Table Of ContentLecture notes on category theory
Ana Agore
2017-2018
Master in Mathematics - Vrije Universiteit Brussel and Universiteit Antwerpen
(cid:13)c Ana Agore
Contents
1 Categories and functors 1
1.1 Definition and first examples . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 Special objects and morphisms in a category . . . . . . . . . . . . . . . . 4
1.3 Some constructions of categories . . . . . . . . . . . . . . . . . . . . . . . 6
1.4 Functors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
1.5 Natural transformations. Representable functors . . . . . . . . . . . . . . 16
1.6 Yoneda’s lemma . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
1.7 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
2 Limits and colimits 27
2.1 (Co)products, (co)equalizers, pullbacks and pushouts . . . . . . . . . . . . 27
2.2 (Co)limit of a functor. (Co)complete categories . . . . . . . . . . . . . . . 36
2.3 (Co)limits by (co)equalizers and (co)products . . . . . . . . . . . . . . . . 41
2.4 (Co)limit preserving functors . . . . . . . . . . . . . . . . . . . . . . . . . 45
2.5 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48
3 Adjoint functors 51
3.1 Definition and a generic example . . . . . . . . . . . . . . . . . . . . . . . 51
3.2 More examples and properties of adjoint functors . . . . . . . . . . . . . . 54
3.3 The unit and counit of an adjunction . . . . . . . . . . . . . . . . . . . . . 57
3.4 Another characterisation of adjoint functors . . . . . . . . . . . . . . . . . 62
i
3.5 Equivalence of categories . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66
3.6 (Co)reflective subcategories . . . . . . . . . . . . . . . . . . . . . . . . . . 70
3.7 Localization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75
3.8 Freyd’s adjoint functor theorem . . . . . . . . . . . . . . . . . . . . . . . . 80
3.9 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85
Bibliography 89
Introduction to category theory ii Ana Agore
1
Categories and functors
1.1 Definition and first examples
Definition 1.1.1 A category C consists of the following data:
(1) A class Ob C whose elements A, B, C,... ∈ ObC are called objects;
(2) For every pair of objects A, B, a set Hom (A,B), whose elements will be called
C
morphisms from A to B;
(3) For every triple of objects A, B, C, a composition law
Hom (A,B)×Hom (B,C) → Hom (A,C)
C C C
(f, g) → g◦f
(4) For every object A, a morphism 1 ∈ Hom (A,A), called the identity on A
A C
such that the following axioms hold:
(1) Associativelaw: givenmorphismsf ∈ Hom (A,B), g ∈ Hom (B,C), h ∈ Hom (C,D)
C C C
the following equality holds:
h◦(g◦f) = (h◦g)◦f.
(2) Identity law: given morphisms f ∈ Hom (A,B) and g ∈ Hom (B,C) the following
C C
identities hold:
1 ◦f = f, g◦1 = g.
B B
Remark 1.1.2 Probably the most unfamiliar thing in Definition 1.1.1 is that the objects
form a class, not a set. The reason for this choice is that we sometimes want to speak
1
CHAPTER 1. CATEGORIES AND FUNCTORS
aboutthecategoryofsets, havingasobjectsallpossiblesets, andwerunintoawell-known
set-theory problem namely that considering ”the set of all sets” leads to a contradiction
(Russell’s paradox). One way to get around this issue is by using the Godel-Bernays set-
theory which introduces the notion of a class. The connection between sets and classes
is given by the following axiom: a class is a set if and only if it belongs to some (other)
class. Informally, a class can be though of as a ”big set”.
Definition 1.1.3 A category C is called a small category if its class of objects Ob C is
a set.
Examples 1.1.4 1) Any set X can be made into a category, called the discrete cat-
egory on X and denoted by C , as follows:
X
ObC := X
X
(cid:26)
∅ if x (cid:54)= y
Hom (x, y) = , for every x, y ∈ X.
CX {1 } if x = y
x
2) More generally, any pre-ordered set1 (poset for short) (X, (cid:54)) defines a category
PO(X, (cid:54)) as follows:
ObPO(X, (cid:54)) := X
(cid:26) ∅ if x (cid:10) y
Hom (x, y) = , for every x, y ∈ X.
PO(X,(cid:54)) {u } if x (cid:54) y
x,y
The composition of morphisms is given as follows u ◦ u = u while the
y,z x,y x,z
identity on x is u .
x,x
3) A monoid (M,·) can be seen as a category M with a single object denoted by ∗ and
the set of morphisms Hom (∗, ∗) = M. The composition of morphisms in M is
M
given by the multiplication of M and the identity on ∗ is just the unit 1 .
