’’B

’

A’’

’

g

is the pullback of

A

(id, f )

c

E A—B

A

(id, g)

if it exists, and in a category with a terminal object, the product of objects A

and B is the pullback of

B

c

E1

A

if it exists.

1.7 Limits 45

5. (a) Let C be a category and A an object of C . Show that the product of two

objects in the category C /A of objects over A is their pullback over A in C.

(b) Show that the functor C /A ’ C which takes B ’ A to B creates

’ ’

pullbacks.

6. Call a non-empty graph connected if it is not the disjoint union of two

non-empty subgraphs.

(a) Show that the forgetful functor A /A ’ A preserves the limits of dia-

’

grams over connected graphs (which are called connected diagrams).

(b) Show that the category of ¬elds and homomorphisms of ¬elds has limits

of connected diagrams and no others.

7. Show that if D is left exact and F : D ’ C preserves ¬nite limits, then the

’

comma category (C , F ) is left exact.

8. Prove Proposition 1.

9. Let A be a topological space and let O(A) denote the set of open sets of

A partially ordered by inclusion considered as a category. Show that O(A) has

¬nite limits. Does O(A) have all limits?

10. A monomorphism is regular if it is the equalizer of two arrows. (The dual

notion is called regular epi, not ”coregular”). Recall from Section 1.4 that a

regular epimorphism is denoted in diagrams by a double-headed arrow:

’’

’

We have no special notation for regular monos nor for ordinary epis. The reason

for this asymmetry is basically one of convenience. In most of the situations in

this book we are interested in ordinary monos, but only regular epis. Actually, in

toposes where much of our attention will be focused, all epis and all monos will

be regular.

(a) Show that any arrow whose domain is the terminal object 1 is a regular

mono.

(b) Show that the pullback of a regular mono is a regular mono.

11. Let D: I ’ Set be a diagram in Set. Let — be a ¬xed one-element set.

’

Show that the set of all cones over D with vertex —, equipped with the correct

projections, can be interpreted as lim D. (This proves that Set is complete).

12. Show that Grp and T are complete.

op

46 1 Categories

13. Let D: I ’ C be a diagram, and let A be an object of C . Then DA =

’

Hom(A, D(’)) is a diagram in Set. Let Cone(A, D) denote the set of cones over

D with vertex A. Show that the limit of DA in Set is the cone ±: (Cone(A, D)

’ DA with ±i (for i an object of I ) de¬ned by ±i(β: A ’ D) = βi, for

’ ’

β ∈ (Cone(A, D).

14. Show that representable functors preserve limits. (Hint: Use Exercises (11)

and (13) A direct proof is also possible.)

15. Let D: I ’ C be a diagram and let ±: W ’ D be a cone over D. For any

’ ’

object A of C , let Hom(A, ±): Hom(A, W ) ’ Hom(A, D(’)) denote the cone

’

with vertex Hom(A, W ) which is de¬ned by Hom(A, ±)i = Hom(A, ±i). Show

that if Hom(A, ±) is a limit of Hom(A, D(’)) for every object A of C , then ±: W

’ D is a limit of D. (Of course the converse is true by Exercise 14.)

’

16. Let C be the category of in¬nite sets and maps between them. Show that

the covariant powerset functor P which takes a map to its image function makes

P(A — B) isomorphic to PA — PB for any objects A and B but does not preserve

products.

17. Suppose that the category A has ¬nite limits. Show that the kernel pair

of any arrow is an equivalence relation. Hint: you will have to use the universal

mapping properties of limits.

18. A more general de¬nition of equivalence relation is this: a pair u: E ’ A,

’

v: E ’ A of arrows is jointly monic if for any f , g: B ’ E, uf = ug and

’ ’

vf = vg imply that f = g. Such a pair makes E an equivalence relation on A if

for each object B the subset of Hom(B, A) — Hom(B, A) induced by Hom(B, E)

is an equivalence relation (in the usual sense) on Hom(B, A). Show that this is

equivalent to the de¬nition in the text when the product A — A exists in the

category.

19. Show that a relation (u, v): R ’ A — A in a category with ¬nite limits is

transitive if and only if, for the pullback

ER

P

c c

EA

R

it is true that (v —¦ pi , u —¦ p2 ) ∈ R.

1.7 Limits 47

20. Show that h, k: K ’ A is a kernel pair of f : A ’ B if and only if this

’ ’

diagram is a pullback:

kE

K A

f

h

c c

EB

A

f

21. (a) Show that the underlying functor from the category of groups creates

limits.

(b) Do the same for the category of compact Hausdor¬ spaces and continuous

maps.

22. Show that if F : C ’ D is an equivalence of categories, and U : D ’ A

’ ’

creates limits, then U F re¬‚ects limits.

23. Show that if E is an equivalence relation on A, then E — E is an equivalence

relation on A — A.

