. groups A as above.
2.5 Cech homology groups as Functors
In the final subsection of the last chapter of [7], Eilenberg and Mac Lane presented a
new treatment of the Cech homology theory in terms of functors.
Definition 2.5.1. An open covering U of a topological space X is a finite collection
U = {A1, A2, ..., An}
of open sets whose union is in X.
The sets Ai i = 1, ..., n may appear with repetitions, and some of them may be
empty. If U1 and U2 are two such coverings, write U1 < U2 whenever U2 is a refinement
of U1, that is, whenever each set of the covering U2 is contained is some set of the
covering U1. With this definition the coverings U of X form a directed set 2, say P .
2A directed set is a set of elements p1, p2,.. with a reflexive and transitive binary relation suchthat for each pair of elements p1, p2 2 P there exists an element p3 2 P with p1 < p3 and p2 < p2
which is denoted by C(X).
38
Let C(X) be the set of coverings of a topological space X. Let ⇠1 : X1 ! X2 be
a continuous mapping of the space X1 into the space X2. Given a covering
U = {A1, ..., An} 2 C(X2),
define
C(⇠)U = {⇠�1(A1), ..., ⇠�1(An)} 2 C(X1)
and one obtains an order preserving mapping
C(⇠) : C(X2) ! C(X1).
One may easily verify that the functions C(X) and C(⇠) define a contravariant functor
C on the category X of topological spaces to the category D of directed sets.
Given a covering U of a topological space X, let N(U) be the nerve 3 of U . If
two coverings U1 < U2 are given, one may select for each set of the covering U2, a set
of the covering U1 containing it. This leads to a simplicial mapping of the complex
N(U2) into the complex N(U1) and therefore gives a transformation
k : N(U2) ! N(U1).
This transformation k is called a projection. The projection k is not uniquely defined
by U1 and U2, but it is known that any two projections k1 and k2 are chain homotopic
and consequently the induced homomorphisms
Hq(k, eG) : Hq(N(U2), G) ! Hq(N(U1), G) (2.5.1)
Hq(k, eG) : Hq(N(U1), G) ! Hq(N(U2), G) (2.5.2)
3The simplicial complex formed from a family of objects by taking sets that have non-emptyintersections.
39
of the homology and cohomology groups do not depend upon the particular choice of
the projection k.
Given a topological group G, consider the collection of the homology groups
Hq(N(U), G) for U 2 C(X). These groups together with the mappings (2.5.1) form
a inverse system of groups defined on the directed set C(X). This inverse system
is denoted by Cq(X,G) and is treated as an object function of the category Inb of
directed sets.
For a discrete topological group G, the cohomology groups Hq(N(U), G) together
with the mappings (2.5.2) form a direct system of groups Cq(X, G) defined on the
directed set C(X). The system Cq(X, G) is treated as an object function of the
category Dir of directed sets.
The functions Cq(X, G) and Cq(X, G) are the object functions of the functors
Cq
and Cq. Now, by defining the mapping functions Cq(⇠, �) and Cq(⇠, �) for given
mappings
⇠ : X1 ! X2, � : G1 ! G2,
one gets the order preserving mapping
C(⇠) : C(X2) ! C(X1) (2.5.3)
which with each covering
U = {A1, ..., An} 2 C(X2)
associates the covering
V = C(⇠)U = {⇠�1A1, ..., ⇠�1An} 2 C(X1).
40
Thus to each set of the covering V corresponds uniquely a set of the covering U , this
yields a simplicial mapping
k : N(V ) ! N(U),
which leads to the homomorphisms
Hq(k, �) : Hq(N(V ), G1) ! H1(N(U), G2), (2.5.4)
Hq(k, �) : Hq(N(U), G1) ! Hq(N(V ), G2). (2.5.5)
The mappings (2.5.3� 2.5.5) define the transformations:
Cq(⇠, �) : C
q(X1, G1) ! C
q(X2, G2) in Inb,
Cq(⇠, �) : Cq(X2, G1) ! Cq(X1, G2) in Dir.
Therefore Cq
defines a functor covariant in both X and Da with values in Inb, while
Cq defines a functor contravariant in X and covariant in D0a with values in Dir.
