(b) An arbitrary object A is a sheaf if and only if whenever B0 ’ B is dense

’

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

’ ’

Remark: It follows readily from this proposition that if j is the topology of

Example (a) of Section 6.1, then the category of sheaves on X is the same as the

category of j-sheaves in the presheaf category.

Proof. If A is a sheaf, these follow from Propositions 4 and 7.

208 6 Permanence Properties of Toposes

Now suppose that the map extension condition holds. Let d0 , d1 : RA ’ A be

’

the kernel pair of the map A ’ SA. The equalizer of d0 and d1 is the diagonal of

’

A — A, which is dense in RA. Then by the version of the map extension condition

in (b), d0 = d1 ; hence A is separated.

Suppose m: A ’ B is monic with B separated. Since any subobject of a

’

separated object is separated (Exercise 5), we may replace B by the j-closure of

A and suppose without loss of generality that A )’ B is dense. The diagram

’

E EB

A

c c

E EB

A

in which the vertical arrows are identities has a diagonal ¬ll-in making the upper

triangle commute given by the map extension condition. As for the lower triangle,

it commutes when restricted to the dense subobject A. With B separated, this

implies that the lower triangle commutes, whence A = B, as required.

Theorem 9. For any object A in a topos, F A is a sheaf, and F is a functor

which is left adjoint to the inclusion of the full subcategory of sheaves in the topos.

Proof. F A is clearly separated; that it is a sheaf then follows from Propositions 7

and 8.

Any map A ’ B to a sheaf gives a unique map SA ’ B, which by Propo-

’ ’

sition 7 extends to a unique map F A ’ F B = B; this gives the adjunction.

’

To show that F is a functor, it is su¬cient to use pointwise construction of

adjoints (Section 1.9).

Exercises 6.2

1. Give an example of a presheaf which is j-closed in the sense of Example (a)

of Section 6.1 which is nevertheless not a sheaf.

2. Show that the product of dense monos is dense and the product of closed

monos is closed.

3. Suppose B0 ’ B is a j-dense inclusion with B j-separated. Prove that a map

’

B ’ A whose restriction to B0 is monic is itself monic.

’

6.3 Sheaves form a topos 209

4. Show that in a serially commutative diagram

E EB

EA

E

c c

c Ec c

EB

EA

E

with both rows kernel pair/coequalizers and E = E —A (A — A), then B ’ B

’

is monic.

5. (a) Show that if A ’ B is monic and B is separated then A is separated.

’

(b) Show that if A ’ B is a dense mono where B is a sheaf then B = F A.

’

6. Show that if j is the topology of Example (iii) of Section 6.1 and E is the

category of sheaves on a topological space, then Shj (E ) is the category of sheaves

on the complement of the open set U .

7. Show that S is the object map of a functor which is left adjoint to the inclusion

of separated objects.

6.3 Sheaves form a topos

The full subcategory of sheaves for a topology jin a topos E is denoted Ej . In

this section we will prove that Ej is a topos. We will also prove that F , which

we now know is left adjoint to inclusion, is left exact, so that the inclusion is a

geometric morphism.

The power object for sheaves

Using Yoneda, let „ A: PA ’ PA denote the map induced by the natural

’

transformation j(A — ’): Sub(A — ’) ’ Sub(A — ’). Then evidently „ A is

’

an idempotent endomorphism of PA. Let Pj A denote the splitting object”the

equalizer of „ A and the identity. Then for any object B, Hom(B, Pj A) consists

of the j-closed subobjects of A — B.

Proposition 1. Let B0 )’ B be a j-dense mono. Then for any object A,

’

pulling back along A — B0 ’ A — B gives a one to one correspondence between

’

j-closed subobjects of A — B and j-closed subobjects of A — B0 .

Proof. Since A — B0 is dense in A — B, it is su¬cient to prove this when A = 1.

If B1 )’ B0 is j-closed, then B2 = jB(B1 ) )’ B is a j-closed subobject of B.

’ ’

210 6 Permanence Properties of Toposes

Lemma 2(a) of Section 6.1 says that B0 © jB(B1 ) = jB0 (B1 ), which is B1 since B1

is closed in B0 . If B3 © B0 = B1 also, we would have two di¬erent factorizations

of B1 )’ B as dense followed by closed, which is impossible by Exercise 2 of

’

Section 5.5.

Proposition 2. Pj A is a j-sheaf.

Proof. We use the characterization of sheaves of Proposition 8 of Section 6.2,

which requires us to show that when B0 is a dense subobject of B, the map

Hom(B, Pj A) )’ Hom(B0 , Pj A)

’

is an isomorphism. The left side (respectively the right side) represents the set

of j-closed subobjects of A — B (respectively A — B0 ). By Proposition 1, those

two sets are in bijective correspondence via pullback.

Theorem 3. Ej is a topos in which, for a sheaf A, Pj A represents the subobjects.

Proof. It follows from the diagonal ¬ll-in property (Lemma 2(d) of Section 6.1)

that a subobject of a sheaf is a sheaf if and only if it is j-closed. Thus for a sheaf

A, Pj A represents the subsheaves of A. Finite limits exist because the inclusion

of the subcategory has a left adjoint.

