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In the rest of this section we will develop the idea of element further and use
it to de¬ne subobjects, which correspond to subsets of a set.

Monomorphisms and epimorphisms

An arrow f : A ’ B is a monomorphism (or just a “mono”, adjective

“monic”), if f (i.e., Hom(T, f )) is injective (one to one) on elements de¬ned on
each object T ”in other words, for every pair x, y of elements of A de¬ned on T ,
f (x) = f (y) implies x = y.
In terms of composition, this says that f is left cancelable, i.e, if f —¦ x = f —¦ y,
then x = y. This has a dual concept: The arrow f is an epimorphism (“an
epi”, “epic”) if it is right cancelable. This is true if and only if the contravariant
functor Hom(f, T ) is injective (not surjective!) for every object T . Note that
surjectivity is not readily described in terms of generalized elements.
In Set, every monic is injective and every epic is surjective (onto). The same
is true of Grp, but the fact that epis are surjective in Grp is moderately hard to
prove (Exercise 6). On the other hand, any dense map, surjective or not, is epi
in the category of Hausdor¬ spaces and continuous maps.
An arrow f : A ’ B which is “surjective on elements”, in other words for

which Hom(T, f ) is surjective for every object T , is necessarily an epimorphism
and is called a split epimorphism. An equivalent de¬nition is that there is an
arrow g: B ’ A which is a right inverse to f , so that f —¦ g = idB . The Axiom of

Choice is equivalent to the statement that every epi in Set is split. In general, in
categories of sets with structure and structure preserving functions, split epis are
surjective and (as already pointed out) surjective maps are epic (see Exercise 11),
22 1 Categories
but the converses often do not hold. We have already mentioned Hausdor¬ spaces
as a category in which there are nonsurjective epimorphisms; another example
is the embedding of the ring of integers in the ¬eld of rational numbers in the
category of rings and ring homomorphisms. As for the other converse, in the
category of groups the (unique) surjective homomorphism from the cyclic group
of order 4 to the cyclic group of order 2 is an epimorphism which is not split.
An arrow with a left inverse is necessarily a monomorphism and is called a
split monomorphism. Split monos in T are called retractions; in fact the
op
word “retraction” is sometimes used to denote a split mono in any category.
The property of being a split mono or split epi is necessarily preserved by any
functor. The property of being monic or epic is certainly not in general preserved
by any functor. Indeed, if F f is epi for every functor F , then f is necessarily a
split epi. (Exercise 10.)
Notation: In diagrams, we usually draw an arrow with an arrowhead at its
tail:

)’’
to indicate that it is a monomorphism. The usual dual notation for an epimor-
phism is

’’

However in this book we reserve that latter notation for regular epimorphisms to
be de¬ned in 1.8.

Subobjects

We now de¬ne the notion of subobject of an object in a category; this idea
partly captures and partly generalizes the concept of “subset”, “subspace”, and
so on, familiar in many branches of mathematics.
If i: A0 ’ A is a monomorphism and a: T ’ A, we say a factors through
’ ’
i (or factors through A0 if it is clear which monomorphism i is meant) if there is
an arrow j for which
T
d
da
j (1)
d
c ‚
d
EA
A0
i
1.4 Elements and Subobjects 23
commutes. In this situation we extend the element point of view and say that
the element a of A is an element of A0 (or of i if necessary). This is written
“a ∈T A0 ”. The subscript A is often omitted if the context makes it clear.
A

Lemma 1. Let i: A0 ’ A and i : A0 ’ A be monomorphisms in a category C .
’ ’
Then A0 and A0 have the same elements of A if and only if they are isomorphic
in the category C /A of objects over A, in other words if and only if there is an
isomorphism j: A ’ A for which

A0
d
di
j (2)
d
c ‚
d
EA
A0
i
commutes.
Proof. Suppose A0 and A0 have the same elements of A. Since i ∈A0 A0 , it
A
factors through A0 , so there is an arrow j: A0 ’ A0 such that (2) commutes.