The Cech homology and cohomology functors are therefore defined as
Hq
= Lim Cq, Hq = Lim!Cq.
Hq
is covariant in X and Da with values in Da, while Hq is contravariant in X,
covariant in D0a with values in D0a. The object functions Hq(X,G) and Hq(X, G)
are the Cech homology and cohomology groups of the topological space X with the
group G as coe�cients (for a detailed discussion of the categories Dir and Inb see [7,
pg.272–281]).
Chapter 3
Further Developments
3.1 Introduction
The first article of Eilenberg and Mac Lane [7] on categories and functors was not
followed by an intensive development of the theory. On the contrary, even the authors
did not divert themselves from their current research in order to develop the ideas
involved in that article. Several years needed before category theory began to be
considered as an object of mathematical interest deserving specific research.
After the contributions of Eilenberg and Mac Lane to the first stages of category
theory, the next milestone in the development of the theory was the publication of a
handful of articles which transformed the theory into an autonomous research field
by the end of the fifties. First, the definition and study of “Abelian Categories”
in separate articles by Alexander Grothendieck and David Buchsbaum. Second, the
introduction of “Adjoint Functors” by Daniel Kan. Third, and somewhat later, the
41
42
first foundational attempts based on categorical ideas by William Lawvere [5, pg.351–
364]. The aim of this chapter is to present the development of the notion of “Ad-
joint Functors” which is due to Kan [10] and its applications on functors involving
C.S.S. Complexes which is due to the same author [11].
3.2 Adjoint Functors in One Variable
Definition 3.2.1. Let X and Z be categories, let S : X ! Z and T : Z ! X be
covariant functors and let
↵ : Hom(S(X), Z) ! Hom(X, T (Z))
be a natural equivelence. Then S is called “the left adjoint of T under ↵” and T “the
right adjoint of S under ↵”, written ↵ : S a T .
An important property of two adjoint functors is that each of them determines
the other up to a unique natural equivalence. This is expressed by the following
uniqueness theorem.
Theorem 3.2.1. Let S, S 0 : X ! Z and T, T 0 : Z ! X be covariant functors and let
↵ : S 0 a T 0. Let � : S 0 ! S be a natural transformation. Then there exists a unique
natural transformation ⌧ : T ! T 0 such that the following diagram is commutative
Hom(S 0(X), Z) Hom(X, T 0(Z))↵0
//
Hom(S(X), Z)
Hom(S 0(X), Z)
Hom(�(X),Z)
✏✏
Hom(S(X), Z) Hom(X, T (Z))↵ // Hom(X, T (Z))
Hom(X, T 0(Z))
Hom(X,⌧(Z))
✏✏
If � is a natural equivalence, then so is ⌧ .
43
Proof. Suppose ⌧ : T ! T 0 is a natural transformation such that the above diagram
is commutative. Then for every object A 2 X and B 2 Z and for every map
f 2 H(A, T (B))
⌧B � f = Hom(A, ⌧B)f = ↵0(Hom(�A, B)↵�1f) = ↵0(↵�1f � �A).
In particular if A = T (B) and f = 1T (B), then
⌧B = ↵0(↵�11T (B) � �T (B)). (3.2.1)
Consequently if a natural transformation ⌧ : T ! T 0 exists such that the above
diagram is commutative, then it must satisfy (3.2.1) and therefore is unique.
It follows from the naturality of ↵0 that for every map g : B ! B0 2 Z the
following diagrams are commutative
T (A)
T 0(B0)
↵0(g�↵�11T (B)��T (B))
��???
????
????
?T (A) T 0(B)
↵0(↵�11T (B)��T (B))// T 0(B)
T 0(B0)
T 0g
������
����
����
T (B)
T 0(B0)
↵0(↵�11T (B0)��T (B0)�S0Tg)
��???
????
????
