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Proposition 3. Let A be a site and j the corresponding topology on SetA .
Then a presheaf F is a j-sheaf if and only if it is an S-sheaf for every sieve S of
the topology.
Proof. Let F be an S-sheaf for every cover S of the site. Let G0 ā’ G be a dense
ā’
inclusion of presheaves. We must by Theorem 8 of Section 6.2 construct for any
Ī±0 : G0 ā’ F a unique extension Ī±: G ā’ F .
ā’ ā’
Let A be an object of A . Since G0 is dense in G, for all a ā GA there is a cover
{Ai ā’ A} with the property that a|Ai ā G0 Ai for each i. Thus Ī±0 A(a|Ai ) ā F Ai
ā’
for each i. Moreover,

(a|Ai )|Ai Ć—A Aj = a|Ai Ć—A Aj = (a|Aj )|Ai Ć—A Aj
230 6 Permanence Properties of Toposes
so applying Ī±0 A, it follows that Ī±0 A(a|Ai ) ā F A and thus determines the required
arrow Ī±.
To prove the converse, ļ¬rst note that the class of covers in a Grothendieck
topology is ļ¬ltered with respect to reļ¬nement: if {Aj ā’ A} and {Ak ā’ A} are
ā’ ā’
covers then {Aj Ć—A Ak ā’ A} is a cover reļ¬ning them.
ā’
Now suppose F is a sheaf. For each object A, deļ¬ne F + A to be the colimit
over all covers of A of the equalizers of

F Ai ā’ā’ F (Ai Ć—A Aj )
ā’ā’

It is easy to see that F + is a presheaf. We need only show that F = F + .
To do that it is suļ¬cient to show that F + is separated and F ā’ F + is dense.
ā’
The latter is obvious. As for the former, it follows from the deļ¬nition of j that
F + is separated if and only if whenever {Ai ā’ A} is a cover and a1 , a2 ā F + A
ā’
with a1 |Ai = a2 |Ai for all i, then a1 = a2 . Since F is a sheaf, F + satisļ¬es this
condition.
Now suppose a1 , a2 ā F + A and there is a cover {Ai ā’ A} for which a1 |Ai =
ā’
a2 |Ai for all i. (In this and the next paragraph, restriction always refers to F + ).
By deļ¬nition of F + , there is a cover {Aj ā’ A} for which a1 |Aj ā F Aj and a
ā’
cover {Ak ā’ A} for which a2 |Ak ā F Ak for all k.
ā’
Next let {Al ā’ A} be a cover simultaneously reļ¬ning {Ai ā’ A}, {Aj
ā’ ā’
ā’ A} and {Ak ā’ A}. Then a1 |Al = a2 |Al for all l because {Al ā’ A} reļ¬nes
ā’ ā’ ā’
{Ai ā’ A}. a1 |Al ā F Al and a2 |Al ā F Al because {Al ā’ A} reļ¬nes both {Aj
ā’ ā’
ā’ A} and {Ak ā’ A}. Thus the two elements are equal in F + A because they
ā’ ā’
are equal at a node in the diagram deļ¬ning F + A. Hence F + is separated, as
required.

Special types of Grothendieck topologies

A topology for which every representable functor is a sheaf is called standard,
or subcanonical. Note that for a standard topology, if L is sheaļ¬ļ¬cation and Y
is the Yoneda embedding, then LY = Y . Thus when a site is standard, it is fully
embedded by y into its own category of sheaves. This embedding plays a central
role in the proof of Giraudā™s Theorem in Section 6.8 and in the construction of
cocone theories in Chapter 8.
The canonical Grothendieck topology on a left exact category is the ļ¬nest
(most covers) subcanonical topology.
Proposition 4. The following are equivalent for a Grothendieck topology.
(a) The covers are eļ¬ective epimorphic families.
6.8 Giraudā™s Theorem 231
(b) The representable functors are sheaves for the topology.
(c) The functor y = LY , where L is sheaļ¬ļ¬cation, is full and faithful.
Proof. (a) is equivalent to (b) by deļ¬nition. If (b) is true, LY = Y and Y is full
and faithful, so (c) is true.
If (c) is true, for a given cover let R be the subfunctor of Y A constructed
above. Then R is j-dense in Y A. Apply L to the diagram
E E YA
R

c
YB
Then the top line becomes equality so we have, from (c),

Hom(LY A, LY B) ā¼ Hom(A, B)
=
ā¼ Hom(Y A, Y B) ā’ Hom(R, Y B) ā’ Hom(LR, LY B)
ā’ ā’
=
ā¼ Hom(LY A, LY B)
=
with composites all around the identity. Hence (a), and therefore (b), is true.

