B

Then there is a map from B to A0 making both triangles commute.

Proof. The inverse image (pullback) of A0 is closed in B by Lemma 2(e) and

dense because it contains B0 , so that the inverse image is B. The conclusion now

follows easily.

Naturality of j for spatial sheaves

Here we outline the proof that the map j of Example (a) is natural. We must

prove that if F and F are presheaves and »: F ’ F is a natural transformation,

’

then

jF E

Sub F Sub F

T T

(5)

Sub » Sub »

E Sub F

Sub F

jF

commutes.

If G is a subfunctor of F , then G = Sub »(G ) if for each open U , GU is the

inverse image of G U along »U . This is because limits are constructed pointwise

in a functor category like E .

Using this notation, it is necessary to show that for every open U , the inverse

image of jF (G )(U ) along »U is jF (G)(U ). To see this, suppose y ∈ jF (G )(U )

and »U (x) = y for some x ∈ F U . Then on some cover {Ui } of U , y|Ui ∈ G Ui for

every i. Then by de¬nition, x|Ui ∈ GUi and so x ∈ jF (G)(U ). Conversely, it is

clear that if x ∈ jF (G)(U ) then »(x) ∈ jF (G )(U ).

6.2 Sheaves for a Topology 203

Exercises 6.1

1. Find an idempotent endomorphism of the three element chain which is in-

¬‚ationary but not order-preserving and one which is order-preserving but not

in¬‚ationary.

2. Suppose that for each object A of a topos there is an idempotent, in¬‚ationary,

order-preserving map jA: Sub A ’ Sub A with the property that whenever

’

AE EA

0

c c

BE EB

0

is a pullback, then there is an arrow jA(A0 ) ’ jB(B0 ) for which

’

jA(A0E EA

)

c c

jB(B0E EB

)

is also a pullback.

Show that these functions constitute a natural endomorphism of P and so

induce a topology on the topos.

3. Prove that the natural transformation j of Example (a) is a topology.

4. Prove that a topology on a category in which subobjects are representable can

be given as an endomorphism j of the representing object which is idempotent,

takes true to true, and commutes with intersection.

5. Use the results and exercises of Section 5.6 to show that there is a topology j

on any topos such that for any subobject A0 ⊆ A, jA0 = ¬¬A0 .

6.2 Sheaves for a Topology

In this section we de¬ne what it means for an object in a topos to be a sheaf for

a topology on the topos, and construct an “associated sheaf functor”.

204 6 Permanence Properties of Toposes

Separated objects

Let j be a topology on a topos. An object A is j-separated (or simply

“separated” if j is understood) if A is a closed subobject of A—A via the diagonal.

We will form the separated quotient of an object A.

Proposition 1. Let R(A) be the closure of A in A — A. Then for any object

B, an element (f, g) ∈B RA if and only if the equalizer of f and g is dense in B.

Proof. If B0 is that equalizer then the outer square and hence by Lemma 1 of

Section 6.1 the right hand square of

BE E jB(B0E EB

)

0

(f, g)

c c c

E E R(A)

E E A—A

A

are pullbacks. The conclusion is now evident.

Corollary 2. R(A) is an equivalence relation on A.

Proof. Re¬‚exivity and symmetry are clear. If (f, g) and (g, h) are elements of

R(A) de¬ned on B then the equalizer of f and h contains the intersection of those

of f and g and g and h, each of which is dense. But by Lemma 2(f) of 6.1 the

intersection of two dense subobjects is dense.

In Abelian groups we form the torsion-free quotient by factoring out the ele-

ments which are “almost zero”. In analogy with this construction, we will form

the separated quotient of A by identifying pairs of elements which are “almost

equal”. Thus we form the quotient

R(A) ’ A ’’ S(A)

’ ’

’’

which we can do because equivalence relations in a topos are e¬ective. Note that

A is separated if and only if A = SA.

S(A) is j-separated. If A )’ B, then S(A) )’ S(B).

Proposition 3. ’ ’

Proof. The diagram

E E A—A

RA

c c

E E SA — SA

SA

6.2 Sheaves for a Topology 205

is a pullback (standard because RA is the kernel pair of A ’ SA). It follows

’

from the fact that the pullback of an epi is an epi that the image of RA in SA—SA

is the diagonal SA. If we apply j we get the pullback

E E A—A

RA = j(A — A)(RA)

c c

E E SA — SA

j(SA — SA)(SA)

and the vertical arrows are epic so that j(SA — SA)(SA) = SA. Thus SA is

separated. As for the second assertion, when A )’ B is monic, the diagram

’

E A—A

A

c c

c c

E B—B

B

is a pullback. Then apply j to get a pullback

E A—A

RA

c c

c c

E B—B

RB

Since RA and RB are equivalence relations on A and B, respectively, it follows

from Exercise 4 that SA ’ SB is mono.

’

Sheaves for a topology

An object in a topos is absolutely closed for a topology j if it is j-closed

as a subobject of any separated object. An object A in a topos is a sheaf for a

topology j if it is j-separated and absolutely closed.

For any object A, let F A denote the object S(j(PSA)(SA)) (using the sin-

gleton map to include SA in PSA). We will show that F A is a sheaf and that

F is the left adjoint of the inclusion of the full subcategory of sheaves. F is the

associated sheaf functor (or shea¬¬cation).

206 6 Permanence Properties of Toposes

Proposition 4. If A is separated, the map A ’ F A is a j-dense mono. If A

’

is a sheaf, then A ’ F A is an isomorphism.

’

Proof. Let A be separated. A is included in j(PA)(A), so by Proposition 3,

SA = A is included in S(j(PA)(A)) which is F A because A is separated.

To show that the inclusion is dense, let B = j(PA)(A) and let C be the inverse

image of A along the map B ’ F A, as in the diagram

’

EC E B = j(PA)(A)

A

c c

c c

E FA

A

Apply j to this diagram using Lemma 1 of Section 6.1 and the top row becomes

the identity on B so the bottom row must also become the identity because the

vertical arrows are epic.

Lemma 5. Let B0 ’ B be a j-dense inclusion. Then any map B0 ’ A can

’ ’

be extended to a map B ’ F A.

’

Proof. In the diagram

(dense)

BE EB

0

c

A

©

c c

SA E j(SA)

E E PSA

(closed)

c

F A = S(j(SA))

(in which j means j(PSA)), the rightmost vertical arrow exists because power

objects are injective (Exercise 3 of Section 2.1) so the diagonal arrow exists by

Lemma 2(d) of Section 6.1.

Proposition 6. Two maps to a separated object which agree on a dense subobject

are equal.

6.2 Sheaves for a Topology 207

Proof. Consider the diagram

(dense) E

BE B

0

c c

E E A—A

A

(diagonal)

where the right arrow is induced by the given arrows. By Lemma 2(d) of Sec-

tion 6.1, that arrow factors through the diagonal, as required.

Proposition 7. Let A be separated and B0 ’ B be a j-dense inclusion. Then

’

any map B0 ’ F A can be extended to a unique map B ’ F A.

’ ’

Proof. Consider the diagram

E B0 (dense) B

E

C

c c

E FA

A

where the square is a pullback. The composite along the top is a dense inclusion

by Proposition 4, and the fact that the composite of dense maps is dense. The

requisite map from B to F A exists by Lemma 5. That map and the map from

B0 to F A agree on C and so are equal by Proposition 6. The uniqueness follows

similarly.

The following proposition shows that the essence of being a sheaf has survived

our process of abstraction.

Proposition 8.

(a) A separated object A is a sheaf if and only if whenever B0 ’ B is dense

’

then any map B0 ’ A has an extension to a map B ’ A.