of its cokernel pair. An easy diagram chase shows that the upper row must be

an equalizer, too.

Since a topos has colimits, it has an initial object 0. As in Set, where 0 is the

empty set, the initial object is the codomain of just one arrow, its own identity

arrow. An initial object with this property up to equivalence is said to be strict.

Proposition 7. If f : A ’ 0, then A ∼ 0.

’ =

Proof. Given f : A ’ 0, (f, idA ): A ’ 0—A is split monic. Since A—’ commutes

’ ’

with colimits, A — 0 0, so A ’ 0 is a split monic. But any map to 0 is epic,

’

so A is isomorphic to 0.

190 5 Properties of Toposes

Corollary 8. Any map 0 ’ A is monic.

’

Proof. Its kernel pair is 0 —A 0 ’ 0.

’

Exercises 5.5

1. A factorization system in a category consists of two classes M and E of

arrows with the properties that

(i) Both M and E contain all identity arrows and are closed under composi-

tion with isomorphisms on both sides.

(ii) Every arrow can be factored as an arrow of E followed by an arrow of M .

(iii) (“Diagonal ¬ll-in property”). In any diagram

eE

A I

c c

EB

J m

with m ∈ M and e ∈ E, there is a unique arrow from I to J making both triangles

commute.

Formulate the way in which the factorization in (ii) is unique and prove it.

2. (a) Show that in any factorization system any map which satis¬es the

diagonal ¬ll-in property with respect to every map of M belongs to E. That is,

if f has the property that whenever there is a commutative square

fE

A B

c c

ED

C e

with m ∈ M , then there is an arrow from B to C making both triangles commute,

then f belongs to E. (Hint: Let f = m —¦ e with e ∈ E and m ∈ M . Consider the

square

fE

· ·

g

e id

c

© c

E·

· m

5.5 Exactness Properties of Toposes 191

and then the square

e E·

·

gm

e m

id

©

c c

©

E·

· m

and use the uniqeness property of the diagonal ¬ll-in.)

Conclude

(b) Every map in M © E is an isomorphism and every isomorphism belongs

to M © E.

(c) E is closed under composition.

(d) In a pushout square,

e E·

·

c c

E·

·

e

if e ∈ E, then so is e .

(e) If M consists of monos, then every split epi belongs to E.

Of course, the duals of all these properties also hold.

3. Show that in a factorization system in which either the class E consists of all

the epis or the class M consists of all the monos, the uniqueness of the diagonal

¬ll-in does not have to be assumed, only its existence.

4. Show that in a regular category the monos and regular epis form a factorization

system. (Hint: The hypothesis is too strong; all that is needed is that a pullback

of a regular epi be an epi.)

5. The image of an arrow f : C ’ B in a category, if it exists, is a subobject

’

I of B through which f factors, with the property that if f factors through any

subobject J of B, then I is a subobject of J. Prove that a topos has images.

6. Prove that the construction in diagram (3) makes B ∪C the least upper bound

of the subobjects B and C.

192 5 Properties of Toposes

7. Show that there is a factorization system in the category of topological spaces

in which E consists of quotient maps with dense images and M consists of in-

jective maps to closed subspaces. Show that there are maps in E which are not

epimorphisms.

8. Find at least three other factorization systems in the category of topological

spaces and continuous maps.

9. Show that in a topos, f + g is an isomorphism if and only if f and g are.

10. Prove that an object in a topos is the initial object if and only if it has

exactly one subobject.

11. The group Z2 — Z2 has three subgroups of order two.

(a) Show that the union of any two of them in the subobject lattice of Z2 — Z2

is the whole group.

(b) Show that the pushout in the category of all groups of any two subgroups

of order 2 over their intersection is an in¬nite group generated by two generators

of order 2. (Hint: The group of isometries of the metric space of integers is

in¬nite and is generated by two elements of order 2, namely rotation around 0

and rotation around 1/2.)

12. Let E be a topos and O the full subcategory of subobjects of 1.

(a) Show that O is a re¬‚ective subcategory of E . (Hint: Take an object X

to the image of X ’ 1.)

’

(b) Show that the left adjoint L of the inclusion I in (a) preserves products.

(Hint: Use the fact that the pullback of an epi is an epi.)

13. Use the previous exercise and the results of Section 5.3 to show that for any

object A of a topos E , the canonical functor Sub(A) ’ E /A has a left adjoint

’

which preserves products.

14. Let f : A ’ B be an arrow in a topos. Let

’ f and f be the left and

right adjoints of the pullback functor f — of Corollary 7.

(a) Show that f induces an arrow f ’1 : Sub B ’ Sub A by pulling back which

’

is the restriction of f — .

(b) Show that the restriction of f is a right adjoint ∀f : Sub A ’ Sub B

’

to f ’1 .

(c) Show that f ’1 has a left adjoint ∃f . (Hint: De¬ne ∃f to be ∃i where ∃ is

de¬ned as in Section 2.4 and i is the inclusion of Imf in B.)

