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ā’

is a full and faithful functor from C to Func(C op , Set).

Proof. It is easy to verify that the maps deļ¬ned in the Theorem are functors.
The fact that the ļ¬rst one is full and faithful follows from the Yoneda Lemma
with Hom(A, ā’) in place of F . The other proof is dual.
The induced maps in the Theorem deserve to be spelled out. If f : S ā’ ā’
T , the natural transformation corresponding to f given by (i) has component
Hom(f, A): Hom(T, A) ā’ Hom(S, A) at an object A of C ā”this is composing by
ā’
T
f on the right. If x ā A, the action of Hom(f, A) āchanges the parameterā in
A along f .
The other natural transformation corresponding to f is Hom(T, f ): Hom(T, A)
ā’ Hom(T, B); since the Yoneda embedding is faithful, we can say that f is
ā’
essentially the same as Hom(ā’, f ). If x is an element of A based on T , then
Hom(T, f )(x) = f ā—¦ x. Since āf is essentially the same as Hom(ā’, f )ā, this
justiļ¬es the notation f (x) for f ā—¦ x introduced in Section 1.4.
The fact that the Yoneda embedding is full means that any natural transfor-
mation Hom(ā’, A) ā’ Hom(ā’, B) determines a morphism f : A ā’ B, namely
ā’ ā’
the image of idA under the component of the transformation at A. Spelled out,
this says that if f is any function which assigns to every element x: T ā’ A an
ā’
element f (x): T ā’ B with the property that for all t: S ā’ T , f (x ā—¦ t) = f (x) ā—¦ t
ā’ ā’
(this is the ācoherence conditionā mentioned in Section 1.4) then f āisā (via the
Yoneda embedding) a morphism, also called f to conform to our conventions,
from A to B. One says such an arrow exists āby Yonedaā.
In the same vein, if g: 1 ā’ A is a morphism of C , then for any object T , g
ā’
determines an element g( ) of A deļ¬ned on T by composition with the unique
element from T to 1, which we denote ( ). This notation captures the perception
that a global element depends on no arguments. We will extend the functional
notation to more than one variable in Section 1.7.
28 1 Categories
Universal elements

Another special case of the Yoneda Lemma occurs when one of the elements
of F deļ¬ned on Hom(A, ā’) is a natural isomorphism. If Ī²: Hom(A, ā’) ā’ F is ā’
such a natural isomorphism, the (ordinary) element u ā F A corresponding to it
is called a universal element for F , and F is called a representable functor,
represented by A. It is not hard to see that if F is also represented by A , then
A and A are isomorphic objects of C . (See Exercise 4, which actually says more
than that.)
The following lemma gives a characterization of universal elements which in
many books is given as the deļ¬nition.
Lemma 3. Let F : C ā’ Set be a functor. Then u ā F A is a universal element
ā’
for F if and only if for every object B of C and every element t ā F B there is
exactly one arrow g: A ā’ B such that F g(u) = t.
ā’
Proof. If u is such a universal element corresponding to a natural isomorphism
Ī²: Hom(A, ā’) ā’ F , and t ā F B, then the required arrow g is the element
ā’
ā’1
(Ī² B)(t) in Hom(A, B). Conversely, if u ā F A satisļ¬es the conclusion of the
Lemma, then it corresponds to some natural transformation Ī²: Hom(A, ā’) ā’ Fā’
by the Yoneda Lemma. It is routine to verify that the map which takes t ā F B
to the arrow g ā Hom(A, B) given by the assumption constitutes an inverse in
Func(C , Set) to Ī²B.
In this book, the phrase āu ā F A is a universal element for F ā carries with
it the implication that u and A have the property of the lemma. (It is possible
that u is also an element of F B for some object B but not a universal element
in F B.)
As an example, let G be a free group on one generator g. Then g is the
āuniversal group elementā in the sense that it is a universal element for the
underlying set functor U : Grp ā’ Set (more precisely, it is a universal element
ā’
in U G). This translates into the statement that for any element x in any group
H there is a unique group homomorphism F : G ā’ H taking g to x, which is
ā’
exactly the deļ¬nition of āfree group on one generator gā.
Another example which will play an important role in this book concerns
the contravariant powerset functor P: Set ā’ Set deļ¬ned in Section 1.2. It is
ā’
straightforward to verify that a universal element for P is the subset {1} of
the set {0, 1}; the function required by the Lemma for a subset B0 of a set
B is the characteristic function of B0 . (A universal element for a contravariant
functor, as hereā”meaning a universal element for P: Setop ā’ Setā”is often called
ā’
a ācouniversal elementā.)
1.6 Pullbacks 29
Exercises 1.5

1. Find a universal element for the functor

Hom(ā’, A) Ć— Hom(ā’, B): Setop ā’ Set
ā’

for any two sets A and B. (If h: U ā’ V , this functor takes a pair (f, g) to
ā’
(h ā—¦ f, h ā—¦ g).)

