ńņš. 3 |

and adjoints. (Limits and adjoints are deļ¬ned later in this chapter.) By the use

of this technique, category theory has made mathematically precise the unity of a

variety of concepts in diļ¬erent branches of mathematics, such as the many prod-

uct constructions which occur all over mathematics (described in Section 1.7) or

the ubiquitous concept of isomorphism, discussed below. Besides explicating the

unity of concepts, categorical techniques for deļ¬ning concepts without mention-

ing elements have enabled mathematicians to provide a useful axiomatic basis for

algebraic topology, homological algebra and other theories.

Despite the possibility of giving element-free deļ¬nitions of these constructions,

it remains intuitively helpful to think of them as being deļ¬ned with elements.

Fortunately, this can be done: In Section 1.4, we introduce a more general notion

of element of an object in a category (more general even when the category is

Set) which in many circumstances makes categorical deļ¬nitions resemble familiar

deļ¬nitions involving elements of sets, and which also provides an explication of

the old notion of variable quantity.

Isomorphisms and terminal objects

The notion of isomorphism can be given an element-free deļ¬nition for any

category: An arrow f : A ā’ B in a category is an isomorphism if it has an

ā’

inverse, namely an arrow g: B ā’ A for which f ā—¦ g = idB and g ā—¦ f = idA . In

ā’

other words, both triangles of the following diagram must commute:

fE

A B

Ā

gĀ

id id

Ā

c Ā

Ā© c

EB

A

f

In a group regarded as a category, every arrow is invertible, whereas in a poset

regarded as a category, the only invertible arrows are the identity arrows (which

are invertible in any category).

It is easy to check that an isomorphism in Grp is what is usually called an

isomorphism (commonly deļ¬ned as a bijective homomorphism, but some newer

texts give the deļ¬nition above). An isomorphism in Set is a bijective function,

and an isomorphism in T is a homeomorphism.

op

Singleton sets in Set can be characterized without mentioning elements, too.

A terminal object in a category C is an object T with the property that for

every object A of C there is exactly one arrow from A to T . It is easy to see

1.1 Deļ¬nition of category 5

that terminal objects in Set, T and Grp are all one element sets with the only

op,

possible structure in the case of the last two categories.

Duality

If C is a category, then we deļ¬ne C op to be the category with the same objects

and arrows as C , but an arrow f : A ā’ B in C is regarded as an arrow from B

ā’

op

to A in C . In other words, for all objects A and B of C ,

HomC (A, B) = HomC op (B, A)

If f : A ā’ B and g: B ā’ C in C , then the composite f ā—¦ g in C op is by deļ¬nition

ā’ ā’

the composite g ā—¦ f in C . The category C op is called the opposite category of

C.

If P is a property that objects or arrows in a category may have, then the

dual of P is the property of having P in the opposite category. As an example,

consider the property of being a terminal object. If an object A of a category C

is a terminal object in C op , then HomC (B, A) has exactly one arrow for every

object B of C . Thus the dual property of being a terminal object is the property:

Hom(A, B) has exactly one arrow for each object B. An object A with this

property is called an initial object. In Set and T the empty set is the initial

op,

object (see āFine pointsā on page 7). In Grp, on the other hand, the one-element

group is both an initial and a terminal object.

Clearly if property P is dual to property Q then property Q is dual to property

P. Thus being an initial object and being a terminal object are dual properties.

Observe that being an isomorphism is a self-dual property.

Constructions may also have duals. For example, the dual to the category of

objects over A is the category of objects under A. An object is an arrow from

A and an arrow from the object f : A ā’ B to the object g: A ā’ C is an arrow

ā’ ā’

h from B to C for which h ā—¦ f = g.

Often a property and its dual each have their own names; when they donā™t

(and sometimes when they do) the dual property is named by preļ¬xing āco-ā.

For example, one could, and some sources do, call an initial object ācoterminalā,

or a terminal object ācoinitialā.

Deļ¬nition of category by commutative diagrams

The notion of category itself can be deļ¬ned in an element-free way. We

describe the idea behind this alternative deļ¬nition here, but some of the sets we

6 1 Categories

construct are deļ¬ned in terms of elements. In Section 1.6, we show how to deļ¬ne

these sets without mentioning elements (by pullback diagrams).

Before giving the deļ¬nition, we mention several notational conventions that

will recur throughout the book.

1. If X and Y are sets, p1 : X Ć—Y ā’ X and p2 : X Ć—Y ā’ Y are the coordinate

ā’ ā’

projections.

2. If X, Y and Z are sets and f : X ā’ Y , g: X ā’ Z are functions,

ā’ ā’

(f, g): X ā’ Y Ć— Z

ā’

is the function whose value at a ā X is (f (a), g(a)).

3. If X, Y , Z, and W are sets and f : X ā’ Z, g: Y ā’ W are functions, then

ā’ ā’

f Ć— g: X Ć— Y ā’ Z Ć— W

ā’

is the function whose value at (a, b) is (f (a), g(b)). This notation is also

used for maps deļ¬ned on subsets of product sets (as in 4 below).

A category consists of two sets A and O and four functions d0 , d1 : A ā’ O,

ā’

u: O ā’ A and m: P ā’ A, where P is the set

ā’ ā’

{(f, g) | d0 (f ) = d1 (g)}

of composable pairs of arrows for which the following Diagrams 1ā“4 commute.