Exercises 6.7

1. Prove that the empty sieve is an eļ¬ective epimorphic family over the initial
object, and that it is universally so if and only if the initial object is strict.

2. Show that the following are equivalent for a Grothendieck topology.
(a) The covers are epimorphic families.
(b) The representable functors are separated presheaves for the topology.
(c) The functor y = LY , where L is sheaļ¬ļ¬cation, is faithful.

6.8 Giraudā™s Theorem
Any topos E is a site with respect to the canonical topology. Giraudā™s Theorem
as originally stated says that certain exactness conditions on a category plus the
requirement that it have a set of generators (deļ¬ned below) are equivalent to
its being the category of sheaves over a small site. It is stated in this way, for
example, in Makkai and Reyes [1977], p. 53 or in Johnstone [1977], p. 17.
Our Theorem 1 below is stated diļ¬erently, but it is in essence a strengthening
of Giraudā™s Theorem.
232 6 Permanence Properties of Toposes
If C is a category, a subset U of the objects of C is a set of (regular)
generators for C if for each object C of C the set of all morphisms from objects
of U to C forms a (regular) epimorphic family. We say that U is a conservative
generating family if for any object C and proper subobject C0 ā C, there is an
object G ā U and an element c āG C for which c āG C0 . See Exercise 4 for the
/
relationships among these conditions.
Sums in a category are disjoint if for any objects A and B, the commutative
diagram
0
Ā d
d
Ā
A B
d Ā
Ā‚Ā
A+B
(in which the arrows to A + B are the canonical injections) is a pullback, and the
canonical injections are monic. Sums are universal if they are preserved under
pullback.
Recall from Exercise 4 of 5.5 that a regular category is a category with ļ¬nite
limits in which regular epis are stable under pullback.
A Grothendieck topos is a category which
(i) is complete;
(ii) has all sums and they are disjoint and universal;
(iii) is regular with eļ¬ective equivalence relations; and
(iv) has a small set of regular generators.
We do not yet know that a Grothendieck topos is a topos, but that fact will
emerge.
Theorem 1. (Giraud) The following are equivalent:
(a) E is a Grothendieck topos;
(b) E is the category of sheaves for a small site;
(c) E is a topos with arbitrary sums and a small set of generators.
Proof. That (b) implies (c) is clear because a sheaf category is a reļ¬‚ective sub-
category of a functor category. Earlier results, taken together with Exercise 4,
show that (c) implies (a). For example, we have shown in Theorem 7 of 2.3 that
equivalence relations are eļ¬ective. In Corollary 8 of 5.3 we showed that pullbacks
have adjoints from which the disjoint stable sums and epis follow.
The proof that (a) implies (b) is immediate from the following proposition.
6.8 Giraudā™s Theorem 233
Proposition 2. Let E be a Grothendieck topos, and C a left exact subcategory
containing a (regular) generating family of E , regarded as a site on the topology
of all sieves which are regular epimorphic families in E . Then E is equivalent
to Sh(C ). Moreover, if C is closed under subobjects, then the topology on C is
canonical.
Proof. The proof consists of a succession of lemmas, in which E and C satisfy the
hypotheses of the Proposition. We begin by showing that the regular epimorphic
families form a topology.
Lemma 3. Every regular epimorphic family in a Grothendieck topos is stable.
Proof. A regular epimorphic family {Ei ā’ E} in a category with all sums is
ā’
characterized by the fact that

(Ei Ć—E Ej ) ā’ā’ Ei ā’ E
ā’
ā’ā’

is a coequalizer. Given the stable sums, this is equivalent to the assertion that

