ńņš. 43 |

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) ā¼ Hom(A, B)

=

ā¼ Hom(Y A, Y B) ā’ Hom(R, Y B) ā’ Hom(LR, LY B)

ā’ ā’

=

ā¼ 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 āG C for which c āG 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

Ā© Ā‚

A B

d Ā

dĀ©

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) ā¼ 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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