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c c
E„¦
1
Because Hom(A, ’) preserves pullbacks and Sub(A) ∼ Hom(A, „¦), the order
=
relation
¤A = {(B, C) | B ⊆ C ⊆ A}
on Sub(A) for any object A is then obtained by pulling back:
E Hom(A, „¦) — Hom(A, „¦)
¤A

’A (2)
c c
E Hom(A, „¦)
1

The following lemma follows immediately from the de¬nitions.
Lemma 1. Let x and y be elements of „¦ de¬ned on A. Let the corresponding
subobjects of A be B and C respectively. Then:
(a) x ¤ y if and only if B ¤A C.
(b) The subobject corresponding to x § y is B § C.
(c) The subobject corresponding to x ∨ y is B ∨ C.
(d) The subobject corresponding to x ’ y is B ’ C.
196 5 Properties of Toposes
Lemma 2. If B and C are subobjects of A, then the following are equivalent:
(e) B ¤A C
(f ) B ’A C = A.
Proof. B ¤A C is equivalent to x ¤ y by Lemma 1(a). That is equivalent to
x ’ y = true (diagram (1)), which is equivalent to B ’ C = A by Lemma 1(d).

Lemma 3. If B, C and D are subobjects of A, then
(g) D § (B ’A C) = D § B ’D D § C, and
(h) D ¤ B ’A C if and only if D § B ¤D D § C.
Proof. Intersecting by D is pulling back along the inclusion of D in A, and a
pullback of a pullback is a pullback; that proves (a). By Lemma 2, D ¤A B ’A C
if and only if D § (B ’A C) = D. By (a) and Lemma 1 applied to Sub(D), that
is true if and only if D § B ¤D D § C.
From now on, we will drop the subscript A on the relation ¤A and the oper-
ation ’A on Sub(A).
Theorem 4. For any object A in a topos E , Sub(A) is a Heyting algebra with
the operations de¬ned above.
Proof. The minimum is evidently the subobject 0 ’ A. Sub(A) is a lattice, so

all that is necessary is to prove that D ¤ B ’ C if and only if D § B ¤ C. If
D ¤ B ’ C, then D §B ¤ D §C ¤ C by Lemma 2(b). Conversely, if D §B ¤ C,
then clearly D § B ¤ D § C because § is the greatest lower bound operation;
then the result follows from Lemma 2(b) again.
Corollary 5. „¦ is a Heyting algebra, with minimum the unique element 0 ’ „¦

and §, ∨ and ’ as de¬ned above.
Proof. This follows immediately from Lemma 1.
When the Heyting algebra in „¦ is that of a Boolean algebra, we call the topos
Boolean. This is equivalent to saying that every subobject of an object has a
complement.
The category of sheaves over a topological space is a topos whose subobject
classi¬er is the sheaf whose value at an open set U is the set of open subsets
of U , with restriction given by intersection. (Exercise 4 of Section 2.3.) The
natural Heyting algebra structure on that sheaf is the Heyting algebra structure
of Corollary 5.
5.6 The Heyting Algebra Structure on „¦ 197
Exercises 5.6

1. Prove the following facts about Heyting algebras:
(a) F ’ F is the maximum of the lattice.
(b) If a ¤ b then b ’ c ¤ a ’ c and c ’ a ¤ c ’ b.
(c) a ’ b = T if and only if a ¤ b.
(d) b ¤ a ’ b.
(e) a § (a ’ b) = a § b. Hence a § (a ’ b) ¤ b.
(f ) a § ¬a = F .
(g) a ¤ ¬¬a.
(h) ¬a = ¬¬¬a
(i) a ¤ b implies ¬b ¤ ¬a

2. Show that a ¬nite lattice can be made into a Heyting algebra by a suitable
choice of “’” if and only if it is distributive. Show also that every chain is a
Heyting algebra. What is the double negation of an element in the latter case?

