Given an epi A ’ B and a global element of B, the pullback

’

EA

C

c c

c c

EB

1

and the map 1 ’ C provides a map 1 ’ A which shows that Hom(1, A)

’ ’

’ Hom(1, B) is surjective, as required.

’

The preservation of epimorphic families follows from the preservation of sums

and epimorphisms.

Finally, given a pushout

E EB

A

c c

E ED

C

the Booleanness implies that B = A + A for some subobject A of B, whence

D = C + A , which we know is preserved by Hom(1, ’).

(d) If A ⊆ B = A+A and A = 0 then the existence of A ’ 1 ’ A provides

’’

a splitting for the inclusion A ’ B. (See Exercise 3 of Section 2.1.)

’

As for (e), let A be an object di¬erent from 0 and 1. Since A = 0, there is

a map 1 ’ A and since A = 1, that map is not an isomorphism so that there

’

is a second map 1 ’ A that does not factor through the ¬rst. Any map out

’

of 1 is a monomorphism. Since 1 has no non-zero subobjects, these determine

disjoint subobjects of A each of which is isomorphic to 1. Since they are disjoint,

their sum gives a mono of 2 into A. Hence it is su¬cient to show that 2 is a

cogenerator. Given a parallel pair

f

’’C

’

B ’’

’

g

with f = g, let v: 1 ’ B be a map with f —¦ v = g —¦ v. Then we have the map

’

[f —¦ v, g —¦ v]: 1 + 1 ’ C which is a mono by exactly the same reasoning and, since

’

2 is injective, splits. But this provides a map h: C ’ 2 for which h —¦ f —¦ v is the

’

one injection of 1 into 2 and h —¦ g —¦ v is the other, so that they are di¬erent. Hence

h —¦ f = h —¦ g which shows that 2 is a cogenerator.

7.1 Freyd™s Representation Theorems 243

Embedding theorems

Theorem 3. Every small topos has an exact embedding into a Boolean topos.

This embedding preserves epimorphic families and all colimits.

Proof. Let E be the topos. For each pair f, g: A ’ B of distinct arrows we will

’

construct a left exact, colimit preserving embedding of E into a Boolean topos

which keeps f and g distinct, and then take the product of all the Boolean toposes

so obtained. It is an easy exercise (Exercise 6) that the product (as categories)

of Boolean toposes is a Boolean topos.

The map from E to E /A which takes an object C to C — A ’ A certainly

’

distinguishes f and g. In the category E /A, the diagonal arrow A ’ A — A’

followed by A — f (respectively A — g) gives a pair of distinct global elements

of B whose equalizer is a proper subobject U of 1. The topology j induced by

U as in Example (c) of Section 6.1 makes the equalizer 0 in Shj (E /A), which is

clearly a nondegenerate topos. The double negation sheaves in that category is

a Boolean topos with the required property (Exercise 3). The arrows that f and

g go to are still distinct because ¬¬0 = 0.

The limit preservation properties follow from the fact that the map E ’ E /A

’

has both adjoints and the associated sheaf functor is left exact and has a right

adjoint.

Theorem 4. Every small Boolean topos B has a logical embedding to a product

of small well-pointed toposes.

Proof. The argument goes by constructing, for each nonzero object A of B, a

logical morphism T : B ’ C (where C depends on A) with C well-pointed and

’

T A = 0. This will show that the the mapping of B into the product of all the

categories C for all objects A is an embedding (Exercise 4).

The proof requires the following lemma.

Lemma 5. For every small Boolean topos B and nonzero object A of B there

is a small topos B and a logical morphism T : B ’ B with T A = 0 and such

’

that for all objects B of B either T B = 0 or T B has a global element.

Proof. Well order the objects of B taking A as the ¬rst element. Let B0 = B

and suppose that for all ordinal numbers β < ±, Bβ has been constructed, and

whenever γ < β, a family of logical morphisms uβγ : Bγ ’ Bβ is given such that

’

(i) uββ = 1 and

244 7 Representation Theorems

(ii) for δ ¤ γ ¤ β, uβγ —¦ uγδ = uβδ .

(Such a family is nothing but a functor on an initial segment of ordinals

regarded as an ordered category and is often referred to as a coherent family).

If ± is a limit ordinal, let B± be the direct limit of the Bβ for β < ±. If ±

is the successor of β then let B be the least object of B which has not become

0 nor acquired a global section in Bβ . Let B be its image in Bβ and let B± be

Bβ /B. Stop when you run out of objects. Since toposes are de¬ned as models of

a left exact theory and logical functors are morphisms of that theory, it follows

from Theorem 4 of Section 4.4 that the direct limit is a topos. By Exercise 1,

the last topos constructed by this process is the required topos. It is easy to see

that the functors in the cone are logical.

To prove Theorem 4, form the direct limit C of B, B , B (forming B using

the image of A in B ) and so on. The image of A in C will be nonzero, and every

nonzero object of C has a global element.

The product of all these categories C for all objects A of B is the required

topos.

Theorem 6. Every small topos has an exact embedding into a product of well-

pointed toposes.

Theorem 7. [Freyd™s Embedding Theorem] Every small topos has an embed-

ding into a power of Set that preserves ¬nite limits, ¬nite sums, epimorphisms,

and the pushout of a monomorphism.

Proof. Every well-pointed topos has a functor to Set, namely Hom(1, ’), with

those properties.

Exercises 7.1

1. Prove that a Boolean topos is well-pointed if and only if every nonzero object

has a global section.

2. Prove that well-pointed toposes are not the models of an LE theory.

3. Show that the category of sheaves for the topology of double negation (Ex-

ercise 5 of Section 6.1) is a Boolean topos. (Hint: In any Heyting algebra

¬¬a = ¬¬¬¬a.)