5.6 The Heyting Algebra Structure on „¦ 193

15. (The doctrinal diagram). Show that in the following diagram in which the

arrows are all de¬ned above,

(a) ∃f —¦ L ∼ L —¦ f,

=

(b) I —¦ ∀f ∼ f —¦ I,

=

(c) f — —¦ I ∼ I —¦ f ’1 , and

=

(d) f ’1 —¦ L ∼ L —¦ f — .

=

f

© f— E E/B

E/A

Ts T

f

I L I L

∃f

c© c

f ’1 E Sub B

Sub A

s

∀f

5.6 The Heyting Algebra Structure on „¦

Heyting algebras

Intuitionistic logic is a weakening of classical logic intended to allow only

“positive” proofs; in other words, proof by contradiction is ruled out. Just as the

concept of Boolean algebra results from abstracting the properties of “and”, “or”

and “not” in classical logic, the concept of Heyting algebra arises by abstract-

ing the properties of “and”, “or” and “implies” (the latter must be taken as a

primitive in intuitionistic logic) in the logic developed by Brouwer, Heyting and

others (see A. S. Troelstra [1977]). Intuitionistic logic arises naturally in toposes;

it is not the result of a philosophical position on the part of those who do topos

theory, although many people interested in constructive mathematics have been

attracted to the subject.

A Heyting algebra is a lattice with some extra structure. We denote the

in¬mum of a and b in a lattice by a § b and the supremum by a ∨ b. The ordering

is denoted by ¤. The maximum and minimum elements of the lattice, if they

194 5 Properties of Toposes

exist, are T and F respectively. Then a Heyting algebra is a lattice with a

minimum and an additional binary operation “’” satisfying the requirement

that for all elements a, b and c, c ¤ a ’ b if and only if a § c ¤ b.

This de¬nition has a number of consequences. A Heyting algebra has a maxi-

mum element, namely F ’ F . The operation ’ has the properties that if a ¤ b

then b ’ c ¤ a ’ c and c ’ a ¤ c ’ b; this last fact means that for ¬xed a,

a ’ ’ is a functor (regarding the lattice as a category) which is right adjoint

to a § ’. Thus a Heyting algebra is a lattice with minimum which is Cartesian

closed as a category.

For any element a in a Heyting algebra, one de¬nes ¬a as a ’ F . The

operation ¬ has only some of the properties of the classical “not”. For example,

a ¤ ¬¬a, but one cannot prove that ¬¬a ¤ a, that a ∨ ¬a = T , or the DeMorgan

laws. In the exercises, you are asked to verify these and other facts about Heyting

algebras.

There are two important types of examples of Heyting algebras.

(i) Any Boolean algebra is naturally a Heyting algebra, de¬ning a ’ b to be

¬a ∨ b.

(ii) The lattice of open sets of any topological space X is a Heyting algebra.

Here, meet and join are intersection and union, and A ’ B is the interior of

(X ’ A) ∪ B. A spxecial case of this, useful for constructing counterexamples, is

the Sierpinski space: the set S = {0, 1} with the empty set, S and {1} as the

only open sets.

The lattice of open dense subsets of a topological space, together with the

empty set, also forms a Heyting algebra in the same way.

More information on Heyting algebras may be found in Rasiowa and Sikorski

[1963], where they are called “pseudocomplemented lattices”.

The Heyting algebra structure on „¦

For an object A of a topos, intersection and union of subobjects make Sub(A)

a lattice. Both operations induce functions Sub(A) — Sub(A) ’ Sub(A), thus

’

functions from

Hom(A, „¦) — Hom(A, „¦) ∼ Hom(A, „¦ — „¦)

=

to Hom(A, „¦). These two functions are natural in A: Suppose X and Y are sub-

objects of B and f : A ’ B. We must show that Sub(f )(X ∪ Y ) = Sub(f )(X) ∪

’

Sub(f )(Y ) and similarly for intersection. Now Sub(f )(X) is computed by pulling

back along f . The intersection and union are respectively a pullback and a

pushout in E (diagram 5, Section 5.5) and the pullback functor has both a left

5.6 The Heyting Algebra Structure on „¦ 195

and a right adjoint by Corollary 7, Section 5.3. Thus it takes limits and colimits

in E /B to limits and colimits in E /A. Colimits in E /A are the same as colimits

in E . As for limits, if D is a diagram in E /A, a limit of D in E /A is the same as

the limit of D ’ A in E , which is the sort of limit we have here. Thus Sub(f )

’

preserves intersection and union, as required.

Since the induced maps Hom(A, „¦ — „¦) ’ Hom(A, „¦) are therefore natural,

’

Yoneda gives us binary operations §: „¦ — „¦ ’ „¦ and ∨: „¦ — „¦ ’ „¦.

’ ’

The order relation and the arrow operation are obtained this way: The equal-

izer

[(B, C) | B © C = B] = [(B, C) | B ¤ C]

of the two maps § and the ¬rst projection from „¦ — „¦ to „¦ is a subobject of

„¦ — „¦ and so corresponds to a pullback

E „¦—„¦

¤

’ (1)