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(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

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