(yCk ) ’’ (yCi ) ’ F

’

’’

Proof. This is a trivial consequence of the fact that every functor, hence every

sheaf is a colimit of a diagram of representables.

Lemma 11. If C is closed in E under subobjects, then every cover in C is also

a cover in E .

Proof. Let {Ci ’ C} be a cover in C . As seen in Exercise 4 of 5.5, a regular

’

category has a factorization system using regular epis and monos. The regular

image in E of Ci ’ C is a subobject C0 ⊆ C which, by hypothesis, belongs

’

to C . If C0 = C, the family is not regular epimorphic.

Now we can ¬nish the proof of Proposition 2. We already know that y is full

and faithful. Let F be a sheaf and represent it by a diagram as in Lemma 10

above. The fact that y preserves sums implies that there is a coequalizer

yE1 ’ yE0 ’ F

’ ’

’’

and the fact that y is full means the two arrows from E1 to E0 come from maps

in E . Letting E be the coequalizer of those maps in E , it is evident that yE ∼ F .

=

Hence y is an equivalence.

The last sentence of the proposition follows from Lemma 11.

It should remarked in connection with this theorem that if E has a small reg-

ular generating set, C can be taken to be small by beginning with the generating

236 6 Permanence Properties of Toposes

set and closing it under ¬nite products and subobjects. However, nothing in the

proof requires C to be small and it could even be taken to be E itself. Of course,

in that case, it may be thought that the functor category and the re¬‚ector L may

not necessarily exist, but this is a philosophical, not a mathematical objection.

For those who care, we remark that it is a topos in a larger universe. More-

over, the “re¬‚ection principle” guarantees that the theorem is valid even in the

category of all sets.

Theorem 12. Any left exact cocontinuous functor between Grothendieck toposes

is the left adjoint part of a geometric morphism.

Proof. We need only show that the right adjoint exists. But Grothendieck toposes

are complete with generators so the Special Adjoint Functor Theorem guarantees

the existence.

A category whose Yoneda embedding has a left adjoint is called total. If

the left adjoint is also left exact, the category is called lex total. Freyd and

Street have characterized Grothendieck toposes as lex total categories satisfying a

mild size restriction [Street, 1981]. Street [1983] characterizes lex total categories

in terms of conditions on epimorphic families generalizing the requirements in

Giraud™s theorem of having universal e¬ective epimorphisms. Total categories are

cocomplete in a strong sense. Street [1983] says of them that they are “...precisely

the the algebraic and topological categories at which traditional category theory

was aimed.”

Some of the more obvious relations among these various notions appear in the

theorem below.

Theorem 13. Let E be a category. Then each of the following properties implies

the next. Moreover, (iv) plus the existence of a generating set implies (i).

(i) E is a Grothendieck topos,

(ii) E is lex total,

(iii) y: E ’ Sh(E ) is an equivalence,

’

(iv) E is a complete topos.

Proof. We prove that (ii) implies (iii), which requires the most argument, and

leave the others as exercises.

Suppose E is lex total. Then Y : E ’ Psh(E ) has a left exact left adjoint Y # .

’

It follows from results of Section 6.5 that E is a cocomplete topos. Let I: Sh(E )

’ Psh(E ) be the inclusion with left adjoint L, and as before, let y = LY . We will

’

use the fact that y preserves colimits; this follows from the fact that it preserves

sums and (regular because E is a topos) epis, and, as we will see in Section 7.6,

Propositions 1 and 2, that is enough in a countably complete topos. To show

6.8 Giraud™s Theorem 237

that y is an equivalence, it is enough to show that every sheaf F is isomorphic

to yE for some object E of E . Since every presheaf is a colimit of representables

IF = colim Y Ei , so

F = LIF = colim LY Ei = colim yEi = y colim Ei

as required.

We make no attempt here to investigate the reverse implications. Exercises 6

and 7 give an example of a complete topos that lacks a small generating set.

Exercises 6.8

1. We have de¬ned a sieve and thus a cover in terms of collections {Ai ’ A}. ’

There is another way. Let us say that a Giraud sieve on A is a subfunctor

of the representable functor Hom(’, A). Say that a collection of Giraud sieves

forms a Giraud topology if it includes the identity sieve on every object and is

invariant under pullback and composition. Further say that a sieve (in the sense

used previously in this book) {Ai ’ A} is saturated if whenever an arrow B

’

’ A factors through one of the arrows in the sieve, it already is one of the arrows

’

in the sieve. Show that there is a one-to-one correspondence between saturated

sieves and Giraud sieves on an object A. Conclude that topologies and Giraud

topologies are equivalent.

