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©

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

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