Exactness of F

We show here that the functor F is exact, which is equivalent to showing that

the inclusion of Ej into E is a geometric morphism. We begin with:

Proposition 4. The separated re¬‚ector S preserves products and monos.

Proof. That S preserves monos is Proposition 3 of Section 6.2.

The product of dense monos is dense (Exercise 2 of Section 6.2). It follows that

for an object A, A—A ’ RA—RA is dense, so that RA—RA ⊆ R(A—A). Since

’

products commute with re¬‚exive coequalizers, we have the following commutative

diagram, in which both rows are coequalizers:

E E SA — SA

E A—A

RA — RA

c c

= =

c c c

E E S(A — A)

E A—A

R(A — A)

Since A—A is dense in RA—RA and RA—RA is closed in A—A—A—A (Exercise 2

of Section 6.2), it follows from Lemma 2(h) of Section 6.1 that RA — RA =

R(A — A). The required isomorphism follows immediately from the uniqueness

of coequalizers.

6.4 Left exact cotriples 211

Proposition 5. F preserves products and monos.

Proof. By the preceding proposition it is enough to show that F restricted to the

category of separated objects preserves products and monos. That it preserves

monos is obvious. Now let A and B be separated. It is clear from Exercise 2 of

Section 6.2 that A — B is dense in F A — F B and that the latter is a sheaf. It

follows from that and from Exercise 5 of Section 6.2 that F A — F A is F (A — B).

Theorem 6. F is left exact.

Proof. F is a left adjoint so preserves cokernel pairs. The theorem then follows

from Proposition 5 above and Theorem 6 of Section 5.5.

Remark. S is not usually left exact. This shows that the full exactness prop-

erties of a topos are required in this theorem. Various other combinations of

exactness properties have been proposed as a non-additive analog of a category

being Abelian (sets of properties which taken together with additivity would im-

ply abelianness) but none of these proposals would appear to allow the proof of

a theorem like Theorem 6 of Section 5.5.

Let E and F be toposes. Recall from Section 2.2 that a functor u: E ’ F

’

— —

is a geometric morphism if u has a left adjoint u and u is left exact. We will

see in Section 7.3 that morphisms of sites induce geometric morphisms. At this

point we wish merely to observe that the left adjoint F constructed above is left

exact, so that:

Corollary 7. If j is a topology on the topos E and Ej the category of j-sheaves,

then the inclusion Ej ’ E is a geometric morphism.

’

6.4 Left exact cotriples

A cotriple G = (G, , δ) in which G is a left exact functor is called a left exact

cotriple. In this section, we will prove:

Theorem 1. Let E be a topos and G a left exact cotriple in E . Then the category

EG of coalgebras of G is also a topos.

The proof requires a sequence of propositions. In these propositions, E and G

satisfy the requirements of the theorem. Note that G, being left exact, preserves

pullbacks and products.

It is not hard to show that when a cotriple is left exact we can speak of a

subobject of a coalgebra being a subcoalgebra without ambiguity. See Exercise 1.

212 6 Permanence Properties of Toposes

Proposition 2. Let (A, ±) be a coalgebra for G and B a subobject of A. Then

B is a subcoalgebra if and only if the inverse image of GB along ± is B.

Proof. If the inverse image of GB along ± is B, let the coalgebra structure β: B

’ GB be the restriction of ± to B. This satis¬es the required coalgebra identities

’

because G2 B ’ G2 A is monic. To say that inclusion is a coalgebra map requires

’

this diagram

iE

E

B A

±

β (1)

c c

E GA

GB

Gi

to commute, but in fact it is a pullback by assumption.

Conversely, suppose we are given β for which (1) commutes. We must show

that for any (variable) element ±: T ’ A of A for which ±(a) ∈ GB is actually

’

in B. This follows from the fact that a = A(±(a)) (where is the counit of the

cotriple) which is an element of B because a natural transformation between left

exact functors takes an element of a subobject to an element of the corresponding

subobject.

Now let (A, ±) and (b, β) be coalgebras. Let

¦(±, β): Sub(A — B) ’ Sub(A — B)

’

be de¬ned by requiring that for a subobject C of A — B,

E E A—B

¦(C)

(2)

±—β

c c

E E GA — GB

GC

where we write ¦(C) for ¦(±, β)(C), is a pullback.

Proposition 3. Let (A, ±) and (B, β) be coalgebras and C a subobject of A — B.

Then ¦C is a subcoalgebra of A — B. It is the largest subcoalgebra contained in

C; in particular, C is a subcoalgebra if and only if C = ¦C.

Proof. Since ± — β is a coalgebra structure on A — B (Exercise 3), it su¬ces by

Proposition 2 to show that we can ¬ll in the upper left diagonal arrow in the

6.4 Left exact cotriples 213

diagram below so that the diagram commutes.

E EA—B

¦(C)

±—β

©

E E GA — GB

G(¦(C))

G± — Gβ ±—β

c c

E G2 A — G2 B

2

GC

s

d

d

d

δA — δB

δ

d

c dc

E GA — GB

GC

This follows immediately from the fact that all squares in the diagram commute

and the inner square is a pullback. The rest is left as Exercise 4.