the sequence B ⊆ Bω ⊆ Bωω ⊆ Bωωω ⊆ · · ·, and repeating the above argument

gives the required result.

Proposition 11. An N-standard topos is standard.

Proof. The conclusion is valid as soon as it is valid in every countable subtopos.

Propositions 6 and 7 reduce it to the case of a 2-valued N-standard Boolean

topos. Accordingly, let B be such a topos, B be an object of B and R ⊆ B — B

270 7 Representation Theorems

be a re¬‚exive, symmetric relation on B. Using the mapping properties of N, let

f : N ’ 2B—B be the unique map such that the diagram

’

EN

N

f f

1

d

˜∆™ d

dc

‚ c

E 2B—B

2B—B R—¦’

commutes. Here ˜∆™ corresponds to the diagonal of B — B and R —¦ ’ is the

internalization of the operation on subobjects of B — B which is forming the

composition with R. It is constructed using the Yoneda lemma.

The transpose of f is a map g: N—B —B ’ 2 which classi¬es Q = [(n, b1 , b2 ) |

’

(b1 , b2 ) ∈ R(n) ]. The image E of Q in B — B under the projection is intuitively

the set of all [(b1 , b2 )] which are in some R(n) . The diagram

n—B—BE E2

B—B N—B—B

(—)

c c c

EQ E1

(n)

R

shows that every R(n) is contained in E. We claim that E is an equivalence

relation. First observe that in a 2-valued N-standard Boolean topos the only

global sections of N are the standard ones (that is are either 0 or one its successors;

this is, of course, where the present use of “standard” comes from). For when

1 has no proper non-0 subobjects, any two global sections are equal or disjoint.

Moreover, in an N-standard topos, no subobject of N, hence no global section

can be disjoint from every standard global section. Hence, every global section

is a standard one. Second observe that in a well-pointed topos, two subobjects

of an object are equal if and only if they admit the same global sections. Thus

to show that E —¦ E ⊆ E, it is su¬cient to show it on global sections. A global

section of E lifts to one on Q and a global section on E —¦ E lifts to a pair of

sections (n, b1 , b2 ) and (m, b2 , b3 ) such that (b1 , b2 ) ∈ R(n) and (b2 , b3 ) ∈ R(m) . We

can easily show by induction that

R(n) —¦ R(m) ⊆ R(m+n) ⊆ E

from which it follows that E is an equivalence relation.

7.6 Countable Toposes and Separable Toposes 271

To complete the argument, we observe that since the natural numbers cover

N and products have adjoints, the maps n — B — B cover N — B — B as n varies

over the standard natural numbers. Since (*) is a pullback, it follows that the

various R(n) cover Q and a fortiori cover E. Hence B is standard.

Proposition 12. [Freyd] Every countable standard topos can be embedded ex-

actly into a power of Set.

Proof. By Proposition 5 a countable standard topos has an exact embedding into

an N-standard topos which by Proposition 6 can be exactly embedded into an

N-standard Boolean topos. By replacing the target by a countable subtopos, we

can apply Proposition 10 to embed it logically”hence exactly”into a product of

N-standard well-pointed toposes. In a well-pointed topos, the functor Hom(1, ’)

is an exact Set-valued functor. Putting this all together, we have the desired

conclusion.

Recall that a separable topos is the category of sheaves on a site which is

both countable and for which there is a set of countable covers that generate the

topology.

Theorem 13. [Makkai and Reyes] If E is a separable Grothendieck topos, then

there is a faithful family of Set-valued left exact functors on E which have right

adjoints.

Set-valued functors on a topos which are left exact and have a right adjoint

are called points. This theorem says that a separable Grothendieck topos has

enough points.

Proof. Let E be a separable topos. For each object C of E and proper subobject

E of C, we construct inside E a countable subcategory C with certain properties.

It should be a topos; it should contain the image of E )’ C, the NNO of E (so

’

C is standard), and a generating set for E . It should contain all the objects used

in the countably many sieves {Ai ’ A} that generate the topology. It should

’

also include the sum S of the objects that appear in each such sieve, together

with the coproduct injections, the induced map S ’ A and the unique arrow S

’

’ N for which

’

E1

Ai

(—)

c c

EN

A

is a pullback. Since we are closing a countable category under ¬nitary and count-

able operations, the resulting category is countable, by exactly the same kind of

272 7 Representation Theorems

argument that you use to show that the free group generated by a countable set

is countable. Note that S is not in general the categorical sum in C .

