subobject classi¬er in a category of G-sets is a two element set with the trivial

action, and comments similar to those just made apply here too. As we will see,

in most toposes „¦ is not nearly so simple.

The following result is a consequence of Exercise 10, page 45.

Corollary 5. Every monomorphism in a topos is regular.

The singleton map

We will de¬ne a special arrow {}: A ’ PA which in the case E = Set is the

’

map taking x to the singleton set containing x. Its importance lies in the fact that

composing with {} internalizes the construction of the graph of a function, for if

f : B ’ A, then {}f : B ’ PA corresponds to the subobject [(b, a) | a = f (b)]

’ ’

of B — A, which in Set is in fact the graph of f . (Thus by this de¬nition, the

graph of f in Set is the set of ordered pairs (b, f (b)) regarded as a subobject of

B — A, so that the graph carries with it the information about the codomain of

f as well as its domain.)

The fact that {}f should correspond to the graph of f suggests the way to

construct {}. Let γ be the natural transformation from HomE (’, A) to Sub(’ —

A) de¬ned by having γB take f : B ’ A to the subobject (idB , f ): B ’ B — A.

’ ’

Observe in the ¬rst place that if (idB , f ) —¦ u = (idB , f ) —¦ v, then (u, f u) = (v, f v)

so that (idB , f ) is indeed monic. To show that γ is a natural transformation

translates into showing that for any arrow g: B ’ B, ’

Sub(g — A)(idB , f ) = (idB , f g)

By de¬nition of Sub, that requires showing that the following diagram is a

pullback, which is easy.

(idB , f ) E

B B—A

T T

g g—A

E B —A

B

(idB , f g)

90 2 Toposes

Now let γ be the natural transformation

γ: HomE (’, A) ’ Sub(’ — A)

’ HomE (’, PA)

Let {}: A ’ PA be the corresponding arrow given by the Yoneda Lemma. Recall

’

that according to the proof of the Yoneda Lemma, for f : B ’ A, γB(f ) = {}f .

’

The following proposition just says that a morphism is determined by its

graph.

Proposition 6. {} is monic.

Proof. If f, f : B ’ A are two morphisms for which (idB , f ): B ’ B — A and

’ ’

(idB , f ): B ’ B — A give equivalent subobjects, then there is an isomorphism

’

j of B for which (j, f j) = (idB , f ), whence f = f . Since by construction {}f

corresponds by adjunction to the subobject (idB , f ): B ’ B — A, {} must be

’

monic.

Equivalence relations

As observed in Exercise 6 of Section 1.8, the kernel pair of a regular epimor-

phism is an e¬ective equivalence relation. In a topos, the converse is true:

Theorem 7. In a topos, every equivalence relation is e¬ective.

Proof. Let E be an equivalence relation on A. E is a subobject of A — A, so

corresponds to an arrow [ ]E : A ’ PA (which in Set is the class map). An

’

element of A de¬ned on T is sent to the subobject of T — A (element of PA

de¬ned on T ) which is the pullback of the diagram

E

c

E A—A

T —A

a — idA

Thus if a ∈T A and (t, a ) ∈V T — A, then (t, a ) ∈V [a]E if and only if

(a —¦ t, a ) ∈V E. This fact is used twice in the proof below.

To show that E is the kernel pair of [ ]E , we must show that if a1 and a2

are elements of A de¬ned on T then [a1 ]E = [a2 ]E if and only if (a1 , a2 ) ∈ E.

To see this, let (t, a): V ’ T — A be an element of T — A de¬ned on V . The

’

corresponding subset of Hom(V, T — A) is

[a1 ]E = {(t, a) | (a1 —¦ t, a) ∈ E};

2.3 Properties of Toposes 91

[a2 ]E is de¬ned similarly. If [a1 ]E = [a2 ]E , let V = T , t = idT , and a = a2 .

By re¬‚exivity, (idT , a2 ) ∈ [a2 ]E , hence belongs to [a1 ]E . Therefore (a1 , a2 ) =

(a1 —¦ idT , a2 ) ∈ E.

For the converse, suppose (a1 , a2 ) ∈ E. Then for all t: V ’ T , (a1 —¦ t, a2 —¦ t) ∈

’

E. Suppose (t, a) ∈ [a1 ]E (de¬ned on V ), then (a1 —¦ t, a) ∈ E and therefore

by symmetry and transitivity, (a2 —¦ t, a) ∈ E. Hence [a1 ]E ⊆ [a2 ]E . The other

inclusion follows by symmetry. This shows that E is the kernel pair of [ ]E .