3. (Freyd) Show that the category of Heyting algebras is a Mal™cev category:
that means that there is a ternary operation µ(a, b, c) with the properties that
µ(a, a, c) = c and µ(a, b, b) = a. To de¬ne µ, ¬rst de¬ne

a ” b = (a ’ b) § (b ’ a)

and then let
µ(a, b, c) = ((a ” b) ” c) § (a ” (b ” c))
6
Permanence Properties of Toposes
This chapter is concerned with certain constructions on a topos which yield a
topos. We have already seen one such construction: a slice E /A of a topos E is
a topos. The most important construction in this chapter is that of the category
of “sheaves” in a topos relative to a “topology”. When the topos is a category of
presheaves on a space and the topology is the “canonical” one, the “sheaves” are
ordinary sheaves. The category of sheaves in a topos (relative to any topology,
canonical or not) turns out to be a topos.
The concept of topology is an abstraction of the concept of all coverings, which
at one level of abstraction is a “Grothendieck topology” and at a higher level is
a “topology on a topos”. An important connection with logic is signalled by the
fact that the double negation operator on a topos is a topology in this sense.
We ¬nd it convenient here to start with the more abstract (but easier to
understand) idea of a topology on a topos ¬rst. Later in the chapter we talk about
Grothendieck topologies and prove Giraud™s Theorem (Theorem 1 of Section 6.8)
which characterizes categories of sheaves for a Grothendieck topology.
We will also consider categories of coalgebras for a left exact cotriple in a topos,
and of algebras for an idempotent left exact triple. Both these categories are also
toposes (the latter are actually sheaves for a topology) and the constructions
yield an important factorization theorem (Section 6.5) for geometric morphisms.


6.1 Topologies
A topology on a category with pullbacks is a natural endomorphism j of the
contravariant subobject functor which is
(i) idempotent: j —¦ j = j,
(ii) in¬‚ationary: A0 ⊆ jA0 for any subobject A0 of an object A (where we
write jA0 for jA(A0 ) as we will frequently in the sequel), and
(iii) order-preserving: if A0 and A1 are subobjects of A and A0 ⊆ A1 , then
jA0 ⊆ jA1 .
See Exercise 1 for the independence of (ii) and (iii).


198
6.1 Topologies 199
When j is a topology on a category in which subobjects are representable
by an object „¦, then using the Yoneda Lemma, j induces an endomorphism of
„¦ which is idempotent, in¬‚ationary and order-preserving, and conversely such
an endomorphism induces a topology on the category. A topology in this sense
on „¦ can be given an equational de¬nition in terms of intersection and truth
(Exercise 4).
A subobject A0 of an object A is j-closed in A if jA(A0 ) = A0 and j-dense
in A if jA(A0 ) = A. Observe that jA(A0 ) is j-closed by idempotence, and A
is j-dense in A because j is in¬‚ationary. When j is understood, we often write
“dense ” and “closed”.
A topology is super¬cially like a closure operator on a topological space. How-
ever, it does not preserve ¬nite unions (in fact we will see later that it does
preserve ¬nite intersections) and to this extent the terminology “dense” and
“closed” is misleading. However, it is standard in the literature, so we retain
it.
Let™s start with some examples.
(a) This example shows how the pasting property of a sheaf motivated the def-
inition of topology. Let X be a topological space and E the category of presheaves
(functors from the opposite of the open set lattice to Set) on X. Let F be a
presheaf. De¬ne an endofunction jF of the set of subfunctors of F by requir-
ing that for an open set U of X and a subfunctor G of F , jF (G)(U ) is the set
of elements x ∈ F U for which there is a cover {Ui ’ U } such that for all i,

x|Ui ∈ G(Ui ).
It is easy to see that if U ⊆ V and y ∈ jF (G)(V ) then the restriction of y in
F U is in jF (G)(U ), so that jF (G) is really a subfunctor of F . Then the maps jF
are the components of a natural endomorphism of the subobject functor which
is a topology on E .
To verify this requires proving that j is a natural transformation and that it
satis¬es (i)-(iii) of the de¬nition of topology. We prove the hardest, naturality,
at the end of this section and leave the rest to you.
(b) In any topos, ¬¬ is a topology (Exercise 5). The proof is implicit in the
results and exercises to Section 5.6. We will see that when a topos is regarded as a
theory, then the sheaves for the double-negation topology force a Booleanization
of the theory. Those familiar with logic should note that the word “force” is used
advisedly (see Tierney [1976]).
(c) In any topos, if U )’ 1, there is a “least destructive” topology j for which