Now apply Freyd™s Theorem 12 to get a faithful family of exact functors

C ’ Set. Each such functor takes (*) to a similar diagram from which it is

’

clear that each such functor preserves the sum An . Since that sum maps

epimorphically onto A, each of the functors preserves the covers which generate

the topology and hence extends to a point of E .

7.7 Barr™s Theorem

By adapting the proof of Theorem 4 of Section 7.1, we can obtain the following

proposition, from which we can deduce an embedding theorem for any Grothen-

dieck topos.

Proposition 1. For every small Boolean topos B there is a small Boolean

topos B in which subobjects of 1 generate, and a logical morphism B ’ B

’

which preserves epimorphic families.

Proof. . Well-order the set of objects of B which have global support. Given

a Boolean topos B and an object A of global support, the map B ’ B/A is ’

faithful by Exercise 1 of Section 5.2, and preserves epimorphic families because

it has a right adjoint. Moreover, the image of A has a global section. Thus

we can construct a well-ordered sequence of Boolean toposes B± together with

faithful logical morphisms Tβ,± : B± ’ Bβ whenever ± ¤ β as follows. Begin by

’

letting B0 = B. Having constructed B± together with the appropriate transition

functor, let A be the least object in the well-ordering which has global support

and lacks a global section. Take B±+1 = B± /A. At a limit ordinal take a direct

limit. In the latter case preservation of epimorphic families follows from Lemma 9

of Section 7.6. Taking the direct limit of this sequence, we get a Boolean topos

B and a faithful logical morphism B ’ B which preserves epimorphic families,

’

again by Lemma 9. Moreover, every object of B with global support has a global

section in B.

By iterating this construction a countable number of times, and taking the

directed limit of the resulting sequence, we get a topos B and a logical morphism

B ’ B which is faithful and preserves epimorphic families. In B, every object

’

with global support has a global section. If A is an arbitrary object of B with

support S, let T be the complement of S in the subobject lattice of 1. Then the

object A + T has a global section which restricts to a section of A ’ S. Hence

’

the subobjects of 1 generate.

7.7 Barr™s Theorem 273

Theorem 2. [Barr] Every Grothendieck topos has a left exact cotripleable em-

bedding into the topos of sheaves on a complete Boolean algebra.

Proof. To show that an exact functor is cotripleable, it is su¬cient to show that

it is faithful and has a right adjoint. Let E be a Grothendieck topos. Let C be

a small topos contained in E which contains a set of generators for E so that

E is the category of canonical sheaves on C as in Theorem 1 of 6.8. Combining

Theorem 3 of 7.1 with the above proposition we conclude that C can be embedded

into a Boolean topos B # which is generated by its subobjects of 1 so that the

embedding is left exact and preserves epimorphic families. If B is the completion

of the Boolean algebra B of subobjects of 1 in B # , we claim that B # is embedded

in the category Sh(B) of sheaves on B and this embedding is exact and preserves

epimorphic families. Assuming this true, the result follows from Theorem 2 of

7.3 plus Theorem 1 of 6.8.

To complete the argument, we must show that B # is embedded in Sh(B).

There is a functor B # ’ Sh(B) given by representing an element b ∈ B as

’

sup bi , with bi in B and de¬ning, for an object C of B # the presheaf on B given

by colim Hom(bi , C). These are not necessarily sheaves, so we must follow this by

the associated sheaf functor. To see that the resultant composite is faithful, let

C )’ D be a non-isomorphic mono in B # and let E be a complement of C in

’

D. Then E is not the initial object of B # so that it has a section over a non-zero

object of B and hence is not the initial presheaf. The associated sheaf contains

a quotient of E and cannot be the initial sheaf. But it remains a complement of

the sheaf associated to C so that the inclusion of C into D does not induce an

isomorphism and the functor is faithful.

Note that the existence of the right adjoint to this left exact embedding means

that the embedding is the left adjoint part of a geometric morphism. Part of the

signi¬cance of this result arises from the following.

Theorem 3. Let E be the category of sheaves on a complete Boolean algebra.

Then E satis¬es the Axiom of Choice.

Proof. We begin by observing that if B is a complete Boolean algebra, a functor

F : B op ’ Set is a sheaf if and only if whenever b = bi is a disjoint union, F b ∼

’ =

F bi , the isomorphism being the canonical map. The condition is necessary for

then the bi cover b and the intersection of any two of them is empty. To see that

the condition is su¬cient it is enough to note that every cover has a re¬nement

which is disjoint. For if b = bi is not disjoint, choose a simple ordering of the

index set and replace bi by bi ’ bj , the union taken over the j i.