M
4) The category Set of sets has the class of all sets as objects while Hom (A, B) is
Set
the set of all maps from A to B. Composition is given by the usual composition of
maps and the identity on any set A is the identity map 1 .
A
5) FinSet is the category whose objects are finite sets, and where Hom (A, B)
FinSet
is just the set of all maps between the two finite sets A and B.
6) Consider RelSet to be the category defined as follows:
ObRelSet := ObSet
Hom (A, B) = P(A×B) = {f | f ⊆ A×B}, for every A, B ∈ ObSet.
RelSet
The composition of morphisms in RelSet is given as follows: given f ⊆ A×B
and g ⊆ B × C we define g ◦ f = {(a, c) ∈ A × C | ∃b ∈ Bsuch that(a, b) ∈
fand(b, c) ∈ g}. Finally, the identity is defined as 1 = {(a, a) | a ∈ A}.
A
1AsetX iscalledpre-ordered ifitisendowedwithabinaryrelation(cid:54)whichisreflexiveandtransitive.
Introduction to category theory 2 Ana Agore
1.1. DEFINITION AND FIRST EXAMPLES
7) Grp is the category of groups, where Ob Grp is the class of all groups while
Hom (A, B) is the set of all group morphisms from A to B. Similarly, Mon
Grp
denotes the category of monoids with monoid morphisms between them.
8) Ab is the category of abelian groups with group morphisms between them.
9) Rng is the category of rings with ring morphisms between them.
10) Ring (resp. Ringc) is the category of (commutative) unitary rings with unit pre-
serving ring morphisms between them.
11) Field is the category of fields with field morphisms between them.
12) For a ring R, we denote by M the category having as objects all left R-modules
R
and with the morphisms between two R-modules given by all R-linear maps. If
R is non-commutative one can also define analogously the category of all right
R-modules M .
R
13) Top is the category of topological spaces where Ob Top is the class of all topological
spaces while Hom (A, B) is the set of continuous maps between A and B. Top
Top ∗
stands for the category of pointed topological spaces, that is the objects are pairs
(A, a ) where A is a topological space and a ∈ A while the morphisms between
0 0
two such pairs (A, a ) and (B, b ) are just continuous maps f : A → B such that
0 0
f(a ) = b .
0 0
14) Haus (respectively KHaus) is the category of Hausdorff (compact) topological
spaces where Ob Haus (Ob KHaus) is the class of all hausdorff (compact) topo-
logical spaces while Hom (A, B) (Hom (A, B)) is the set of continuous
Haus KHaus
maps between A and B.
Remark 1.1.5 Notice that although we sometimes work with categories whose objects
are sets, the morphisms need not be functions. This situation is best illustrated in Ex-
ample 1.1.4, 6).
Definition 1.1.6 Let C, C(cid:48) be two categories. We shall say that C(cid:48) is a subcategory of C
if the following conditions are satisfied:
(i) Ob C(cid:48) ⊆ Ob C;
(ii) Hom (A,B) ⊆ Hom (A,B) for every A, B ∈ ObC(cid:48);
C(cid:48) C
(iii) The composition of morphisms in C(cid:48) is induced by the composition of morphisms
in C;
(iv) The identity morphisms in C(cid:48) are identity morphisms in C.
Moreover, C(cid:48) is said to be full if for every pair (A, B) of objects of C(cid:48) we have:
Hom (A,B) = Hom (A,B)
C(cid:48) C
Introduction to category theory 3 Ana Agore
CHAPTER 1. CATEGORIES AND FUNCTORS
Examples 1.1.7 1) The category FinSet is a full subcategory of Set;
2) The category Ab is a full subcategory of Grp;
3) The category Haus is a full subcategory of Top;
4) Ring is a subcategory of Rng but not a full subcategory;
5) Set is a subcategory of RelSet but not a full subcategory.
1.2 Special objects and morphisms in a category
Definition 1.2.1 Let C be a category and f ∈ Hom (A,B).
C
1) f is called a monomorphism if f ◦g = f ◦g implies g = g for every C ∈ ObC
1 2 1 2
and every g , g ∈ Hom (C,A);
1 2 C
2) f is called an epimorphism if h ◦f = h ◦f implies h = h for every C ∈ ObC
1 2 1 2
and every h , h ∈ Hom (B,C);
1 2 C
3) f is called an isomorphism if there exists f(cid:48) ∈ Hom (B,A) such that f ◦f(cid:48) = 1
C B
and f(cid:48)◦f = 1 . In this case we say that A and B are isomorphic objects and we
A
∼
denote this by A = B.