j0 = U , namely that which for a subobject A0 )’ A has jA(A0 ) = A0 ∪ A — U

(note A — U )’ A — 1 = A). This has the property that if U ¤ V ¤ 1 then V is

closed in 1.
200 6 Permanence Properties of Toposes
(d) Topologies exist in categories which are not toposes, too. There is a
topology on the category of Abelian groups which assigns to each subgroup B of
an Abelian group A the subgroup
{a ∈ A | there is a positive integer n for which na ∈ B}
This subgroup is the kernel of the composite
A/B
A ’ A/B ’
’ ’
t(A/B)
where t denotes the torsion subgroup.
We should think of jA(B) as the set of all elements of A which are “almost
in” B. Equivalently, it may be thought of as elements which are “almost zero”
mod B. Topologies on additive categories are often called torsion theories .

Properties of topologies

We state here some technical lemmas which will be used many times later.
Lemma 1. If
BE EB
0


˜500 f (1)
c c
E EA
A0
is a pullback, then there is a (necessarily unique) arrow jB(B0 ) ’ jA(A0 ) for

which
jB(B0E EB
)

f (2)
c c
jA(A0E EA
)
is also a pullback.
Proof. Follow A0 around the two paths of the following diagram.
E Sub B
Sub A

jA jB
c c
E Sub B
Sub A

Exercise 2 gives a converse to Lemma 1.
6.1 Topologies 201
Lemma 2. Let C be a category with a topology j, A be an object of C and
B, C, D be subobjects of A. Then
(a) If C ⊆ B, then jB(C) = B © jA(C).
(b) If C ⊆ B, then jB(C) ⊆ jA(C).
(c) B ⊆ jA(B) is dense and jA(B) ⊆ A is closed.
(d) The “diagonal ¬ll-in property” of a factorization system is satis¬ed: if
E EC
D
c c
(3)
c c
E EA
B
is a commutative square of monos with the top arrow dense and the bottom arrow
closed, then C ⊆ B.
(e) If f : A ’ A is any map in C and B is dense (resp. closed) in A, then

f ’1 (B) is dense (resp. closed) in A .
(f) If B and C are both dense (resp. closed) in A then B © C is dense (resp.
closed) in A.
(g) If C ⊆ B ⊆ A and both inclusions are dense (resp. closed), then C is
dense (resp. closed) in A.
(h) jA(B) is characterized uniquely by the facts that B is dense in jA(B) and
jA(B) is closed in A.
Proof. (a) is a special case of Lemma 1 (if you ever get stuck trying to prove
something about a topology, try using the fact that a topology is a natural trans-
formation) and (b) is immediate from (a). (c) is immediate from (a) applied to
B ⊆ jA(B). For (d), apply j in the diagram to get
E E jC(D) =E C
E
D
c c c


c c c
E E jA(B)
E EA
B=
from which the conclusion is immediate. The “dense” half of (e) is a special case
of Lemma 1, and the other half is true in any factorization system (Exercise 2 of
Section 5.5). Exactly the same is the case for (f), while both parts of (g) are true
in any factorization system. Finally (h) follows from the uniqueness of image in a
factorization system. The factorization system is on the category with the same
objects as C and the monos as maps.
202 6 Permanence Properties of Toposes
Proposition 3. Let B and C be subobjects of A. Then jA(B © C) = jA(B) ©
jA(C).
Proof. It follows from Lemma 2(f) that B © C is dense in jA(B) © jA(C) and that
jA(B) © jA(C) is closed in A. By Lemma 2(h), this characterizes jA(B © C).
Proposition 4. In a category with pullbacks which has a topology j, suppose the
left vertical arrow in the following commutative square is a dense mono, and the
right vertical arrow is a closed mono.
E Ao
B0
c c
(4)
c c

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