2. Let C be a site and E = Sh(C ).

(a) Show that there is a presheaf which assigns to each object the set of

Giraud sieves on that object and that that is the subobject classi¬er „¦ in the

presheaf category.

(b) Show that the presheaf „¦j that assigns to each object the set of Giraud

covers is a sheaf and in fact the subobject classi¬er in the sheaf category.

(c) Show that the topology j (which, recall, can be viewed as an endomor-

phism of „¦) is the classifying map of „¦j .

3. Let C be a site and E = Shv(C ). Prove that a Grothendieck topology on

E is contained in the ¬¬ topology if and only if no cover in the Grothendieck

topology is empty.

4. (a) Show that in a category C a set U of objects is a set of generators if and

only if when f, g: C ’ B are distinct then there is an element c ∈G C for some

’

G ∈ U for which f (c) = g(c).

(b) Show that any regular generating family is conservative.

238 6 Permanence Properties of Toposes

(c) Show that in a category with equalizers, any conservative generating

family is a generating family.

(d) Show that in a topos any generating family is conservative. (Hint: every

mono is regular.)

(e) Show that in a complete topos, every epimorphic family is regular; hence

the converse of (b) is true and every generating family is regular.

5. Let E be a countably complete (hence countably cocomplete) topos and R ⊆

A — A a relation on an object A. If S1 and S2 are two relations on A, then the

composite of S1 and S2 is, as usual, is the image of the pullback S1 —A S2 where

S1 is mapped to A by the second projection and S2 by the ¬rst.

(a) Let S = R ∪ ∆ ∪ Rop . Show that S is the re¬‚exive, symmetric closure of

R.

(b) Let E be the union of the composition powers of S. Show that E is the

equivalence relation generated by R.

(Hint: Pullbacks, unions and countable sums are all preserved by pulling

back.)

6. By a measurable cardinal ± is meant a cardinal number for which there

is an ultra¬lter f with the property that if any collection {Ui }, i ∈ I is given

for which #(I) < ±, and each Ui ∈ f, then ©Ui ∈ f. Call such an f an ±-

measure. It is not known that measurable cardinals exist, but for this exercise,

we will assume not only that they exist, but that there is a proper class of them.

So let ±1 , ±2 , . . . , ±ω , . . . (indexed by all ordinals) be an increasing sequence of

measurable cardinals and f1 , f2 , . . . , fω , . . . a corresponding sequence of ultra¬lters.

Assume the following (known) property of measurable cardinals: they are strongly

inaccessible, meaning they cannot be reached by operations of product, sum or

exponentiation involving fewer, smaller cardinals.

De¬ne, for any set S and ultra¬lter f a functor F S = colim S U , U ∈ f. Note

that the diagonal map S ’ S U induces a function S ’ F S, which is, in fact,

’ ’

the component at S of a natural transformation.

(a) Show that the functor F is left exact.

(b) Show that if ± is measurable, f an ±-measure and #(S) < ±, then S

’ F S is an isomorphism.

’

(c) Show that if Fi is de¬ned as above, using the ultra¬lter fi on ±i , then

there is a sequence Hi of left exact endofunctors on Set de¬ned by Hj+1 = Fj —¦ Hj

and Hj = colim Hk , k < j when j is a limit ordinal.

6.8 Giraud™s Theorem 239

(d) Show that for any set S there is an i dependent on S such that for j > i,

Gj (S) ’ Gi (S) is an isomorphism.

’

(e) Conclude that there is a left exact functor H: Set ’ Set whose values

’

are not determined by the values on any small subcategory.

(f ) Show that there is a left exact cotriple on Set — Set whose functor part is

given by G(S, T ) = (S — HT, T ) for which the category of algebras does not have

a set of generators and hence is not a Grothendieck topos.

7. Let C be a proper class and G be the free group generated by C. (If you don™t

like this, we will describe an alternate approach later. Meantime continue.) Let

E be the category of those G-sets which have the property that all but a small

subset of the elements of C act as the identity automorphism. Show that E is a

complete topos which does not have a small generating set.

An alternate approach to the same category is to take as an object a 3-tuple

(S, C0 , f ) in which S is a set, C0 a sub-set of C and f : C0 ’ Aut(S) a function.

’

Morphisms are de¬ned so as to make this the category of G-sets as de¬ned above.

Although set-theoretically unassailable, this approach seems conceptually much

less clear.