For example, the left diagram of 2 below says that d0 ā—¦ p1 = d0 ā—¦ m. We will treat

diagrams more formally in Section 1.7.

uE 'u

A O A

d Ā

(1)

d0d Ā d1

idO

d Ā

dc

Ā‚Ā Ā©

O

This says that the source and target of idX is X.

p2 E p1 E

P A P A

m m

d0 d1 (2)

c c c c

EO EO

A A

d0 d1

1.1 Deļ¬nition of category 7

This says that the source of f ā—¦ g is that of g and its target is that of f .

(u ā—¦ d0 , 1) E ' (1, u ā—¦ d1 )

A P A

d Ā

d Ā

d Ā (3)

m

idA d Ā idA

d Ā

d Ā

dc

Ā‚Ā Ā©

A

This characterizes the left and right identity laws.

In the next diagram, Q is the set of composable triples of arrows:

Q = {(f, g, h) | d1 (h) = d0 (g) and d1 (g) = d0 (f )}

1Ć—m

E

Q P

(4)

m

mĆ—1

c c

EA

P m

This is associativity of composition.

It is straightforward to check that this deļ¬nition is equivalent to the ļ¬rst one.

The diagrams just given actually describe geometric objects, namely the clas-

sifying space of the category. Indeed, the functions between O, A, P and Q

generated by u, d0 , d1 , m and the coordinate maps form a simplicial set trun-

cated in dimension three. The reader needs no knowledge of simplicial sets for

this text.

Fine points

Note that a category may be empty, that is have no objects and (of course)

no arrows. Observe that a subcategory of a monoid regarded as a category may

be empty; if it is not empty, then it is a submonoid. This should cause no more

diļ¬culty than the fact that a submonoid of a group may not be a subgroup. The

basic reason is that a monoid must have exactly one object, while a subcategory

need not have any.

It is important to observe that in categories such as Set, Grp and T in which

op

the arrows are actually functions, the deļ¬nition of category requires that the

function have a uniquely speciļ¬ed domain and codomain, so that for example

8 1 Categories

in T the continuous function from the set R of real numbers to the set R+ of

op

nonnegative real numbers which takes a number to its square is diļ¬erent from the

function from R to R which does the same thing, and both of these are diļ¬erent

from the squaring function from R+ to R+ .

A deļ¬nition of āfunctionā in Set which ļ¬ts this requirement is this: A func-

tion is an ordered triple (A, G, B) where A and B are sets and G is a subset of

A Ć— B with the property that for each x ā A there is exactly one y ā B such that

(x, y) ā G. This is equivalent to saying that the composite

G ā‚ā’ A Ć— B ā’ A

ā’

is an isomorphism (the second function is projection on the ļ¬rst coordinate).

Then the domain of the function is the set A and the codomain is B. As a

consequence of this deļ¬nition, A is empty if and only if G is empty, but B may

or may not be empty. Thus there is exactly one function, namely (ā…, ā…, B), from

the empty set to each set B, so that the empty set is the initial object in Set,

as claimed previously. (Note also that if (A, G, B) is a function then G uniquely

determines A but not B. This asymmetry is reversed in the next paragraph.)

An equivalent deļ¬nition of function is a triple (A, Gā— , B) where Gā— is the

quotient of the disjoint union A + B by an equivalence relation for which each

element of B is contained in exactly one equivalence class. In other words, the

composite

B ā’ A + B ā’ā’ Gā—

ā’ ā’

is an isomorphism, where the ļ¬rst arrow is the inclusion into the sum and the

second is the quotient mapping. This notion actually corresponds to the intu-

itive picture of function frequently drawn for elementary calculus students which

illustrates the squaring function from {ā’2, ā’1, 0, 1, 2} to {0, 1, 2, 3, 4} this way:

ā’2 4

2

ā’1 1

1

0 0

2

3

The set G is called the graph and Gā— the cograph of the function. We will see

in Section 1.8 that the graph and cograph are dual to each other.

1.1 Deļ¬nition of category 9

Exercises 1.1

1. Show that the following deļ¬nition of category is equivalent to the deļ¬nition

given in this section. In this deļ¬nition, to say that an element e has the identity

property means that for all f and g, e ā—¦ f = f whenever e ā—¦ f is deļ¬ned and

g ā—¦ e = g whenever g ā—¦ e is deļ¬ned.

This is the alternative deļ¬nition: A category is a set with a partially deļ¬ned

binary operation denoted āā—¦ā with the following properties:

(a) the following statements are equivalent:

(i) f ā—¦ g and g ā—¦ h are both deļ¬ned;

(ii) f ā—¦ (g ā—¦ h) is deļ¬ned;

(iii) (f ā—¦ g) ā—¦ h is deļ¬ned;

(b) if (f ā—¦ g) ā—¦ h is deļ¬ned, then (f ā—¦ g) ā—¦ h = f ā—¦ (g ā—¦ h);

(c) for any f , there are elements e and e with the identity property for which

e ā—¦ f and f ā—¦ e are deļ¬ned.

2. Verify that the following constructions produce categories.

(a) For any category C , the arrow category Ar(C ) of arrows of C has as

objects the arrows of C , and an arrow from f : A ā’ B to g: A ā’ B is a pair

ā’ ā’

of arrows h: A ā’ A and k: B ā’ B making the following diagram commute:

ā’ ā’

hE

A A

g

f

ńņš. 3 |