10. Show that every category is equivalent to its skeleton (see Exercise 7 of

Section 1.1).

11. Show that equivalence is an equivalence relation on any set of categories.

(This exercise is easier to do after you do Exercise 7 of Section 1.3).

12. (a) Make the statement “a preordered set can be regarded as a category

in which there is no more than one arrow between any two objects” precise by

de¬ning a subcategory of the category of categories and functors that the category

of preordered sets and order-preserving maps is equivalent to (see Exercise 9 of

Section 1.1).

(b) Show that, when regarded as a category, every preordered set is equivalent

to a poset.

13. An atom in a Boolean algebra is an element greater than 0 but with no

elements between it and 0. A Boolean algebra is atomic if every element x

of the algebra is the join of all the atoms smaller than x. A Boolean algebra

is complete if every subset has an in¬mum and a supremum. A CABA is a

complete atomic Boolean algebra. A CABA homomorphism is a Boolean algebra

16 1 Categories

homomorphism between CABA™s which preserves all infs and sups (not just ¬nite

ones, which any Boolean algebra homomorphism would do). Show that the op-

posite of the category of sets is equivalent to the category of CABA™s and CABA

homomorphisms.

14. An upper semilattice is a partially ordered set in which each ¬nite sub-

set (including the empty set) of elements has a least upper bound. Show that

the category of upper semilattices and functions which preserve the least upper

bound of any ¬nite subset (and hence preserve the ordering) is equivalent to

the category of commutative monoids in which every element is idempotent and

monoid homomorphisms.

15. Show that the arrow and twisted arrow categories of Exercise 2 of Section 1.1

are comma categories.

16. Show neither that the category Set of sets nor the category Ab of abelian

groups is equivalent to its opposite category. (Hint: Find a property of the

category for which the dual property is not satis¬ed.)

1.3 Natural transformations

In topology, a homotopy from f : A ’ B to g: A ’ B is given by a path in B

’ ’

from f x to gx for each element x ∈ A such that the paths ¬t together continuously.

A natural transformation is analogously a deformation of one functor to another.

If F : C ’ D and G: C ’ D are two functors, »: F ’ G is a natural

’ ’ ’

transformation from F to G if » is a collection of arrows »C: F C ’ GC, one

’

for each object C of C , such that for each arrow g: C ’ C of C the following

’

diagram commutes:

»C E

FC GC

Fg Gg

c c

E GC

FC

»C

The arrows »C are the components of ».

The natural transformation » is a natural equivalence if each component

of » is an isomorphism in D.

The natural map of a vector space to its double dual is a natural transforma-

tion from the identity functor on the category of vector spaces and linear maps to

the double dual functor. When restricted to ¬nite dimensional vector spaces, it

1.3 Natural transformations 17

is a natural equivalence. As another example, let n > 1 be a positive integer and

let GLn denote the functor from the category of commutative rings with unity

to the category of groups which takes a ring to the group of invertible n — n

matrices with entries from the ring, and let Un denote the group of units functor

(which is actually GL1 ). Then the determinant map is a natural transformation

from GLn to Un. The Hurewicz transformation from the fundamental group of a

topological space to its ¬rst homology group is also a natural transformation of

functors.

Functor categories

Let C and D be categories with C small. The collection Func(C , D) of func-

tors from C to D is category with natural transformations as arrows. If F and

G are functors, a natural transformation » requires, for each object C of C , an

element of HomD (F C, GC), subject to the naturality conditions. If C is small,

there is no more than a set of such natural transformations F ’ G and so this

’

collection is a set. If »: F ’ G and µ: G ’ H are natural transformations, their

’ ’

composite µ —¦ » is de¬ned by requiring that its component at C to be µC —¦ »C.

Of course, Func(C , D) is just HomCat (C , D), and so is already a functor in each

variable to Set. It is easy to check that for any F : D ’ E ,

’

Func(C , F ): Func(C , D) ’ Func(C , E )

’

is actually a functor and not only a Set-function, and similarly for Func(F, C ),

so that in each variable Func is actually a Cat-valued functor.

We denote the hom functor in Func(C , D) by Nat(F, G) for functors F, G: C

’ D. A category of the form Func(C , D) is called a functor category and is

’

frequently denoted D C especially in the later chapters on sheaves.

Notation for natural transformations

Suppose there are categories and functors as shown in this diagram:

F

“» ‚D

HE KE

B C E

G

Note that in diagrams, we often denote a natural transformation by a double

arrow: »: F ’ G.

18 1 Categories

Suppose »: F ’ G is a natural transformation. Then » induces two natural

’

transformations K»: KF ’ KG and »H: F H ’ GH. The component of K»

’ ’

at an object C of C is

K(»C): KF C ’ KGC

’

Then K» is a natural transformation simply because K, like any functor, takes

commutative diagrams to commutative diagrams. The component of »H at an

object B of B is the component of » at HB. »H is a natural transformation

because H is de¬ned on morphisms.

We should point out that although the notations K» and »H look formally

dual, they are quite di¬erent in meaning. The ¬rst is the result of applying

a functor to a value of a natural transformation (which is a morphism in the

codomain category) while the second is the result of taking the component of a

natural transformation at a value of a functor. Nonetheless, the formal properties

of the two quite di¬erent operations are the same. This is why we use the parallel

notation when many other writers use distinct notation. (Compare the use of

f, v for f (v) by many analysts.) Thus advances mathematics.

Exercise 6 below states a number of identities which hold for natural transfor-

mations. Some of them are used later in the book, particularly in triple theory.

Exercises 1.3

1. Show how to describe a natural transformation as a functor from an arrow

category to a functor category.