Remark 1.2.2 The notions of monomorphism and epimorphism are generalizations to
arbitrary categories of the familiar injective and surjective maps from Set. However,
althoughinSetamorphism(function)isanisomorphismifandonlyifitisbothinjective
and surjective this is no longer true in an arbitrary category. More precisely, a morphism
that is both a monomorphism and an epimorphism need not be an isomorphism (see, for
instance, Example 1.2.3, 4)).
Examples 1.2.3 1) In the categories Set of sets, Grp of groups, Ab of abelian
groups, M of left R-modules, Top of topological spaces monomorphism (epimor-
R
phisms) coincide with the injective (surjective) morphisms. We will only prove that
monomorphisms in Set coincide with injective maps. Indeed, suppose f : A → B
is an injective map and g, h : C → A such that f ◦ h = f ◦ g. Then, we have
(cid:0) (cid:1) (cid:0) (cid:1)
f h(c) = f g(c) for any c ∈ C and since f is injective we get h(c) = g(c) for any
c ∈ C, i.e. g = h as desired. Assume now that f : A → B is a monomorphism
and let a, a(cid:48) ∈ A such that f(a) = f(a(cid:48)). We denote by i : {∗} → A, respec-
a
tively i : {∗} → A the maps given by i (∗) = a, i (∗) = a(cid:48). Then we also have
a(cid:48) a a(cid:48)
f◦i = f◦i and since f is a monomorphism we obtain i = i . Therefore a = a(cid:48)
a a(cid:48) a a(cid:48)
and f is indeed injective.
Introduction to category theory 4 Ana Agore
1.2. SPECIAL OBJECTS AND MORPHISMS IN A CATEGORY
2) In the category Div of divisible2 groups, the quotient map q : Q → Q/Z is obviously
not injective but it is a monomorphism. Indeed, let G be another divisible group
and f, g : G → Q two morphisms of groups such that q ◦f = q ◦g. By denoting
h = f −g we obtain q◦h = 0. Now for any x ∈ G we have q(h(x)) = 0 and thus
h(x) ∈ Z. Suppose there exists some x ∈ G such that h(x ) (cid:54)= 0. We can assume
0 0
without loss of generality that h(x ) ∈ N∗. Since we work with divisible groups, we
0
can find some y ∈ G such that x = 2h(x )y . By applying h to the above equality
0 0 0 0
yields:
h(x ) = 2h(x )h(y )
0 0 0
which is an obvious contradiction since h(x) ∈ Z for all x ∈ G. Hence we get h = 0
which implies f = g and we proved that q is indeed a monomorphism in Div.
3) In the category Ringc of unitary commutative rings, the inclusion i : Z → Q is
obviously not surjective but it is an epimorphism. Indeed, let R be another ring
together with two ring morphisms f, g : Q → R such that f ◦i = g ◦i. Consider
now z ∈ Z∗; then we have 1 = f(1) = f(z)f(1/z) and therefore f(1/z) = 1/f(z).
Similarly we can prove that g(1/z) = 1/g(z) and since f and g coincide on Z we
get f(1/z) = g(1/z). Now for any z(cid:48) ∈ Z we have:
f(z(cid:48)/z) = f(z(cid:48))f(1/z) = g(z(cid:48))g(1/z) = g(z(cid:48)/z)
Therefore f = g which implies that i is an epimorphism in Ringc.
4) Itcanbeeasilyseenthattheinclusioni : Z → Qisamonomorphisminthecategory
Ringc ofunitarycommutativeringsandalsoanepimorphismbyExample1.2.3, 3).
Therefore, it provides an example of a morphism which is both a monomorphism
and an epimorphism but not an isomorphism.
5) Let (T,τ) be a topological space such that τ is different from the discrete topol-
ogy. Consider now the set T endowed with the discrete topology P(T). Then the
identity Id : (T,P(T)) → (T,τ) is obviously bijective and a continuous map be-
T
tween the two topological spaces. Therefore, Id is a morphism in Top but not an
T
isomorphism although it is a bijective map.
6) In the category PO(X, (cid:54)) associated a partially ordered set3 (X, (cid:54)), any isomor-
phism is an identity morphism. Indeed suppose f : x → y is an isomorphism; this
implies that x (cid:54) y. If g : y → x is the inverse of f then we also have y (cid:54) x. Due to
the antisymmetry of (cid:54) we obtain x = y. Therefore f : x → x must be the identity
on x.
2An abelian group (G, +) is called divisible if for every positive integer n and every g ∈ G, there
exists h∈G such that nh=g.