?T (B) T (B0)
Tg // T (B0)
T 0(B0)
↵0(↵�11T (B0)��T (B0))
������
����
����
Also, in view of the naturality of ↵ and � one easily sees that the following diagram
is also commutative
S 0T (B0) ST (B0)�T (B0)
//
S 0T (B)
S 0T (B0)
S0Tg
✏✏
S 0T (B) ST (B)�T (B)// ST (B)
ST (B0)
STg
✏✏ST (B0) B0
↵�11T (B0)
//
ST (B)
ST (B0)
ST (B)
ST (B0)
ST (B) B↵�11T (B) // B
B0
g
✏✏
44
Consequently,
⌧B0 � Tg = ↵0(↵�11T (B0) � �T (B0)) � Tg
= ↵0(↵�11T (B0) � �T (B0) � S 0Tg)
= ↵0(g � ↵�11T (B) � �T (B))
= T 0g � ↵0(↵�11T (B) � �T (B))
= T 0g � ⌧B0.
(3.2.2)
i.e. the function ⌧ defined by (3.2.1) is a natural transformation.
Now let � be a natural equivalence and let ⌧ 0 be a natural transformation induced
by ��1. Then ⌧⌧ 0 and ⌧ 0⌧ are natural transformations induced by ���1 and ��1�,
and the uniqueness of ⌧⌧ 0 and ⌧ 0⌧ together with the fact that ���1 and ��1� are
identities yields that ⌧⌧ 0 and ⌧ 0⌧ are also identities, i.e. ⌧ is a natural equivalence
with inverse ⌧ 0.
One of the main features of “adjointness” is that many duality theorems hold.
Theorem 3.2.2. Let ↵ : S(X) a T (Z) and define for every object A⇤ 2 Xop and
B⇤ 2 Zop a map
↵⇤(B⇤, A⇤) : Hom(T ⇤(B⇤), A⇤) ! Hom(B⇤, S⇤(A⇤))
by
↵⇤(B⇤, A⇤) = ↵�1(A, B). (3.2.3)
Then the function
↵⇤ : Hom(T ⇤(Zop), Xop) ! Hom(Zop, S⇤(Xop))
is a natural equivalence, i.e. ↵⇤ : T ⇤ a S⇤. Also ↵⇤⇤ = ↵.
45
Proof. Let x⇤ : A⇤ ! X 0⇤ 2 Xop and z⇤ : B0⇤ ! B⇤ 2 Zop be maps. Then it follows
from the naturality of ↵ that for every map f ⇤ : Hom(T ⇤(B⇤), A⇤)
↵⇤(B0⇤, A0⇤)Hom(T ⇤z⇤, x⇤)f ⇤ = ↵⇤(B0⇤, A0⇤)(x⇤ � f ⇤ � T ⇤z8)
= (↵�1(A0, B0)(Tz � f � x))⇤
= (↵�1(A0, B0)H(x, Tz)f)⇤
= (Hom(Sx, z)↵�1(A, B)f)⇤
= (z � ↵�1(A, B)f � Sx)⇤
= S⇤x⇤ � ↵⇤(B⇤, A⇤)f ⇤ � z⇤
= Hom(z⇤, S⇤x⇤)↵⇤(B⇤, A⇤)f ⇤.
(3.2.4)
i.e. ↵⇤ is a natural transformation. The fact that ↵ and hence ↵�1 is a natural
equivalence now implies that ↵⇤ is so. The fact that ↵⇤⇤ = ↵ follows immediately
from (3.2.3).
3.3 Adjoint Functors in Several Variables
After the introduction of the notion of the adjoint functor in one variable, Kan [10]
generalised his results to functors of several variables.
A covariant functor S : X, Y ! Z may be regarded as a collection consisting of
1. A covariant functor S(�, A) : X ! Z for every object B 2 Y and
2. A natural transformation S(�, b) : S(�, B) ! S(�, B0) for every map b : B !
B0 2 Y.