Interchanging A0 and A0 we get k: A0 ’ A0 such that i —¦ k = i . Using the fact

that i and i are monic, it is easy to see that j and k must be inverses to each
other, so they are isomorphisms.
Conversely, if j is an isomorphism making (2) commute and a ∈T A0 , so that
A
T
a = i —¦ u for some u: T ’ A0 , then a = i —¦ j —¦ u so that a ∈A A0 . A similar

argument interchanging A0 and A0 shows that A0 and A0 have the same elements
of A.
Two monomorphisms which have the same elements are said to be equiva-
lent. A subobject of A is an equivalence class of monomorphisms into A. We
will frequently refer to a subobject by naming one of its members, as in “Let A0
)’ A be a subobject of A”.

In Set, each subobject of a set A contains exactly one inclusion of a subset
into A, and the subobject consists of those injective maps into A which has that
subset as image. Thus “subobject” captures the notion of “subset” in Set exactly.
Any map from a terminal object in a category is a monomorphism and so
determines a subobject of its target. Because any two terminal objects are iso-
morphic by a unique isomorphism (Exercise 6 of Section 1.1), that subobject
contains exactly one map based on each terminal object. We will henceforth
assume that in any category we deal with, we have picked a particular terminal
object (if it has one) as the canonical one and call it “the terminal object”.
24 1 Categories
Global elements

In the category of sets, an element in the ordinary sense of a set B is essentially
the same thing as an arrow from the terminal object of Set to B. In general,
an arrow in some category from the terminal object to some object is called a
global element of that object, for reasons which will become apparent in the
next paragraph. In most categories which arise in practice, except Set, an object
is not determined by its global elements. For example, in Grp, each group has
exactly one global element.
A more interesting example arises in connection with continuous functions.
This example is worth studying in detail because it illustrates and motivates
much of sheaf theory. Let A be a topological space and let R denote the set of
real numbers. Let O(A) denote the category whose objects are the open sets of
A and whose arrows are the inclusion maps of one open set into another. Let
C: O(A)op ’ Set denote the contravariant functor which takes each open set U

to the set of real-valued continuous functions de¬ned on U , and to each inclusion
of an open set U of A into an open set V associates the map from C(V ) to C(U )
which restricts a continuous function de¬ned on V to U . An important point
about these restriction maps is that they are not in general surjective”that is,
there are in general functions de¬ned on an open set which cannot be extended
to a bigger open set. Think of f (x) = 1/x, for example.
This functor C is an object in the category F = Func(O(A)op , Set). The
terminal object of F is the functor which associates a singleton set to each open
set of A and the only possible map to each arrow (inclusion map) of O(A). It
is a nice exercise to prove that a global element of C is precisely a continuous
real-valued function de¬ned on all of A.

Exercises 1.4

1. Describe initial objects using the terminology of elements, and using the
terminology of indexed families of subsets.

2. Show that in Set, a function is injective if and only if it is a monomorphism
and surjective if and only if it is an epimorphism.

3. Show that every epimorphism in Set is split. (This is the Axiom of Choice.)

4. Show that in the category of Abelian groups and group homomorphisms, a
homomorphism is injective if and only if it is a monomorphism and surjective if
and only if it is an epimorphism.
1.4 Elements and Subobjects 25
5. Show that neither monos nor epis are necessarily split in the category of
Abelian groups.

6. Show that in Grp, every homomorphism is injective if and only if it is a
monomorphism and surjective if and only if it is an epimorphism. (If you get
stuck trying to show that an epimorphism in Grp is surjective, see the hint on
p.21 of Mac Lane [1971].)

7. Show that all epimorphisms are surjective in T but not in the category of
op,
all Hausdor¬ spaces and continuous maps.

8. Show that the embedding of an integral domain (assumed commutative with
unity) into its ¬eld of quotients is an epimorphism in the category of commutative
rings and ring homomorphisms. When is it a split epimorphism?

