’

EG .

Proof. Naturality is equivalent to the requirement that the upper square in the

diagram below must commute. Here, C is a subobject of A — B and C is the

inverse image of C along A — f .

¦(C E EA—B

)

d

d A—f

d

d

‚ ©

E EA—B

¦(C)

±—β ±—β

c c

E GA — GB

GC

d

s

d

GA — Gf

d

c dc

E GA — GB

GC

214 6 Permanence Properties of Toposes

The inner and outer squares are pullbacks by de¬nition. The bottom square is

G applied to the pullback in the de¬nition of the subobject functor and hence

is a pullback because G is left exact. It follows that the upper square composed

with the inner square is a pullback. Since the inner square is too, so is the top

one, which therefore commutes.

Let R: E ’ EG be the right adjoint to U : EG ’ E .

’ ’

Corollary 5. There is a map ¦(±): RPA ’ RPA for which

’

Hom((B, β), ¦(±)) = ¦(±, β)

Proof. Yoneda.

We can now prove Theorem 1. Finite limits exist in EG because they are

created by the underlying functor U : EG ’ E . To prove this you use the same

’

sort of easy argument as in proving Exercise 3.

The power object P(A±) for a coalgebra (A, ±) is de¬ned to be the equalizer

of ¦(±) and the identity map on RPA. Since

Sub(A — B) ∼ Hom(B, PA) ∼ Hom((B, β), RPA)

= =

the theorem follows from Proposition 3 and Corollary 5.

Observe that R is a geometric morphism. Its left adjoint is easily seen to be

faithful, as well. We will see later that any geometric morphism with a faithful

left adjoint arises from a cotriple in this way.

A nice application of the theorem is a new proof, much simpler than that

in Section 2.1, that the functor category SetC is a topos for any small category

C . This follows from two observations: (i) If S is the set of objects of C , then

SetS = Set/S, which is very easily seen to be a topos. (ii) The map SetC ’ SetS

’

induced by the forgetful functor is adjoint tripleable (Section 3.7), hence SetC is

equivalent to the category of coalgebras of an (evidently) left exact cotriple on

Set/S.

Exercises 6.4

1. Let G = (G, , δ) be a left exact cotriple and (A, ±) be a G-coalgebra. Show

that a subobject A0 of A “is” a subcoalgebra if and only if there is a commutative

square

AE EA

0

c c

E GA

GA0

6.5 Left exact triples 215

where the right arrow is the coalgebra structure map, in which case that square

is a pullback. Conclude that there is at most one subcoalgebra structure on a

subobject of an algebra, namely the left arrow in this square.

2. Show that if F and G are product-preserving functors, »: F ’ G a natural

’

transformation, then for any objects A and B for which A — B exists, »(A — B) =

»A — »B.

3. Show that if (G, , δ) is a left exact cotriple on a category with products and

(A, ±) and (B, β) are coalgebras, then so is (A — B, ± — β).

4. Verify the third sentence of Proposition 3. (Hint: Follow the map ¦(C)

’ GC by C.)

’

6.5 Left exact triples

A left exact triple in a topos induces a topology on the topos for which the

objects of the form T A are sheaves. We will use this construction and the topos of

coalgebras of a cotriple discussed in the preceding section to obtain a facotrization

theorem for geometric morphisms.

Given a left exact triple T = (T, ·µ) in a topos E , de¬ne for each object A a

function jA: Sub A ’ Sub A in this way: for a subobject A0 of A, jA(A0 ) is the

’

inverse image of T A0 along ·A. In other words,

jA(A0E EA

)

(1)

·A

c c

T AE E TA

0

must be a pullback. (Note the lower arrow, hence the upper, must be monic

because T is left exact.)

We will prove that these maps jA form a topology on E . The following lemmas

assume that T is a left exact triple, E is a category with ¬nite limits, and j is

de¬ned as above.

Lemma 1. Whenever

AE EA

0

(2)

c c

BE EB

0

216 6 Permanence Properties of Toposes

is a pullback, then so is

jA(A0E EA

)

(3)

c c

E EB

jB(B0 )

Proof. This can be read o¬ the following square in much the same way that the

naturality of ¦ was deduced from (3) of Section 6.4.