Now suppose that for every object B 2 Y , a covariant functor TB : Z ! X and a
46
natural equivalence
↵Y : Hom(S(X, B), Z) ! Hom(X, TB(Z))
are given i.e. ↵B : S(�, B) a TB. Then it follows from Theorem (3.2.1) that for every
map b : B ! B0 2 Y there exists a unique natural transformation Tb : TB0 ! TB such
that the following diagram is commutative
Hom(S(X, B0), Z) Hom(X, TB0(Z))↵B0//
Hom(S(X, B), Z)
Hom(S(X, B0), Z)
OO
Hom(S(X,b),Z)
Hom(S(X, B), Z) Hom(X,TB(Z))↵B // Hom(X,TB(Z))
Hom(X, TB0(Z))
OO
Hom(X,Tb(Z))
Let b0 : B0 ! B00 2 Y . Then the uniqueness of the natural transformation TB, TB0
and TB0B implies that TBTB0 = TB0B. Similarly if 1 : B ! B is the identity map,then
T1 : TB ! TB is the identity natural transformation. Therefore the function T defined
by
T (B, C) = TBC,
T (b, z) = TBz � TbC
for every object B 2 Y and C 2 Z and every map b : B ! B0 2 Y and z : C ! C 0 2
Z, is a functor T : Y, Z ! X, contravariant in Y and covariant in Z.
Clearly the function ↵ defined by
↵(A, B, C) = ↵Y (A, C)
for every object A 2 X, B 2 Y and C 2 Z, is a natural equivalence
↵ : Hom(S(X, Y ), Z) ! Hom(X, T (Y, Z))
thus we have the following theorem
47
Theorem 3.3.1. Let S : X, Y ! Z be a covariant functor and let for every object
B 2 Y be given a covariant functor TB : Z ! X and a natural transformation
↵B : Hom(S(X,B), Z) ! Hom(X, TB(Z)),
i.e. ↵B : S(�, B) a TB. Then there exists a unique functor
T : Y, Z ! X
contravariant in Y and covariant in Z and a unique natural equivalence
↵ : Hom(S(X, Y ), Z) ! Hom(X, T (Y, Z))
such that for every object A 2 X, B 2 Y and C 2 Z
T (B, C) = TBC, ↵(A, B, C) = ↵B(A, C)
i.e. ↵ : S a T .
Definition 3.3.1. Let S : X, Y ! Z be a covariant functor, let T : Y, Z ! X be a
functor contravariant in Y and covariant in Z and let
↵ : Hom(S(X, Y ), Z) ! Hom(X, T (Y, Z))
be a natural equivalence. The S is called “the left adjoint of T under ↵” and T is
called “the right adjoint of S under ↵”, written ↵ : S a T .
As in the case of functors of one variable, adjoint functors in two variables deter-
mine each other up to a unique natural equivalence. This is expressed by the following
uniqueness theorem.
48
Theorem 3.3.2. Let S, S 0 : X, Y ! Z be covariant functors, let T, T 0 : Y, Z ! X be
functors contravariant in Y and covariant in Z and let ↵ : S a T and ↵0 : S 0 a T 0.
Let � : S 0 ! S be a natural transformation. Then there exists a unique natural
transformation ⌧ : T ! T 0 such that the following diagram is commutative
Hom(S 0(X, Y ), Z) Hom(X, T 0(Y, Z))↵0
//
Hom(S(X, Y ), Z)
Hom(S 0(X, Y ), Z)
Hom(�(X,Y ),Z)
✏✏
Hom(S(X, Y ), Z) Hom(X, T (Y, Z))↵ // Hom(X, T (Y, Z))
Hom(X, T 0(Y, Z))
Hom(X,⌧(Y,Z))
✏✏
if � is a natural equivalence, then so is ⌧ .
As in the case of functors of one variable, duality holds for adjoint functors in two
variables.
Theorem 3.3.3. Let ↵ : S(X, Y ) a T (Y, Z) and define for every object A⇤ 2 Xop,
B 2 Y op, and C⇤ 2 Zop a map
↵⇤ : (C⇤, B, A⇤) : Hom(T ⇤(C⇤, B), A⇤) ! Hom(C⇤, S⇤(B, A⇤))
by
↵⇤(C⇤, B, A⇤) = ↵�1(A, B, C).
Then the function
↵⇤ : Hom(T ⇤(Zop, Y ), Xop) ! Hom(Zop, S⇤(Y,Xop))
is a natural equivalence, i.e. ↵⇤ : T ⇤(Zop, Y ) a S⇤(Y, Xop). Also ↵⇤⇤ = ↵.
