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EA
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.

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