2. (a) Show that an action of a group G on a set A is essentially the same thing
as a functor from G regarded as a category to Set.
(b) Show that such an action has a universal element if and only if for any
pair x and y of elements of A there is exactly one element g of G for which gx = y.

3. Are either of the covariant powerset functors deļ¬ned in Exercise 6 of Sec-
tion 1.2 representable?

4. Let F : C ā’ Set be a functor and u ā F A, u ā F A be universal elements for
ā’
F . Show that there is a unique isomorphism Ļ: A ā’ A such that F Ļ(u) = u .
ā’

5. Let U : Grp ā’ Set be the underlying set functor, and F : Set ā’ Grp the
ā’ ā’
functor which takes a set A to the free group on A. Show that for any set A, the
covariant functor HomSet (A, U (ā’)) is represented by F A, and for any group G,
the contravariant functor HomGrp (F (ā’), G) is represented by U G.

1.6 Pullbacks
The set P of composable pairs of arrows used in Section 1.1 in the alternate
deļ¬nition of category is an example of a āļ¬bered productā or āpullbackā. A
pullback is a special case of ālimitā, which we treat in Section 1.7. In this section,
we discuss pullbacks in detail.
Let us consider the following diagram D in a category C .
B

g
(1)
c
EC
A
f
We would like to objectify the set {(x, y) | f (x) = g(y)} in C ; that is, ļ¬nd
an object of C whose elements are those pairs (x, y) with f (x) = g(y). Observe
30 1 Categories
that for a pair (x, y) to be in this set, x and y must be elements of A and B
respectively deļ¬ned over the same object T .
The set of composable pairs of arrows in a category (see Section 1.1) are a
special case in Set of this, with A = B being the set of arrows and f = d0 , g = d1 .
Thus we must consider commutative diagrams like
yE
T B

g
x (2)
c c
EC
A
f

In this situation, (T, x, y) is called a commutative cone over D based on
T , and the set of commutative cones over D based on T is denoted Cone(T, D) .
A commutative cone based on T over D may usefully be regarded as an element
of D deļ¬ned on T . In Section 1.7, we will see that a commutative cone is actually
an arrow in a certain category, so that this idea ļ¬ts with our usage of the word
āelementā.
Our strategy will be to turn Cone(ā’, D) into a functor; then we will say that
an object represents (in an informal sense) elements of D, in other words pairs
(x, y) for which f (x) = g(y), if that object represents (in the precise technical
sense) the functor Cone(ā’, D).
We make will make Cone(ā’, D) into a contravariant functor to Set: If h: W
ā’ T is an arrow of C and (T, x, y) is a commutative cone over (1), then
ā’

Cone(h, D)(T, x, y) = (W, x ā—¦ h, y ā—¦ h)
which it is easy to see is a commutative cone over D based on W .
An element (P, p1 , p2 ) of D which is a universal element for Cone(ā’, D) (so
that Cone(ā’, D) is representable) is called the pullback or the ļ¬ber product
of the diagram D. The object P is often called the pullback, with p1 and p2
understood. As the reader can verify, this says that (P, p1 , p2 ) is a pullback if
p2 E
P B