The proofs of (3.3.1) � (3.3.3) are immediate generalisations of the analogous
theorems for adjoint functors in one variable. It follows from Theorem (3.3.3) that
49
for every Theorem say V involving a natural equivalence ↵ : S(X, Y ) a T (Y, Z), a
dual Theorem V ⇤ may be obtained by applying Theorem V to the natural equivalence
↵⇤ : T ⇤(Zop, Y ) a S⇤(Y, Xop) and then writing the result in terms of the categories
X, Y and Z, the functors S and T and the natural equivalence ↵, i.e. “reversing all
arrows” in the categories Xop and Zop.
Next, consider functors in more than two variables. Let
S : X, K1, K2, ..., Km, L1, L2, ..., Ln ! Z
be a functor covariant in X,K1, ..., Km and contravariant in L1, ..., Ln and let
T : K1, ..., Km, L1, ..., Ln, Z ! X
be a functor contravariant in K1, ..., Km and covariant in L1, ..., Ln, Z. Then S and
T may be considered as functors in two variables as follows:
Let Y be the cartesian product category
Y = (Y
i
Ki)⇥ (Y
j
L⇤j) for i = 1, ...,m and j = 1, ..., n.
Then S may be considered as a covariant functor
S 0 : X, Y ! Z
and T as a functor
T 0 : Y, Z ! X
contravariant in Y and covariant in Z. Therefore the case of functors in more than
two variables may be brought back to that of functors in two variables only.
50
3.4 Applications of Adjoint Functors to C.S.S. Com-
plexes
Based on the theory of adjoint functors, Kan [11] gave a procedure by which functors
and natural transformations may be constructed using c.s.s. complexes. He also
introduced a functor Hv(�,�) from chain complexes to c.s.s. abelian groups with the
following properties:
1. The functor Hv(�,�) sets up a one-to-one correspondence between chain com-
plexes which are zero in dimension < 0 and c.s.s. abelian groups.
2. For every chain complex K
Hn(K) ⇠= ⇡n(Hv(�, K))
3. Let (⇡, n) be a chain complex which has the abelian group ⇡ in dimension n
and 0 in the others. Then
Hv(�, (⇡, n)) = K(⇡, n)
i.e. Hv(�, (⇡, n)) is the Eilenberg-Mac Lane complex1 of ⇡ on level n.
Definition 3.4.1. For each integer n = 0, let [n] denote the ordered set (0, ..., n). By
a monotone function ↵ : [m] ! [n] denote a function such that
↵(i) 5 ↵(j) 0 5 i 5 j 5 m.
Clearly the composition of two monotone functions is again a monotone function
and for every integer n = 0 the identity map en : [n] ! [n] is also monotone. Therefore
1Let G be a group and n a positive integer. A connected topological space is called an Eilenberg-Mac Lane space of type K(G, n) if it has nth homotopy group ⇡n(X) isomorphic to G and all otherhomotopy groups trivial. Such spaces were first introduced by Samuel Eilenberg and Saunders MacLane in 1950 (see [8]).
51
the ordered sets [n] and the monotone functions ↵ : [m] ! [n] form a category, say
V .
Let M be the category of sets and M v the category of contravariant functors
V ! M .
Definition 3.4.2. A c.s.s. complex K is a contravariant functor K : V ! M , i.e.
an object of the category M v. Similarly a c.s.s. map f : K ! L is a natural
transformation from K to L, i.e. a map of the category M v. The elements of the set
K[n] are called n-simplices of K.
Let Z be a category which has direct limits (see [7, pg. 277] and [10, pg.313]).
Then with every covariant functor ⌃ : V ! Z, one can associate two covariant
functors
(�⌦ ⌃) : M v ! Z, Hv(⌃,�) : Z ! M v
where (� ⌦ ⌃) is a left adjoint of Hv(⌃,�). Conversely, every pair of covariant
functors
S : M v ! Z, T : Z ! M v
where S is a left adjoint of T , may (up to natural equivalences) be obtained in this
manner.
Because Z has direct limits, the embedding functor Ed : Z ! Zd has a left
adjoint. Let limd : Zd ! Z be an arbitrary but fixed such left adjoint and let ↵d
be an arbitrary but fixed natural equivalence ↵d : limd(Zd) a Ed(Z). Consider the
functor ⌦d : M v, Zv ! Zd and the natural equivalence
� : Hom(M v ⌦d Zv, Ed(Z)) �! Hom(M v, Hv(Zv, Z)).
