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