p1 g (3)
c c
EC
A
f
1.6 Pullbacks 31
commutes and for any element of D based on T , there is a unique element of P
based on T which makes
T
r
ed rr
ry
ed
rr
ed
prE
edĀ‚ j
r
2
xe P B (4)
e
p1 g
e
e
ec
Ā… c
EC
A
f
commute. Thus there is a bijection between the elements of the diagram D
deļ¬ned on T and the elements of the ļ¬ber product P deļ¬ned on T . When a
diagram like 4 has this property it is called a pullback diagram.
The Cone functor exists for any category, but a particular diagram of the
form 1 need not have a pullback.
Proposition 1. If Diagram 3 is a pullback diagram, then the cone in Diagram 2
is also a pullback of Diagram 1 if and only if the unique arrow from T to P making
everything in Diagram 4 commute is an isomorphism.
Proof. (This theorem actually follows from Exercise 4 of Section 1.5, but we
believe a direct proof is instructive.) Assume that (2) and (3) are both pullback
diagrams. Let u: T ā’ P be the unique arrow given because 3 is a pullback
ā’
diagram, and let v: P ā’ T be the unique arrow given because 2 is a pullback
ā’
diagram. Then both for g = u ā—¦ v: P ā’ P and g = idP it is true that p1 ā—¦ g = p1
ā’
and p2 ā—¦ g = p2 . Therefore by the uniqueness part of the deļ¬nition of universal
element, u ā—¦ v = idP . Similarly, v ā—¦ u = idT , so that u is an isomorphism between
T and P making everything commute. The converse is easy.
The preceding argument is typical of many arguments making use of the
uniqueness part of the deļ¬nition of universal element. We will usually leave
arguments like this to the reader.
A consequence of Proposition 1 is that a pullback of a diagram in a category
is not determined uniquely but only up to a āunique isomorphism which makes
everything commuteā. This is an instance of a general fact about constructions
deļ¬ned as universal elements which is made precise in Proposition 1 of Section 1.7.
32 1 Categories
Notation for pullbacks

We have deļ¬ned the pullback P of Diagram 1 so that it objectiļ¬es the set
{(x, y) | f (x) = g(y)}. This ļ¬ts nicely with the situation in Set, where one
pullback of (1) is the set {(x, y) | f (x) = g(y)} together with the projection
maps to A and B, and any other pullback is in one to one correspondence with
this one by a bijection which commutes with the projections. This suggest the
introduction of a setlike notation for pullbacks: We let [(x, y) | f (x) = g(y)]
denote a pullback of (1). In this notation, f (x) denotes f ā—¦ x and g(y) denotes g ā—¦ y
as in Section 1.4, and (x, y) denotes the unique element of P deļ¬ned on T which
exists by deļ¬nition of pullback. It follows that p1 (x, y) = x and p2 (x, y) = y,
where we write p1 (x, y) (not p1 ((x, y))) for p1 ā—¦ (x, y).
The idea is that square brackets around a set deļ¬nition denotes an object of
the category which represents the set of arrows listed in curly bracketsā”ārepre-
sentsā in the technical sense, so that the set in curly brackets has to be turned
into the object map of a set-valued functor. The square bracket notation is
ambiguous. Proposition 1 spells out the ambiguity precisely.
We could have deļ¬ned a commutative cone over (1) in terms of three arrows,
namely a cone (T, x, y, z) based on T would have x: T ā’ A, y: T ā’ B and z: T
ā’ ā’
ā’ C such that f ā—¦ x = g ā—¦ y = z. Of course, z is redundant and in consequence
ā’
the Cone functor deļ¬ned this way would be naturally isomorphic to the Cone
functor deļ¬ned above, and so would have the same universal elements. (The
component of the natural isomorphism at T takes (T, x, y) to (T, x, y, f ā—¦ x)).
Thus the pullback of (1) also represents the set {(x, y, z) | f (x) = g(y) = z},
and so could be denoted [(x, y, z) | f (x) = g(y) = z]. Although this observation
is inconsequential here, it will become more signiļ¬cant when we discuss more
general constructions (limits) deļ¬ned by cones.
There is another way to construct a pullback in Set when the map g is monic.
In general, when g is monic, {(x, y) | f (x) = g(y)} ā¼ {x | f (x) ā g(B)}, which
=
ā’1
in Set is often denoted f (B). In general, a pullback along a subobject can be
interpreted as an inverse image which as we will see is again a subobject.
The pullback Diagram 3 is often regarded as a sort of generalized inverse image
construction even when g is not monic. In this case, it is called the āpullback of
g along f ā. Thus when P is regarded as the ļ¬ber product, the notion of pullback
is symmetrical in A and B, but when it is regarded as the generalized inverse
image of B then the diagram is thought of as asymmetrical.
A common notation for the pullback of (1) reļ¬‚ecting the perception of a
pullback as ļ¬ber product is āA Ć—C Bā.
1.6 Pullbacks 33
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