EA

jA(A0 )

d

d

d

‚

©

jB(B0 ) E B

(4)

c c

T B0 E T B

d

s

d

c dc

E TA

T A0

In this square, the inner and outer squares are pullbacks by de¬nition and the

bottom square because T is left exact.

Lemma 2. A0 ⊆ jA(A0 ) for any subobject A0 of an object A.

Proof. Use the universal property of pullbacks on ·A0 and the inclusion of A0 in

A.

Lemma 3. For any subobject A0 of an object A, T jA(A0 ) = jT A(T A0 ) = T A0 .

Proof. To prove that T jA(A0 ) = T A0 , apply T to (1) and follow it by µ, getting

µA0E

E T T A0

T jA(A0 ) T A0

c c

(5)

c c c

E TTA E TA

TA

T ·A µA

The right vertical map from T A to T A is the identity, so T jA(A0 ) ’ T T A0

’

’ T A0 ’ T A is the inclusion. Cancelling the top and bottom arrows then

’ ’

shows that the left vertical arrow is an inclusion. This shows that T jA(A0 ) ⊆

T A0 , while the opposite inclusion is evident from Lemma 2.

6.5 Left exact triples 217

To show that jT A(T A0 ) = T A0 , consider the following similar diagram.

·T A0 µA0E

E

T A0 T T A0 T A0

c c

(6)

c c c

E TTA E TA

TA

·T A µT A

In the same way as for (5), the left and right vertical arrows from top to bottom

are identities. This means the outer square is a pullback, so by Exercise 12 of

Section 2.2, the upper square is too, as required.

If A0 )’ A1 )’ A then jA(A0 ) ⊆ jA(A1 ).

Lemma 4. ’ ’

Proof. Easy consequence of the universal property of pullbacks.

Lemma 5. A0 )’ A is j-dense if and only if there is an arrow from A to T A0

’

making I rotated the following, to be compatible

·A0

E

A0 T A0

c c

(7)

c c

E TA

A

·A

commute.

Proof. If jA(A0 ) = A, then there is a pullback diagram of the form

E T A0

A

c c

id (8)

c c

E TA

A

˜·A

(Notice the subtle point here: jA(A0 ) is de¬ned as a subobject of A, which

means that if it equals A the top arrow must be the identity). This gives the

diagonal arrow in (7) and makes the lower triangle commute; but then the upper

one does too since the bottom arrow is monic.

Conversely, if there is such a diagonal arrow, taking it as the left arrow in (8)

is easily seen to make (8) a pullback, as required.

218 6 Permanence Properties of Toposes

Theorem 6. Given a left exact triple T in a topos E , the maps jA de¬ned above

form a topology on E for which each object of the form T A is a j-sheaf.

Proof. That j is a natural transformation follows from Lemma 1 and Exercise 5 of

Section 2.3. Lemma 2 shows that j is in¬‚ationary, Lemma 3 that it is idempotent

(because the diagrams corresponding to (1) for jA and jjA become the same) and

Lemma 4 that it is order-preserving.

We use Proposition 8 of Section 6.2 to show that T A is a sheaf. Assume that

B0 is a dense subobject of an object B and f : B0 ’ T A is given. We must ¬nd

’

a unique extension B ’ T A. This follows from the following diagram, in which

’

g is the arrow given by Lemma 5.

f E TA

B0

T

©

·B0 µA ·T A (9)

B

gd

dc

‚ c

E TTA

T B0

Tf

The required map is µA —¦ T f —¦ g. It is straightforward to show that it is the

unique map which gives f when preceded by the inclusion of B0 in B.

Factorization of geometric morphisms

Now suppose that U = U— : E ’ E is a geometric morphism with inverse

’

image map U — . Like any adjoint pair, U determines a triple T = U— —¦ U — , ·, µ)

on E . Let j be its topology induced as in Theorem 6, and Ej the category of

j-sheaves.

For any object A of E , U A is a j-sheaf.

Proposition 7.

Proof. We again use Proposition 8 of Section 6.2. Suppose that B0 )’ B is

’