52
Composition of the natural equivalence ↵d with the functor ⌦d yields a natural equiv-
alence
↵d⌦d : Hom(limd(Mv ⌦d Zv), Z) �! Hom(M v ⌦d Zv, Ed(Z)).
Composition of the natural equivalences ↵d⌦d and � yields the natural equivalence
� : Hom(limd(Mv ⌦d Zv), Z) �! Hom(M v, Hv(Zv, Z)).
It follows that � is completely determined by the choice of limd and ↵d. Now, denote
by
⌦ : M v, Zv ! Z
the composite functor limd⌦d : M v, Zv ! Z and write S instead of M v. Then � is a
natural equivalence
� : Hom(S ⌦ Zv, Z) ! Hom(S, Hv(Zv, Z)).
Hence, given the functor limd : Zd ! Z and the natural equivalence ↵d : limd a Ed,
one may associate with every object ⌃ 2 Zv
1. The covariant functor
Hv(⌃,�) : Z ! S
which is the right adjoint of
2. The covariant functor
(�⌦ ⌃) : S ! Z
under
3. The natural equivalence
�⌃ = �(S, ⌃, Z) : Hom(S ⌦ ⌃, Z) ! Hom(S, Hv(⌃, Z)),
53
4. The natural transformation induced by �⌃
k⌃ : E(S) ! Hv(⌃, S ⌦ ⌃)
satisfying the relation
�⌃f = Hv(⌃, f) � k⌃K
for every object K 2 ⌃ and A 2 Z and every map f : K ⌦ ⌃ ! A 2 Z, and
5. The natural transformation induced by ��1⌃
µ⌃ : Hv(⌃, Z)⌦ ⌃ ! E(Z)
satisfying the relation
��1⌃ g = µ⌃A � g ⌦ ⌃
for every object K 2 S, A 2 Z and every map g : K ! Hv(⌃, A) 2 S.
The main striking results of Kan [11], are the introduction of the functor Hv(�,�)
from chain complexes to c.s.s abelian groups and the proof that the functor Hv(�, (⇡, n))
(where (⇡, n) is a chain complex) is indeed the Eilenberg-Mac Lane complex of ⇡ on
level n.
For every object A 2 @G (where @G is the category of abelian chain complexes),
the c.s.s complex Hv(�, A) may be converted into a c.s.s. abelian group as follows:
Let �, ⌧ : �[n] ! A be two n-simplices of Hv(�, A). Then the sum �+⌧ : �[n] ! A
is defined by
(� + ⌧)� = �� + ⌧�, � 2 �[n].
For every chain map f : A ! B the c.s.s. map Hv(�, f) : Hv(�, A) ! Hv(�, B) then
becomes a c.s.s. homomorphism. Hence Hv(�,�) may be regarded as a functor
Hv(�,�) : @G ! Gv.
54
Let @G0 be the full subcategory of @G generated by the chain complexes which
are 0 in dimension < 0, i.e. A 2 @G0 if and only if Ai = 0 for i < 0. Let
M : Gv ! @G0
be the functor which assigns to every c.s.s. abelian group G the chain complex
MG = eG and to every c.s.s. homomorphism f : G ! H the chain map Mf : eG ! eH
given by (Mf)� = f� for � 2 eG.
Roughly speaking, the functor Hv(�,�) sets up a one-to-one correspondence be-
tween the objects and maps of @G0 and those of Gv. An exact formulation is given
in the following theorems, in which E denotes the identity functor.
Theorem 3.4.1. There exists a natural equivalence
↵ : MHv(�, @G0) ! E(@G0).
Proof. For each non-degenerate simplex ↵ 2 �[n], let CN↵ be the corresponding
generator of �[n]. Let A 2 @G0 be an object and let G = Hv(�, A). For each simplex
� : �[n] ! A 2 fGn define an element ↵� 2 An by
↵� = �(Cnen),
where en : [n] ! [n] is the identity map, i.e. the only non-degenerate n-simplex of
�[n]. As the addition in G was induced by that of A, it follows that the function
↵ : eGn ! An is a homomorphism for each n.
