EXERCISES IN CATEGORY THEORY 5
1. Equalisers and coequalisers, pullbacks and pushouts
1.1. Some questions about Set.
(1) Let f : X → X be a function. Can you describe Fix(f) = {x ∈ X : f(x) = x} as the
equaliser in Set of f and another function?
(2) Let U, V ⊆ X be subsets of X and consider the inclusions i : U → X and j : V → X.
Show that the intersection of U and V is the pullback of i and j:
U ∩ V
p
//
q

U
i

V
j
// X
Can you find a nice description of the pushout of p and q as a subset of X too?
(3) Let U ⊂ Y and f : X → Y . Describe the preimage f−1
(U) = {x ∈ X : f(x) ∈ U} as
a pullback.
1.2. General categorical questions.
(1) Show that a category with a terminal object and pullbacks also has products.
(2) In a general category, prove that an arrow f : A → B is mono if and only if the square
A
1 //
1

A
f

A
f
// B
(3) Let f, g : A B have coequaliser h : B → C. Show that h is an epi.
1.3. Quotients and coequalisers.
(1) Consider an equivalence relation E on X, denoted E = {(x, y) ∈ X2
: xEy}. Then
we have projections p, q : E X where p(x, y) = x and q(x, y) = y. Show that the
coequaliser of p and q is X → X/E where X/E is the set of equivalence classes of E.
(2) Let C be a category with pullbacks and coequalisers. Given f : A → B consider the
pullback of f with itself:
K(f)
p
//
q

A
f

K(f)
p
//
q
// A
g
>>>>>>>>
f
// B
A
f
// V C
h
??
and then the coequaliser g : A → C of p and q. Show that there exist a unique arrow
h : C → B such that hg = f as drawn above.
Show that in Set the above factorisation agrees with the factorisation of a function
through its image:
X
g
// im(f)
h // Y
Date: October 15, 2014.
1
2 EXERCISES IN CATEGORY THEORY 5
where im(f) = {y ∈ Y : ∃x such that fx = y}.
(You may find it helpful to use that K(f) = {(x, y) : fx = fy} is an equivalence
relation and use the previous question).
(3) If you are enthusiastic, show that in the category of monoids/groups/vector spaces,
the same factorisation coincides with the factorisation of a homomorphism through
its image.
(4) Let f : G → H be a group homomorphism. Show that the kernel of f is the equaliser
ker(f) // A
f
//
0
// B
where 0 : G → H is the homomorphism sending every element to the unit.
Given a group G and normal subgroup N describe the quotient group G/N as a
coequaliser in a similar way.
1.4. Coequalisers and gluing topological spaces. These exercises concern the relationship
between coequalisers, pushouts and gluing in topology.
(1) Let 1 = {∗} be the 1-point space and consider the real interval [0, 1]. There are two
continuous maps f, g : 1 [0, 1] given by f(∗) = 0 and g(∗) = 1 respectively. Show
that the coequaliser in the category of topological spaces of f and g is the circle S1
.
(2) Can you describe the cylinder S1
× [0, 1] as a coequaliser? How about the Mobius
strip or more complex spaces?
Can you describe the 2-dimensional sphere S2
= {(x, y, z) : x2
+ y2
+ z2
= 1} as a
pushout by gluing two disks along their boundary?