ńņš. 6 |

it to deļ¬ne subobjects, which correspond to subsets of a set.

Monomorphisms and epimorphisms

An arrow f : A ā’ B is a monomorphism (or just a āmonoā, adjective

ā’

āmonicā), if f (i.e., Hom(T, f )) is injective (one to one) on elements deļ¬ned on

each object T ā”in other words, for every pair x, y of elements of A deļ¬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 ā—¦ x = f ā—¦ y,

then x = y. This has a dual concept: The arrow f is an epimorphism (āan

epiā, āepicā) 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 Hausdorļ¬ spaces and continuous maps.

An arrow f : A ā’ B which is āsurjective on elementsā, in other words for

ā’

which Hom(T, f ) is surjective for every object T , is necessarily an epimorphism

and is called a split epimorphism. An equivalent deļ¬nition is that there is an

arrow g: B ā’ A which is a right inverse to f , so that f ā—¦ g = idB . The Axiom of

ā’

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 Hausdorļ¬ spaces

as a category in which there are nonsurjective epimorphisms; another example

is the embedding of the ring of integers in the ļ¬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 āretractionā 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:

)ā’ā’

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

phism is

ā’ā’

ā’

However in this book we reserve that latter notation for regular epimorphisms to

be deļ¬ned in 1.8.

Subobjects

We now deļ¬ne the notion of subobject of an object in a category; this idea

partly captures and partly generalizes the concept of āsubsetā, āsubspaceā, and

so on, familiar in many branches of mathematics.

If i: A0 ā’ A is a monomorphism and a: T ā’ A, we say a factors through

ā’ ā’

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 Ā‚

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

āa āT A0 ā. The subscript A is often omitted if the context makes it clear.

A

Lemma 1. Let i: A0 ā’ A and i : A0 ā’ A be monomorphisms in a category C .

ā’ ā’

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 ā’ A for which

ā’

A0

d

di

j (2)

d

c Ā‚

d

EA

A0

i

commutes.

Proof. Suppose A0 and A0 have the same elements of A. Since i āA0 A0 , it

A

factors through A0 , so there is an arrow j: A0 ā’ A0 such that (2) commutes.

ā’

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

ā’

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 āT A0 , so that

A

T

a = i ā—¦ u for some u: T ā’ A0 , then a = i ā—¦ j ā—¦ u so that a āA A0 . A similar

ā’

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 āLet A0

)ā’ A be a subobject of Aā.

ā’

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 āsubobjectā captures the notion of āsubsetā 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 āthe terminal objectā.

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 ā’ Set denote the contravariant functor which takes each open set U

ā’

to the set of real-valued continuous functions deļ¬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 deļ¬ned on V to U . An important point

about these restriction maps is that they are not in general surjectiveā”that is,

there are in general functions deļ¬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 deļ¬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 Hausdorļ¬ spaces and continuous maps.

8. Show that the embedding of an integral domain (assumed commutative with

unity) into its ļ¬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 ā’ B in a

ā’

category C are equivalent:

(i) Hom(T, f ) is surjective for every object T of C .

(ii) There is an arrow g: B ā’ A such that f ā—¦ g = idB .

ā’

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 ā’ Set be a faithful functor and f an arrow of C . (Note that the

ā’

functors we have called āforgetfulāā”we have not deļ¬ned that word formallyā”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 ā’ Set is a functor G with the properties

ā’

(a) GA ā F A for every object A of C .

(b) If f : A ā’ B, then F (f ) restricted to G(A) is equal to G(f ).

ā’

Show that the subfunctors of a functor are the āsameā 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 ā’ Set is an object in the functor category Func(C , Set):

ā’

an āelementā of F is therefore a natural transformation into F . The Yoneda

Lemma, Lemma 1 below, says in eļ¬ect that the elements of a Set-valued functor

F deļ¬ned (in the sense of Section 1.4) on the homfunctor Hom(A, ā’) 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 ā’ B in C , then f induces a natural transformation from Hom(B, ā’)

ā’

to Hom(A, ā’) by composition: the component of this natural transformation at

an object C of C takes an arrow h: B ā’ C to h ā—¦ f : A ā’ C. This construction

ā’ ā’

deļ¬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(ā’, ā’) is contravariant in the ļ¬rst variable (it is a special case

of Hom), the map which takes an object B of C and a functor F : C ā’ Set ā’

to Nat(Hom(B, ā’), F ) is a functor from C Ć— Func(C , Set) to Set. Another such

functor is the evaluation functor which takes (B, F ) to F B, and (g, Ī»), where g: B

ā’ A ā C and Ī»: F ā’ G is a natural transformation, to Gg ā—¦ Ī»B. Remarkably,

ā’ ā’

these two functors are naturally isomorphic; it is in this sense that the elements

of F deļ¬ned on Hom(B, ā’) are the ordinary elements of F B.

Lemma 1. [Yoneda] The map Ļ: Nat(Hom(B, ā’), F ) ā’ F B deļ¬ned by Ļ(Ī») =

ā’

Ī»B(idB ) is a natural isomorphism of the functors deļ¬ned in the preceding para-

graph.

Proof. The inverse of Ļ takes an element u of F B to the natural transformation

Ī» deļ¬ned by requiring that Ī»A(g) = F g(u) for g ā Hom(B, A). The rest of proof

is a routine veriļ¬cation of the commutativity of various diagrams required by the

deļ¬nitions.

The ļ¬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, ā’), where A is an object of C ; this results in the statement that there is

a natural bijection between arrows g: A ā’ B and natural transformations from

ā’

Hom(B, ā’) to Hom(A, ā’).

1.5 The Yoneda Lemma 27

Theorem 2. [Yoneda Embeddings]

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

ā’

Hom(B, ā’) ā’ Hom(A, ā’)

ā’

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

2. The map taking f to the natural transformation

Hom(ā’, A) ā’ Hom(ā’, B)

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