4. (a) Show that an exact functor from a Boolean topos is faithful if and only

if it takes no non-zero object to zero.

(b) Show that an exact functor from a 2-valued topos to any non-degenerate

topos is faithful.

7.2 The Axiom of Choice 245

5. Show that the embeddings of Theorems 6 and 7 re¬‚ect all limits and colimits

which they preserve. (Hint: First show that they re¬‚ect isomorphisms by consid-

ering the image and the kernel pair of any arrow which is not an isomorphism.)

6. (a) Show that the product as categories of toposes is a topos.

(b) Show that the product of Boolean toposes is Boolean.

7. A category has stable sups if the supremum of any two subobjects exists

and is preserved by pullbacks. It has stable images if for any arrow f : A ’ B,

’

Sub f : Sub B ’ Sub A has left adjoint which is preserved by pullbacks. A left

’

exact category with stable sups and stable images is called a logical category.

Show that a category is a pretopos if and only if it is logical, has ¬nite disjoint

sums and e¬ective equivalence relations. (This comes from [Makkai and Reyes,

1977, pp.121“122].)

7.2 The Axiom of Choice

The Axiom of Choice

If f : A ’ B is an arrow in a category, we say that a map g: B ’ A is a

’ ’

section of f if it is a right inverse of f , i.e. f —¦ g = 1. If f has a section, we

say that it is a split epi (it is necessarily epi), although the second word is often

omitted when the meaning is clear. It is easy to see that the Axiom of Choice

in ordinary set theory is equivalent to the statement that in the category of sets,

all epis are split. A section of the map (): A ’ 1 is a global element. A global

’

element of A is thus often called a global section of A.

We say a topos satis¬es the Axiom of Choice (AC) if every epi splits. It is

often convenient to break this up into two axioms:

(SS) (Supports Split): Every epimorphism whose codomain is a subobject of

1 splits.

(IAC) (Internal Axiom of Choice): If f : A ’ B is an epi, then for every

’

C C C

object C, f : A ’ B is an epi.

’

The name “Supports Split” comes from the concept of the support of an

object X, namely the image of the map X ’ 1 regarded as a subobject of 1.

’

An object has global support if its support is 1.

It is an easy exercise that AC implies SS and IAC. It will emerge from our

discussion that SS and IAC together imply AC.

246 7 Representation Theorems

We say f : A ’ B is a powerful epi if it satis¬es the conclusion of IAC. We

’

de¬ne § f by the pullback

E AB

§f

fB

c c

E BB

1

where the lower map is the transpose of the identity. Intuitively, § f is the set of

sections of f .

It is clear that if f B is epi then § f has global support (the converse is also

true, see Exercise 1) and that § f has a global section if and only if f has a section.

In fact, global sections of § f are in one to one correspondence with sections of f .

Proposition 1. For a morphism f : A ’ B in a topos E , the following are

’

equivalent:

(a) f is a powerful epi;

(b) f B is epi;

(c) § f has global support.

(d) There is a faithful logical embedding L: E ’ F into some topos F such

’

that Lf is split epi.

Proof. (a) implies (b) by de¬nition. (b) implies (c) because a pullback of an

epi is epi. To see that (c) implies (d) it is su¬cient to let F be E / § f and L

be § f — ’. By Theorem 6 of Section 5.3, L is faithful and logical. L therefore

preserves the constructions of diagram (1). In the corresponding diagram in F ,

§ Lf has a global section (the diagonal) which corresponds to a right inverse for

Lf . For (d) implies (a), let C be an object and g a right inverse for Lf . Then

g LC is a right inverse for L(f C ), which is isomorphic to (Lf )C . Thus L(f C ) is

epi, which, because L is faithful, implies that f C is epi.

Proposition 2. Given a topos E there is a topos F and a logical, faithful

functor L: E ’ F for which, if f is a powerful epi in E , then Lf is a split epi

’

in F .

Proof. Well order the set of powerful epis. We construct a trans¬nite sequence

of toposes and logical morphisms as follows: If E± is constructed, let E±+1 be

E± / § f± where f± is the powerful epi indexed by ± and the logical functor that

constructed by Proposition 1. At a limit ordinal ±, let E± be the direct limit of

all the preceding logical functors. The required topos F is the direct limit of this

family. As observed in the proof of Lemma 4 of Section 7.1, the direct limit of

toposes and logical morphisms is a topos and the cone functors are logical.

7.2 The Axiom of Choice 247

Corollary 3. Given a topos E , there is a topos F in which every powerful epi

splits and a faithful, logical functor L from E to F .

Proof. Repeat the above process countably often.

Corollary 4. A topos satis¬es the IAC if and only if it has a faithful, logical

embedding into a topos that satis¬es the Axiom of Choice.

Proof. The “if” part is very easy. For if a topos has such an embedding, it is

immediate that every epi is powerful. So suppose E is a topos that satis¬es the

IAC. If we show that every slice and any colimit of such slices (called a limit

slice because it is a limit in the category of geometric morphisms) satis¬es the

IAC, then the topos constructed above will have every epi powerful and every

powerful epi split. The limit part is trivial since every epi in the colimit is an

epi before the colimit is reached (since the functors are all faithful). Thus it is

su¬cient to show that slicing preserves the IAC. If

EC

B

d

gd h

d

d

‚©

A

is an epimorphism in E /A, then f : B ’ C is epi in E . Since E satis¬es the

’

IAC, the object § f has global support. Now consider the diagram of toposes and

logical functors:

E E /§f

E

c c

E E /(§f — A)

E /A

If we apply § f — ’ to (*), we get

§f — f E

§f — B §f — C

d

§f — gd §f — h

d

d

‚©

§f — A

The map § f — f has a splitting in E / § f . It is immediate, using the fact that