©

‚

d

©

A1 —B A2 A2 —B A2

d

d

©

d

‚ ©

A1 —B A2 E A2

c c

EB

A1

s

d

d

s

d

d

A1 —B A d

d

d

c dc

EA

A1

in which A ’ B is a presentation, and then Ai ’ Ai —B A are presentations

’ ’

for i = 1, 2. Furthermore, Ai —B A is a pullback which by (iv) is preserved by f .

The composite arrow in the upper left corner can be seen to be epi using Freyd™s

Theorem 7 of 7.1 (the near-exact embedding into a power of Set) and a diagram

chase in Set.

254 7 Representation Theorems

In the target category D we have a similar diagram:

E f A2

f A1 —f A f A2

d

d

©

‚

d

©

f A2 —f B f A2

f (A1 —B A2 )

d

d

©

‚

d

©

f A1 —f B f A2 E f A2

c c

E fB

f A1

d

s

s

d

d

d

f A1 —f B f A

d

d

c dc

E fA

f A1

The maps f Ai ’ f Ai —f B f A (i = 1, 2) are epi because they are f of the

’

corresponding arrows in the diagram preceding this one. By the same argument

using Freyd™s representation theorem as was used in the earlier diagram,

f A1 —f A f A2 ’ f A1 —f A f A2

’

is epi, so the second factor

f (A1 —B f A2 ) ’ f A1 —f B f A2

’

is also epi.

(vii). f preserves arbitrary pullbacks.

Proof. It follows from (v) and (vi) that f (A1 —B A2 ) ’ f A1 —f B f A2 is an

’

isomorphism. Thus we can apply the argument used in (iii) above to the diagram

E B2

B1 —B B2

c c

EB

B1

7.3 Morphisms of Sites 255

by replacing A by B throughout.

There are other proofs of Proposition 1 known, not quite as long, which depend

on analyzing the form of the Kan extension. See for example Makkai and Reyes

[1977, Theorem 1.3.10].

In the application below, the categories B and D are both functor categories,

and the use of Freyd™s theorem can be avoided in that case by a direct argument.

Theorem 2. Suppose A and C are sites, and f : A ’ C is a morphism of

’

sites. Then there is a functor f# : Sh(A ) ’ Sh(C ) which is left exact and has a

’

right adjoint for which

fE

A C

y y

c c

E Sh(C )

Sh(A )

f#

commutes.

Proof. Form the diagram

LE

YE A op

' Sh(A )

A Set

I

T T

—

f—

f#

f f! f

c c c

LE

E SetC op ' Sh(C )

C Y I

in which f — is the functor composing with f and f! is the left Kan extension. Y

is Yoneda, I is inclusion, and L is shea¬¬cation. y = L —¦ Y . The fact that f is

cover-preserving easily implies that f — takes sheaves to sheaves and so induces a

functor which we also call f — on the sheaf categories. Then by Theorem 2 of 4.3,

f# = L —¦ f! —¦ I is left adjoint to f — .

The commutativity follows from the following calculation:

Hom(f# yA, F )∼ Hom(yA, f — F ) = Hom(LY A, f — F )

=

∼ Hom(Y A, If — F ) ∼ If — F (A) = IF (f A)

= =

∼ Hom(Y f A, IF ) ∼ Hom(LY f A, F ) = Hom(yf A, F )—¦

= =

Since f! is an instance of f# , it commutes with Y . Since Y is left exact,

f! Y = Y f is left exact and Proposition 1 forces f! to be left exact. Thus f# = Lf! I

is the composite of three left exact functors.

256 7 Representation Theorems

Exercise 7.3

1. Let A denote the category whose objects are Grothendieck toposes and whose

morphisms are left exact functors with a right adjoint (that is the adjoints to geo-

metric morphisms). Let B denote the category whose objects are essentially

small sites, meaning those sites which possess a small subcategory with the prop-

erty that every object of the site can be covered by covering sieves with domains

in that subcategory. There is an underlying functor U : A ’ B which associates

’

to each Grothendieck topos the site which is the same category equipped with the

category of epimorphic families (which, in a Grothendieck topos, is the same as

the topology of regular epimorphic families). Show that the category of sheaves

functor is left adjoint to U .

7.4 Deligne™s Theorem

A topos E is coherent if it has a small full left exact generating subcategory

C such that every epimorphic family Ei ’ C (for any object C) contains a

’

¬nite epimorphic subfamily. Johnstone [1977] gives a proof of a theorem due to

Grothendieck that characterizes coherent Grothendieck toposes as those which

are categories of sheaves on a site which is a left exact category with a topology

in which all the covers are ¬nite. In Chapter 8, we will see that coherent to-

poses classify theories constructed from left exact theories by adding some ¬nite

cocones. (In general, geometric theories allow cocones of arbitrary size).

Theorem 1. [Deligne] Let E be a coherent Grothendieck topos. Then E has a

left exact embedding into a product of copies of the category of sets which is the

left adjoint of a geometric morphism.

Proof. Let E0 be the smallest subtopos of E which contains C as well as every

E -subobject of every object of C . E0 is small because each object has only a set

of subobjects (they are classi¬ed by maps to „¦), and we need only repeatedly

close under P, products and equalizers at most a countable number of times, and

take the union.

According to Theorem 7 of Section 7.1, there is a faithful near-exact embed-

ding T from E0 to a power of Set. f = T |C is left exact and preserves ¬nite

epimorphic families which, given the nature of C , implies that it preserves covers

(by cover we mean in the topology of epimorphic families). We interrupt the

proof for

Lemma 2. f preserves noncovers; that is, if a sieve {Ci ’ C } is not a cover,

’

then {f Ci ’ f C} is not a cover.

’

7.5 Natural Number Objects 257

Proof. E = Im(Ci ’ C) exists in E because E is complete. Since E0 is closed

’

under subobjects, that union belongs to E0 ; moreover, by hypothesis, it cannot

be all of C. Therefore since T is faithful, it follows that T E is a proper subobject

of T C, through which all the T Ci factor. Consequently all the f Ci factor through

this same proper subobject of f C.

Let S be the codomain of f . By theorem 2 of Section 7.3, f : C ’ S extends

’

to a left exact functor f# : E = Sh(C ) ’ S which has a right adjoint f — . We

’

claim that f# is faithful. It is enough to show that given a proper subobject E0

of an object E, then f# (E0 ) is a proper subobject of f# (E).

Since C generates, there is an object C of C and an arrow C ’ E which

’

does not factor through E0 . Form the diagram

E EE

E1 EC

Ci

c c

EE EE

0

in which the square is a pullback and the Ci are a cover of E1 and belong to C .

Since C ’ E does not factor throught E0 , E1 is a proper subobject and so the

’

sieve {Ci ’ C} is a noncover. By Lemma 2 and the fact that f# is a left adjoint,

’

we have

E f# EE

E E f# C

f # Ci 1

c c

f# EE E f# E

0

where the square is still a pullback and the sieve is a noncover. It follows that

the top map cannot be an isomorphism.

7.5 Natural Number Objects

In a topos, (A, a, t) is a pointed endomorphism structure, or PE-structure,

if a: 1 ’ A is a global element of A and t: A ’ A is an endomorphism. PE-

’ ’

structures are clearly models of an FP-theory, and f : (A, a, t) ’ (A , a , t ) is a

’

258 7 Representation Theorems

morphism of PE-structures if

tE

A A