9. Show that the following two statements about an arrow f : A ’ B in a

category C are equivalent:

(i) Hom(T, f ) is surjective for every object T of C .
(ii) There is an arrow g: B ’ A such that f —¦ g = idB .


Furthermore, show that any arrow satisfying these conditions is an epimor-
phism.

10. Show that if F f is epi for every functor F , then f is a split epi.

11. Let U : C ’ Set be a faithful functor and f an arrow of C . (Note that the

functors we have called “forgetful””we have not de¬ned that word formally”are
obviously faithful.) Prove:
(a) If U f is surjective then f is an epimorphism.
(b) If f is a split epimorphism then U f is surjective.
(c) If U f is injective then f is a monomorphism.
(d) If f is a split monomorphism, then U f is injective.

12. A subfunctor of a functor F : C ’ Set is a functor G with the properties

(a) GA ⊆ F A for every object A of C .
(b) If f : A ’ B, then F (f ) restricted to G(A) is equal to G(f ).

Show that the subfunctors of a functor are the “same” as subobjects of the
functor in the category Func(C , Set).
26 1 Categories
1.5 The Yoneda Lemma
Elements of a functor

A functor F : C ’ Set is an object in the functor category Func(C , Set):

an “element” of F is therefore a natural transformation into F . The Yoneda
Lemma, Lemma 1 below, says in e¬ect that the elements of a Set-valued functor
F de¬ned (in the sense of Section 1.4) on the homfunctor Hom(A, ’) for some
object A of C are essentially the same as the (ordinary) elements of the set F A.
To state this properly requires a bit of machinery.
If f : A ’ B in C , then f induces a natural transformation from Hom(B, ’)

to Hom(A, ’) by composition: the component of this natural transformation at
an object C of C takes an arrow h: B ’ C to h —¦ f : A ’ C. This construction
’ ’
de¬nes a contravariant functor from C to Func(C , Set) called the Yoneda map
or Yoneda embedding. It is straightforward and very much worthwhile to
check that this construction really does give a natural transformation for each
arrow f and that the resulting Yoneda map really is a functor.
Because Nat(’, ’) is contravariant in the ¬rst variable (it is a special case
of Hom), the map which takes an object B of C and a functor F : C ’ Set ’
to Nat(Hom(B, ’), F ) is a functor from C — Func(C , Set) to Set. Another such
functor is the evaluation functor which takes (B, F ) to F B, and (g, »), where g: B
’ A ∈ C and »: F ’ G is a natural transformation, to Gg —¦ »B. Remarkably,
’ ’
these two functors are naturally isomorphic; it is in this sense that the elements
of F de¬ned on Hom(B, ’) are the ordinary elements of F B.
Lemma 1. [Yoneda] The map φ: Nat(Hom(B, ’), F ) ’ F B de¬ned by φ(») =

»B(idB ) is a natural isomorphism of the functors de¬ned in the preceding para-
graph.

Proof. The inverse of φ takes an element u of F B to the natural transformation
» de¬ned by requiring that »A(g) = F g(u) for g ∈ Hom(B, A). The rest of proof
is a routine veri¬cation of the commutativity of various diagrams required by the
de¬nitions.
The ¬rst of several important consequences of this lemma is the following
embedding theorem. This theorem is obtained by taking F in the Lemma to be
Hom(A, ’), where A is an object of C ; this results in the statement that there is
a natural bijection between arrows g: A ’ B and natural transformations from

Hom(B, ’) to Hom(A, ’).
1.5 The Yoneda Lemma 27
Theorem 2. [Yoneda Embeddings]

1. The map which takes f : A ’ B to the induced natural transformation


Hom(B, ’) ’ Hom(A, ’)


is a full and faithful contravariant functor from C to Func(C , Set).
2. The map taking f to the natural transformation

Hom(’, A) ’ Hom(’, B)

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