Ei ) ā’ā’
( Ei ) Ć—E ( Ei ā’ E
ā’
ā’ā’

is a coequalizer, which implies that Ei ā’ E is a regular epi. Since both sums
ā’
and regular epis are stable, so is this condition.
In the rest of this section, we use Y , L, and y = LY as deļ¬ned in 6.7. We
show eventually that y is left exact and cocontinuous and deduce that it is an
equivalence.
For every object E of E , Y (E) is a sheaf.
Corollary 4.
Proof. Since every cover is a regular epimorphic family in E , it follows from the
deļ¬nition of covers that representable functors are sheaves.
Lemma 5. y(0) = 0.
Proof. The stable sums in E imply that 0 is a strict initial object. For any object
C of C , Y (0)(C) = Hom(C, 0) = ā… unless C = 0 in which case it is a singleton.
If 0 is not an object of C , this is the constantly null functor which is initial
in the functor category, so that its associated sheaf y(0) is initial in the sheaf
category. If 0 is an object of C , then y(0) can readily be seen to be initial, from
the deļ¬nition of initial, as soon as we observe that F (0) = 0 for every sheaf F .
Since 0 is covered by the empty sieve, we have, for any sheaf F , an equalizer

() ā’ā’
F (0) ā’ā’ ()
ā’ā’
in which the empty products have a unique element so that F (0) does too.
234 6 Permanence Properties of Toposes
Lemma 6. y preserves sums.
Proof. We must show that if E = Ei , then Y Ei ā’ Y E is a dense inclusion
ā’
(in the presheaf category) so that the sheaf associated to the ļ¬rst is the second.
It is an inclusion because Ei Ć—E Ej = 0, the injections into the sum are mono, and
Y preserves limits. Evaluating this at C in C (remember that Y is Hom-functor-
valued), we must show that for any f : C ā’ E, there is a cover {Ck ā’ C}
ā’ ā’
such that f |Ck ā Hom(Ck , Ei ). This means that for each k there is an i with
f |Ck ā Hom(Ck , Ei ).
Given f, let
fj E
Cj Ej
c c

c c
E
C Ei
be a pullback. The objects Cj may not be in C , but for each one there is a
regular epimorphic family (hence cover) {Cjk ā’ Cj } with all Cjk in C because
ā’
C contains a generating family. (Observe that the second subscript k varies over
an index set which depends on the ļ¬rst subscript j). Since C = Cj , {Cj ā’ C}
ā’
is a cover of C. Hence the doubly indexed sieve {Cjk ā’ C} is a cover of C in
ā’
C . Furthermore, f |Cjk = fj |Cjk , which is in Hom(Cjk , Ej ).
Lemma 7. y preserves regular epis.
Proof. The argument is similar to the above. Given E ā’ā’ E an element f ā
ā’
Hom(C, E) will not necessarily lift to an element of Hom(C, E ), but will do so
on a cover of C, namely a cover {Ci ā’ C } by objects in C of the object C
ā’
gotten by pulling f back
EC
E
C

f
f
c c
EE
E
E

Lemma 8. y preserves coequalizers.
Proof. It preserves limits because L and Y do. We also know that it preserves
sums and images and hence unions (even inļ¬nite unions). The constructions used
in Exercise 5 are therefore all preserved by y, so y preserves the construction of
6.8 Giraudā™s Theorem 235
equivalence relations. Thus if
h
ā’ ā’ B ā’c C
ā’
A ā’ā’ ā’ā’
ā’
k
is a coequalizer, y takes the kernel pair of c (which is the equivalence relation
generated by h and k) to the kernel pair of yc. Since yc is regular epi, it is the
coequalizer of yh and yk.
Lemma 9. y is full and faithful.
Proof. The faithfulness follows immediately from the fact that C contains a set
of regular generators for E and that the covers are regular epimorphic families.
As for the fullness, the deļ¬nition of y implies that when C is an object of C ,
Hom(yC, yE) ā¼ Hom(C, E). The universal property of colimits, together with
=
the fact that these are preserved by y allows one to extend this conclusion easily
to the case that C belongs to the colimit closure of C which is E .
Lemma 10. Every sheaf F is a coequalizer of a diagram of the kind
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