Next we note that an epimorphism is onto. Let f : F ’’ G be an epimorphism

’

of sheaves and G0 )’ G be the presheaf image of the map. We claim that

’

274 7 Representation Theorems

G0 = G. For let b = bi be a disjoint union. In the diagram

E GE

E0 EG

Fb

c c c

E

E E E

F bi G0 bi Gbi

it is easy to see that when the outer vertical arrows are isomorphisms, so is the

middle one. Thus G0 is a sheaf, which shows that G0 = G. Now to split the epi f ,

we choose a maximal element (J, h) among the partially ordered set of pairs (I, g)

in which I is an ideal of B and g is a splitting of f |I. If J is not the whole of B,

we consider separately the cases that J is or is not principal. In the former case,

let J = (b) and suppose b ∈ J. Then we may replace b by b ’ b and suppose that

/

b is disjoint from b. Given an element of x ∈ Gb , choose an arbitrary element of

y ∈ F b mapping to it and extend h by h(b ∪ c) = y ∪ hc. It is easy to see that

this gives a morphism of presheaves, hence of sheaves. If J is not principal, let

b = J and extend h to b by h(b) = h(J). We see, using the fact that F and

G are sheaves that this extends h. This completes the construction of a section

of f .

7.8 Notes to Chapter 7

The results of Section 7.1, Section 7.5, and the parts of Section 7.6 pertaining to

standard toposes appeared in a remarkable paper that Peter Freyd wrote during

a visit to Australia in 1971 [Freyd, 1972]. These results form the basis for the

modern representation theory of toposes. We have followed Freyd™s exposition

quite closely.

Although Barr™s theorem [Barr, 1974] appeared two years later than Freyd™s

paper, the work was done in ignorance of Freyd™s work with a proof quite di¬erent

from that presented here. The latter is, of course, based on the ideas used by

Freyd. There is an entirely di¬erent proof of this theorem due to Diaconescu

which is given in Johnstone [1977]. The result was in response to a question

of Lawvere™s as to whether the example of a Boolean-valued model of set theory

for which the Boolean algebra lacked points (complete 2-valued homomorphisms)

was essentially the most general example of a topos without points. The result,

that every topos has a faithful point in a suitable Boolean topos, showed that the

Lawvere™s surmise was correct.

Diaconescu [1975] was the ¬rst to show that AC implied Boolean.

7.8 Notes to Chapter 7 275

Deligne™s theorem [SGA 4, 1972] was proved for the purposes of algebraic

geometry. The original proof was completely di¬erent. So far as we know, this

is the ¬rst place in which it has been derived as a simple consequence of Freyd™s

theorem.

Makkai and Reyes [1977] proved the theorem credited to them without refer-

ence to Freyd™s work. Again, this seems to be the ¬rst place in which the proof

is done using Freyd™s results.

Johnstone [1977] has put down Freyd™s representation theorems as “. . . some-

thing of a blind alley”. This chapter clearly demonstrates the utility of the

theorems. It is possible, of course, to want to avoid the use of Freyd™s theorems

out of dislike of the use of representation theory for proving things, or from a

more general preference for elementary or constructive methods. We do not share

those attitudes. We feel it is a matter of taste whether, for example, the proof we

have given of the fact that AC implies Boolean is better or worse than a harder,

but elementary proof. We generally prefer a proof which is more readily under-

stood. (That is not necessarily the same as easier”see our proof of Proposition 1

of Section 7.3).

There are other useful representation theorems. For example, Makkai and

Reyes [1977, Theorem 6.3.1] show that any Grothendieck topos can be embedded

into the topos of sheaves on a complete Heyting algebra by a functor which is

the left adjoint part of a geometric morphism and which preserves all infs as well

as ∀. They also show [Theorem 6.3.3] that when there are enough points the

Heyting algebra can be taken to be the open set lattice of a topological space.

In both Abelian categories and regular categories there is a full embedding

theorem, which states that there is a full, exact embedding into a standard cate-

gory. In the case of Abelian categories, the standard categories are the categories

of R-modules for rings R, while in the regular case it was Set-valued functor cat-

egories. (A Set-valued functor category can be viewed as the category of M -sets

where is M is a monoid with many objects, i.e. a category.) The corresponding

theorem for toposes would be a full, exact embedding into a functor category.

Makkai [unpublished] has given an example of a topos that has no such full em-

bedding. Fortunately, these full embeddings have had very limited usefulness.

The existence of an embedding that re¬‚ects isomorphisms has allowed all the

diagram-chasing arguments that one seems to need.

It should be observed that hypotheses that a topos be small or even countable