As a matter of historical interest, this category was one of the earliest known

examples to show the necessity of the solution set condition in the GAFT. The

evident underlying set functor is easily seen to lack an adjoint while satisfying all

the other conditions. We believe it is due to Freyd.

8. Prove that if U is a set of regular generators for a category C , then every

object of C is a colimit of an indexed family of objects of U .

7

Representation Theorems

7.1 Freyd™s Representation Theorems

In this section we prove a number of theorems due to Freyd representing toposes

into various special classes of toposes. The development follows Freyd [1972] very

closely.

Terminology

We de¬ne several related concepts which will be used in Chapters 7 and 8. We

have put all the de¬nitions here, although they are not all used in this section,

because some of the terminology varies in the literature.

A functor is regular if it is left exact and preserves regular epis (but it

need not preserve all coequalizers.) An exact category is a category which

has ¬nite limits and ¬nite colimits. An exact functor is one which preserves

¬nite limits and colimits, i.e., it is left and right exact. In the past the term

“exact category” has been used to denote a regular category which has e¬ective

equivalence relations (every equivalence relation is the kernel pair of some arrow),

and regular functors have also been called exact functors, but we will not use that

terminology.

A pretopos is a left exact category with e¬ective equivalence relations which

has ¬nite sums which are disjoint and stable and in which every morphism factors

as a composite of a stable regular epimorphism and a monomorphism. The

corresponding type of morphism is a near exact functor which is left exact and

preserves regular epimorphisms and ¬nite sums. The main import of the work

of Makkai and Reyes [1977] is that pretoposes correspond to a broad class of

theories in the sense of model theory in mathematical logic.

Booleanness

A topos E is Boolean if for every subobject A of an object B, A ∨ ¬A ∼ B.

=

In the following proposition, 2 denotes 1 + 1.

The following are equivalent for a topos E :

Proposition 1.

240

7.1 Freyd™s Representation Theorems 241

(a) E is Boolean.

(b) Every subobject of an object in E has a complement.

(c) The Heyting algebra structure on „¦ as de¬ned in Section 5.6 is a Boolean

algebra.

(d) If false: 1 ’ „¦ is the classifying map of the zero subobject, then (true, false): 2

’

’ „¦ is an isomorphism.

’

Proof. (a) implies (b) because for any subobject A, A § ¬A = 0. (b) and (c) are

clearly equivalent by Theorem 4 of Section 5.6.

To see that (b) implies (d), observe that true:1 ’ „¦ has a complement A

’

’ „¦. But A ’ „¦ classi¬es subobjects just as well as true because there is

’ ’

a one to one correspondence between subobjects and their complements so that

Proposition 4 of Section 2.3 shows that A = 1.

Finally, if (d) holds, any map f : E ’ 2 de¬nes a complemented subobject

’

since E is the sum of the inverse images of the two copies of 1 (sums are stable

under pullback).

Warning: Even when 2 is the subobject classi¬er there may be other global

sections 1 ’ 2. In fact, the global sections of 2 can form any Boolean algebra

’

whatever. However, see Proposition 2 and Theorem 4 below. A topos is said to

be 2-valued if the only subobjects of 1 are 0 and 1.

A topos is well-pointed if it is nondegenerate (that is 0 = 1) and 1 is a

generator. By Exercise 4 of Section 6.8, this is equivalent to saying that for each

object A and proper subobject A0 ⊆ A, there is global element of A that does

not factor through A0 . In particular, every non-zero object has a global element.

Proposition 2. Let E be a well-pointed topos. Then

(a) E is 2-valued.

(b) E is Boolean.

(c) Hom(1, ’) preserves sums, epimorphisms, epimorphic families and pushouts

of monomorphisms.

(d) Every nonzero object is injective.

(e) Every object not isomorphic to 0 or 1 is a cogenerator.

Proof. (a) If U were a proper nonzero subobject of 1, the hypothesis would force

the existence of a map 1 ’ U , making U = 1.

’

(b) There is a map (true, false): 1 + 1 ’ „¦. This induces a map from

’

Hom(1, 1 + 1) to Hom(1, „¦) which is an isomorphism by (a). Since 1 is a genera-

tor, it follows that (true, false) is an isomorphism. Booleanness now follows from

Proposition 1.

242 7 Representation Theorems

(c) A map from 1 to ΣAi induces by pulling back on the inclusions of Ai into

the sum a decomposition 1 = ΣUi . By (a), all but one of the Ui must be zero

and the remaining one be 1, which means that the original map factors through