24. Let F : B ’ D and G: C ’ D be functors. Show that the following diagram

’ ’

is a limit in the category of categories. Here (F, G) is the comma category as

de¬ned in Section 1.2.

(F, G)

d

d

d

d

© d

‚

c

B C

Ar(D)

d

d

F G

dom cod d

d

c©

‚

dc

D D

25. Show that if A and B are categories and D: I ’ Func(A , B) a diagram,

’

and for each object A of A the diagram D —¦ ev(A) gotten by evaluating at A has

a limit, then these limits make up the values of a functor which is the limit of D

in the functor category. Conclude that if B is complete, so is Func(A , B).

48 1 Categories

26. Suppose that A is a category and that B is a subcategory of Func(A op , Set)

that contains all the representable functors. This means that the Yoneda embed-

ding Y of A op into the functor category factors through B by a functor y: A op

’ B. Suppose further that a class C of cones is given in A with the property

’

that each functor in B takes every cone in C to a limit cone. Show that y op : A

’ B op takes every cone in to a limit cone.

’

1.8 Colimits

A colimit of a diagram is a limit of the diagram in the opposite category. Spelled

out, a commutative cocone from a diagram D: I ’ C with vertex W is a

’

natural transformation from D to the constant diagram with value W . The

set of commutative cocones from D to an object A is Hom(D, A) and becomes

a covariant functor by composition. A colimit of D is a universal element for

Hom(D, ’).

For example, let us consider the dual notion to “product”. If A and B are

objects in a category, their sum (also called coproduct) is an object Q together

with two arrows i1 : A ’ Q and i2 : B ’ Q for which if f : A ’ C and g: B

’ ’ ’

’ C are any arrows of the category, there is a unique arrow f, g : Q ’ C for

’ ’

which f, g —¦ i1 = f and f, g —¦ i = 2 = g. The arrows i1 and i2 are called the

coproduct injections although they need not be monic. Since Hom(A + B, C) ∼ =

Hom(A, C) — Hom(B, C), f, g represents an ordered pair of maps, just as the

symbol (f, g) we de¬ned when we treated products in Section 1.7.

The sum of two sets in Set is their disjoint union, as it is in Top. In Grp the

categorical sum of two groups is their free product; on the other hand the sum of

two abelian groups in the category of abelian groups is their direct sum with the

standard inclusion maps of the two groups into the direct sum. The categorical

sum in a poset regarded as a category is the supremum. The categorical sum

of two posets in the category of posets and non-decreasing maps is their disjoint

with no element of the one summand related to any element of the second.

The coequalizer of two arrows f, g: A ’ B is an arrow h: B ’ C such that

’ ’

(i) h —¦ f = h —¦ g, and

(ii) if k: B ’ W and k —¦ f = k —¦ g, then there is a unique arrow u: C ’ W for

’ ’

which u —¦ h = k.

The coequalizer of any two functions in Set exists but is rather complicated

to construct. If K is a normal subgroup of a group G, then the coequalizer of

the inclusion of K into G and the constant map at the identity is the canonical

map G ’ G/K.

’

1.8 Colimits 49

The dual concept to “pullback” is “pushout”, which we leave to the reader to

formulate.

The notion of a functor creating or preserving a colimit, or a class of colimits,

is de¬ned analogously to the corresponding notion for limits. A functor that

preserves ¬nite colimits is called right exact, and one that preserves colimits of

all small diagrams is called cocontinuous. In general, a categorical concept that

is de¬ned in terms of limits and/or colimits is said to be de¬ned by “exactness

conditions”.

Regular monomorphisms and epimorphisms

A map that is the equalizer of two arrows is automatically a monomorphism

and is called a regular monomorphism. For let h: E ’ A be an equalizer of

’

f, g: A ’ B and suppose that k, l: C ’ E are two arrows with h —¦ k = h —¦ l. Call

’ ’

this common composite m. Then f —¦ m = f —¦ h —¦ k = g —¦ h —¦ k = g —¦ m so that, by

the universal mapping property of equalizers, there is a unique map n: C ’ E ’

such that h —¦ n = m. But k and l already have this property, so that k = n = l.

The dual property of being the coequalizer of two arrows is called regular

epimorphism. In many familiar categories (monoids, groups, abelian groups,

rings, . . . ) the regular epimorphisms are the surjective mappings, but it is less

often the case that the injective functions are regular monomorphisms. Of the four

categories mentioned above, two (groups and abelian groups) have that property,

but it is far from obvious for groups.

Regular categories

A category A will be called regular if every ¬nite diagram has a limit, if

every parallel pair of arrows has a coequalizer and if whenever

fE

A B

g h

c c

ED

C

k

is a pullback square, then h a regular epimorphism implies that g is a regular

epimorphism. Weaker de¬nitions are sometimes used in the literature. The

property required of regular epis is sometimes described by the phrase, “Regular

epis are stable under pullback.” Some related ideas are de¬ned on page 240.

50 1 Categories

In Set and in many other familiar categories (groups, abelian groups, rings,