3A set X is called partially ordered if it is endowed with a binary relation (cid:54) which is reflexive,
antisymmetric and transitive.
Introduction to category theory 5 Ana Agore
CHAPTER 1. CATEGORIES AND FUNCTORS
Definition 1.2.4 Let C be a category.
1) We say that A ∈ ObC is an initial object if Hom (A,B) has exactly one element
C
for each B ∈ ObC;
2) We say that A ∈ ObC is a final object if Hom (C,A) has exactly one element for
C
each C ∈ ObC;
3) If A ∈ ObC is both an initial and a final object we say that A is a zero-object.
Proposition 1.2.5 If A and B are initial (final) objects in a category C then A is
isomorphic to B.
Proof: Since A is initial there exists a unique morphism f : A → B and a unique
morphism from A to A which must be the identity 1 . The same applies for B: there
A
exists a unique morphism g : B → A and a unique morphism from B to B, namely the
identity 1 . Now remark that g ◦f ∈ Hom (A,A) and thus g ◦f = 1 . Similarly we
B C A
get f ◦g = 1 and we proved that A and B are isomorphic. The statement about final
B
objects can be proved in a similar manner.
Examples 1.2.6 1) In the category Set of sets the initial object is the empty set
while the final objects are the singletons, i.e. the one-element sets {x}. Thus Set
has infinitely many final objects and they are all isomorphic;
2) The category Set of sets has no zero-objects. In the categories Grp of groups, Ab
of abelian groups, M of modules, {0} is a zero-object;
R
3) The category Field has neither an initial nor a final object since there are no
morphisms between fields of different characteristics;
4) Let (X, (cid:54)) be a poset and PO(X, (cid:54)) the associated category (see Example 1.1.4,
2)). Then PO(X, (cid:54)) has an initial object if and only if (X, (cid:54)) has a least element
(i.e. some element 0 ∈ X such that 0 ≤ x for any x ∈ X). Similarly, PO(X, (cid:54))
has a final object if and only if (X, (cid:54)) has a greatest element (i.e. some element
1 ∈ X such that x ≤ 1 for any x ∈ X).
1.3 Some constructions of categories
In this section we provide several methods of constructing new categories.
Definition 1.3.1 Given C a category, the dual (opposite) category of C, denoted by Cop,
is obtained as follows:
i) ObCop = ObC;
Introduction to category theory 6 Ana Agore
1.3. SOME CONSTRUCTIONS OF CATEGORIES
ii) Hom (A,B) = Hom (B,A) (i.e. the morphisms of Cop are those of C written in
Cop C
the reverse direction; in order to avoid confusion we write fop : A → B for the
morphism of Cop corresponding to the morphism f : B → A of C);
iii) The composition map
Hom (A,B)×Hom (B,C) → Hom (A,C)
Cop Cop Cop
is defined as follows:
gop◦fop = (f ◦g)op, for all fop ∈ Hom (A,B), gop ∈ Hom (B,C);
Cop Cop
iv) The identities are the same as in C.
Example 1.3.2 Let PO(X, (cid:54)) be the category associated to the poset (X, (cid:54)). Then
PO(X, (cid:54))op = PO(X, (cid:62)), where (cid:62) is the pre-order on X defined as follows: x (cid:62) y if
and only if y (cid:54) x.
Theorem 1.3.3 (Duality principle) Suppose that a statement expressing the exis-
tence of some objects or some morphisms or the equality of some composites is valid
in every category. Then the dual statement (obtained by reversing the direction of every
arrow and replacing every composite f ◦g by the composite g◦f) is also valid in every
category.
Proof: If S denotes the given statement and Sop denotes the dual statement then prov-
ing the statement Sop in a category C is equivalent to proving the statement S in the
category Cop which is assumed to be valid.
Remarks 1.3.4 It is straightforward to see that:
1) fop : A → B is a monomorphism (resp. an epimorphism) in Cop if and only if
f : B → A is an epimorphism (resp. a monomorphism ) in C;
2) C is an initial object (resp. a final object) in Cop if and only if C is a final object
(resp. an initial object) in C.
Definition 1.3.5 Let C and D be two categories. We define the product category C×D
in the following manner:
(cid:0) (cid:1)
i) Ob C ×D = ObC ×ObD, i.e. the objects of C ×D are pairs of the form (C, D)
with C ∈ ObC and D ∈ ObD;
ii) Hom (cid:0)(C,D), (C(cid:48), D(cid:48))(cid:1)= Hom (C,C(cid:48))×Hom (D,D(cid:48));
C×D C D
Introduction to category theory 7 Ana Agore