It follows from the definition of fGn that a simplex � : �[n] ! A is in fGn if and
only if �ei : �[n�1] ! A is the zero map for i 6= 0, i.e. � maps all generators of �[n],
with the possible exception of Cnen and Cne0, into zero. Consequently
55
@n(↵�) = @n(�(Cnen))
=X
(�1)i(�(Cnei))
= �(Cne0)
= (�e0)(Cnen�1)
= (e@�)(Cnen�1)
= ↵(e@�),
(3.4.1)
i.e. the function ↵ : eG ! A is a chain map. It also follows that � is completely
determined by �(Cnen) 2 An. Hence ↵ : eG ! A is an isomorphism. Naturality is
easily verified.
An immediate consequence of Theorem (3.4.1) is that for each object A@ 2 G and
for every integer n = 0
j⇤↵⇤ : ⇡n(Hv(�, A)) ⇠= Hn(A)
i.e. the nth homology group of the c.s.s. group Hv(�, A) is isomorphic with the nth
homology group of the chain complex A.
Now let X be a topological space. The homotopy groups of X are by definition
those of its simplicial singular complex K = Hv(⌃, X), and the singular homology
groups of X are the homology groups of the chain complex CnK (For more on Ho-
mology and Homotopy Theory see [9]). Let � be a 0-simplex of K, let
h⇤ : ⇡n(K,�) ! Hn(CnK)
56
be the Hurewicz homomorphism 2, and let
k : E(S) ! Hv(�, S ⌦ �)
be the natural transformation induced by the natural equivalence � : S ⌦ � a
Hv(�, @G). Then it clearly follows that the following diagram is commutative
⇡n(K,�)
Hn(CnK)
h⇤ ��???
????
??
⇡n(Hv(�, K ⌦ �), )
⇡n(K,�)
??(kK)⇤
����
����
�⇡n(Hv(�, K ⌦ �), )
Hn(CnK)
c⇤�↵⇤� ⇤
✏✏Hn(CnK) Hn(K ⌦ �)oo
c⇤
⇡n(Hv(�, K ⌦ �), )
Hn(CnK)
⇡n(Hv(�, K ⌦ �), )
Hn(CnK)
⇡n(Hv(�, K ⌦ �), ) ⇡n(Hv(�, K ⌦ �)) ⇤ // ⇡n(Hv(�, K ⌦ �))
Hn(K ⌦ �)
↵⇤
✏✏
where ⇤, ↵⇤, c⇤ are isomorphisms and = (kK)� i.e. the map
kK : K ! Hv(�, K ⌦ �)
induces (up to an equivalence) the Hurewicz homomorphism.
Finally, let ⇡ be an abelian group and n an integer = 0. Denote by (⇡, n) the
chain complex with ⇡ in dimension n and 0 in the others.
Theorem 3.4.2. Hv(�, (⇡, n)) = K(⇡, n), the Eilenberg-Mac Lane complex of ⇡ on
level n.
Proof. A q-simplex of K(⇡, n) is an element of Zn(�[q], ⇡) i.e. a chain map �[q] !
(⇡, n) and hence a q-simplex of Hv(�, (⇡, n)) and conversely. Clearly this one-to-one
correspondence commutes with all operators ↵.
2For any space X and any integer k > 0 the homomorphism h⇤ : ⇡k(X) ! Hk(X) from homotopyto reduced homology with integer coe�cients. In fact, it turns out to be equivalent, if k = 1, to thecanonical abelianisation map h⇤ : ⇡1(X) ! ⇡1(X)/[⇡1(X),⇡1(X)].
57
The introduction of the concept of Eilenberg-Mac Lane space by Samuel Eilen-
berg and Saunders Mac Lane in 1950 had a great impact on questions concerning
constructions of topological invariants, i.e. computing the cohomology group for a
given topological space (see [8]). Thus, the introduction of the functor Hv(�, (⇡, n))
and the proof that this functor gives the Eilenberg-Mac Lane complex, led to great
simplifications in computing such spaces by applying the functor Hv(�, (⇡, n)) to a
very simple chain complex without the need of complicated explicit construction.
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