Exercises 2.3

1. Let G: B ’ A be a functor and F : ObA ’ ObB be a function such that

’ ’

for each A ∈ A there is a natural (in B) equivalence

Hom(A, GB) Hom(F A, B)

Use Proposition 1 to show that F has a unique extension to a functor left ad-

joint to G. (This gives a second proof of the pointwise construction of adjoints,

Section 1.9).

2. Prove that in a topos an arrow which is both a monomorphism and an epi-

morphism is an isomorphism. (Hint: Use Corollary 5.)

3. (Interchanging true and false). Describe how to de¬ne P in Set so that the

subobject classi¬er is the set of subsets of a one-element set and the value of true

is the empty set.

4. Let X be a topological space. If U is open in X, de¬ne „¦(U ) to be the set of

open subsets of U . If V ⊆ U , let „¦(U, V )(W ) = W © V .

(a) Show that „¦ is a sheaf, and is the subobject classi¬er in Sh(X).

(b) What is {}?

5. Suppose that for each object A of a topos there is a map jA: Sub(A) ’ Sub(A)

’

with the property that whenever

AE EA

0

c c

BE EB

0

92 2 Toposes

is a pullback, then there is a pullback

jA(A0E EA

)

c c

jB(B0E EB

)

where the top arrow is the inclusion.

Use the Yoneda lemma to show that show that these functions constitute a

natural endomorphism of P.

6. (a) Show that if M is a monoid and E is the topos of left M actions and

equivariant maps, then „¦ is the set of left ideals of M , with action mL = {n |

nm ∈ L}.

(b) Show that in E , „¦ has exactly two global elements.

2.4 The Beck Conditions

The Beck conditions are useful technical conditions concerning inverse images

and forward images induced by inclusions.

A mono a: A0 ’ A induces a set function a —¦ ’: Sub(A0 ) ’ Sub(A) by

’ ’

composition, taking the subobject determined by u: A1 ’ A0 to the subobject

’

determined by a —¦ u.

Proposition 1. [The Beck Condition, external version] Let

f0 E

A0 B0

a b

c c

EB

A

f

be a pullback. Then

Sub f0

Sub A0 ' Sub B0

a—¦’ b—¦’

c c

Sub A ' Sub B

Sub f

2.4 The Beck Conditions 93

commutes.

Proof. This translates into proving that if both squares in the following diagram

are pullbacks, then so is the outer rectangle. That is Exercise 12(a) of Section 2.3.

a EA

E A0

A1

f

c c c

E B0 EB

B1

b

Observe that in Set, A0 is the inverse image of B0 along f .

The object PA is said to “internalize” Sub(A). For a given monic a: A0 ’ A,

’

there is an arrow ∃a: PA0 ’ PA which internalizes a —¦ ’ in the same sense. To

’

construct ∃a, we ¬rst observe that a —¦ ’ induces an arrow

(B — a) —¦ ’: Sub(B — A0 ) ’ Sub(B — A)

’

for any object B.

Proposition 2. (’—a) is a natural transformation from Sub(’—A0 ) to Sub(’—

A).

Proof. Suppose f : B ’ B is given. Then the diagram

’

B —a E B —A

B — A0

f — A0 f —A

c c

E B—A

B — A0

B—a

is a pullback (easy exercise), so that by Proposition 1,

(B — a) —¦ ’E

Sub(B — A0 ) Sub(B — A)

Sub(f — A0 Sub(f — A)

c c

E Sub(B — A)

Sub(B — A0 )

(B — a) —¦ ’

commutes as required for (’ — a) to be a natural transformation.

De¬nition. If a: A0 ’ A is monic, ∃a: PA0 ’ PA is the arrow induced by the

’ ’

natural transformation

HomE (’, PA0 ) Sub(’ — A0 ) ’ Sub(’ — A)

’ HomE (’, PA)

∃a takes an element of PA0 to the element of PA regarded as the same subobject.

94 2 Toposes

Proposition 3. [The Beck condition, internal version] If

f0 E

A0 B0

c c

a (ii)

b

c c

EB

A

f

is a pullback, then

Pf0

PA0 ' PB0

∃a ∃B

c c

PA ' PB

Pf

commutes.

What the Beck condition really says is that if X is a subobject of B0 , f ’1 (X)

is unambiguously de¬ned.

Proof. If (ii) is a pullback, then so is

E B — B0

C — A0

c c

E C —B

C —A

so by Proposition 1,

Sub(C — A0 ) ' Sub(C — B0 )

c c

Sub(C — A) ' Sub(C — B)

commutes. Hence because diagram (1) of Section 2.1 commutes,