2. What is a natural transformation from one group homomorphism to another?

3. Let R: C ’ D be a functor. Show that f ’ Rf is a natural transformation

’

HomC (C, ’) ’ HomD (RC, R(’)) for any object C of C .

’

4. (a) Show that the inclusion of a set A into the free group F A generated by

A determines a natural transformation from the identity functor on Set to the

functor U F where U is the underlying set functor.

(b) Find a natural transformation from F U : Grp ’ Grp to the identity func-

’

tor on Grp which takes a one letter word of F U G to itself. Show that there is

only one such.

5. In Section 1.2, we mentioned three ways of de¬ning the powerset as a func-

tor. (See Exercise 6.) For which of these de¬nitions do the maps which take

each element x of a set A to the set {x} (the “singleton” maps) form a natural

transformation from the identity functor to the powerset functor?

1.3 Natural transformations 19

6. Let categories and functors be given as in the following diagram.

F H

‚ ‚

B C D

G K

Suppose κ: F ’ G and µ: H ’ K are natural transformations.

’ ’

(a) Show that this diagram commutes:

Hκ E HG

HF

µF µG

c c

E KG

KF

Kκ

(b) De¬ne µκ by requiring that its component at B be µGB —¦ HκB, which

by (a) is KκB —¦ µF B. Show that µκ is a natural transformation from H —¦ F

to K —¦ G. This de¬nes a composition operation, called application, on natural

transformations. Although it has the syntax of a composition law, as we will

see below, semantically it is the result of applying µ to κ. In many, especially

older works, it is denoted µ — κ, and these books often use juxtaposition to denote

composition.

(c) Show that Hκ and µG have the same interpretation whether thought of

as instances of application of a functor to a natural transformation, resp. evalu-

ation of a natural transformation at a functor, or as examples of an application

operation where the name of a functor is used to stand for the identity natural

transformation. (This exercise may well take longer to understand than to do.)

(d) Show that application as de¬ned above is associative in the sense that if

(µκ)β is de¬ned, then so is µ(κβ) and they are equal.

(e) Show that the following rules hold, where —¦ denotes the composition of

natural transformations de¬ned earlier in this chapter. These are called Gode-

ment™s rules. In each case, the meaning of the rule is that if one side is de¬ned,

then so is the other and they are equal. They all refer to the diagram below, and

the name of a functor is used to denote the identity natural transformation from

that functor to itself. The other natural transformations are κ: F1 ’ F2 , »: F2

’

’ F3 , µ: G1 ’ G2 , and ν: G2 ’ G3 .

’ ’ ’

F1 G1

F2 “ κ E G2 “ µ E

… …

EE HE

A B C D E

“» ! “ν !

F3 G3

20 1 Categories

(i) (The interchange law) (ν —¦ µ)(» —¦ κ) = (ν») —¦ (µκ)

(ii) (H —¦ G1 )κ = H(G1 κ).

(iii) µ(F1 —¦ E) = (µF1 )E.

(iv) G1 (» —¦ κ)E = (G1 »E) —¦ (G1 κE).

(v) (µF2 ) —¦ (G1 κ) = (G2 κ) —¦ (µF1 ).

7. Show that two categories C and D are equivalent if and only if there are

functors F : C ’ D and G: D ’ C such that G —¦ F is naturally equivalent to

’ ’

idC and F —¦ G is naturally equivalent to idD .

1.4 Elements and Subobjects

Elements

One of the important perceptions of category theory is that an arrow x: T

’ A in a category can be regarded as an element of A de¬ned over T . The

’

idea is that x is a variable element of A, meaning about the same thing as the

word “quantity” in such sentences as, “The quantity x2 is nonnegative”, found

in older calculus books.

One must not get carried away by this idea and introduce elements every-

where. One of the main bene¬ts of category theory is you don™t have to do things

in terms of elements unless it is advantageous to. In 3.3.1 is a construction that

is almost impossible to understand in terms of elements, but is very easy with the

correct conceptual framework. On the other hand, we will see many examples

later in which the use of elements leads to a substantial simpli¬cation. The point

is not to allow a tool to become a straitjacket.

When x: T ’ A is thought of as an element of A de¬ned on T , we say that

’

T is the domain of variation of the element x. It is often useful to think of x

as an element of A de¬ned in terms of a parameter in T . A related point of view

is that x is a set of elements of A indexed by T . By the way, this is distinct from

the idea that x is a family of disjoint subsets of T indexed by A, as mentioned

in 1.1.

The notation “x ∈T A” is a useful quick way of saying that x is an element

of A de¬ned on T . This notation will be extended when we consider subobjects

later in this section.

If x ∈T A and f : A ’ B, then f —¦ x ∈T B; thus morphisms can be regarded

’

as functions taking elements to elements. The Yoneda Lemma, Theorem 2 of the

next section, says (among other things) that any function which takes elements

1.4 Elements and Subobjects 21

to elements in a coherent way in a sense that will be de¬ned precisely “is” a

morphism of the category. Because of this, we will write f (x) for f —¦ x when it is

helpful to think of x as a generalized element.

Note that every object A has at least one element idA , its generic element.

If A is an object of a category C and F : C ’ D is a functor, then F takes

’

any element of A to an element of F A in such a way that (i) generic elements

are taken to generic elements, and (ii) the action of F on elements commutes

with change of the domain of variation of the element. (If you spell those two

conditions out, they are essentially the de¬nition of functor.)

Isomorphisms can be described in terms of elements, too: An arrow f : A ’ B ’

is an isomorphism if and only if f (thought of as a function) is a bijection between

the elements of A de¬ned on T and the elements of B de¬ned on T for all objects

T of C . (To get the inverse, apply this fact to the element idA : A ’ A.) And a

’

terminal object is a singleton in a very strong sense”for any domain of variation

it has exactly one element.