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exactly one Ai .
Given an epi A ’ B and a global element of B, the pullback


c c
c c
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

c 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

B ’’

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
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
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

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

c c
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

(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
gd  h

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

c c
E E /(§f — A)
E /A
If we apply § f — ’ to (*), we get
§f — f E
§f — B §f — C
§f — gd  §f — h

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


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