for which “ —¦ ·“ = id and L —¦ L· = id, from which the adjointness will follow

(Exercise 15 of Section 1.9).

Let F be a presheaf on X. On an open set U of X, de¬ne the natural

transformation ·F by requiring that (·F )U take an element s of F U to the

section s. On the other hand, for a space E over X, the continuous map E is

de¬ned to take an element sx of L(“(E)) to s(x). The necessary veri¬cations are

left to the reader, as is the proof that when F is a sheaf, ·F is an isomorphism,

and when p: E ’ X is a local homeomorphism, E is a homeomorphism over

’

X. The latter two facts prove that L is a natural equivalence between Sh(X) and

LH /X.

The functor “ —¦ L: Psh(X) ’ Sh(X) is called the shea¬¬cation functor.

’

Corollary 4. For any topological space X, LH /X is a topos.

Corollary 5. Sh(X) is a re¬‚ective subcategory of Psh(X).

Proof. The re¬‚ector is the shea¬¬cation functor.

Change of base space

Any continuous function between topological spaces induces a pair of functors

between the sheaf categories which are adjoint.

Given a continuous function f : X ’ Y and a sheaf F on X, the direct

’

image functor f— : Sh(X) ’ Sh(Y ) is de¬ned to be the restriction of

’

Func(f ’1 , Set): Func(O(X)op , Set) ’ Func(O(Y )op , Set)

’

Note that f ’1 is a functor from O(Y )op to O(X)op . Thus f— is composition with

f ’1 .

84 2 Toposes

On the other hand, given a local homeomorphism p: E ’ Y , f — (E) is de¬ned

’

to be the pullback

EE

f — (E)

p

c c

EY

X

f

so that f — (E) = {(x, e) | f x = pe}. It is easy to see that the map (x, e) ’ x is a

’

local homeomorphism. On sections, f — takes a section s of p de¬ned on an open

set V of Y to the map which takes x ∈ f ’1 (V ) to (x, sf x).

Proposition 6. f — is left adjoint to f— . Moreover, f — preserves all ¬nite limits.

Note that f — perforce preserves all colimits since it has a right adjoint.

Proof. f — is the restriction of the pullback functor from T op/Y to T op/X, which

has as a left adjoint composing with f (the proof is easy). Thus it preserves limits

in T op/Y ; but ¬nite limits in LH /Y are the same as in T op/Y (again easy).

There is a natural map from E to f— —¦ f — (E) whose component on an open set

V of Y takes a section s in “(V, E) to the function s —¦ f : f ’1 (V ) ’ E, which by

’

de¬nition is an element of “(V, f— —¦ f — (E)).

On the other hand, let F be a sheaf on X and let x ∈ X. For any open V of Y

for which x ∈ f ’1 (V ), we have a map from “(f ’1 (V )), F ) to the stalk Fx , hence

a map from “(V, f— (F )) (which is, by de¬nition, the same as “(f ’1 (V ))) to Fx .

This directed system de¬nes a map tx from the stalk of f— (F ) at f x to the stalk of

F at x. We then de¬ne a natural transformation from f — f— to the identity which

takes a section s of f — f— (F ) de¬ned on U to the section of F which takes x ∈ U to

tx (s(x)). These two natural transformations satisfy the hypotheses of Exercise 15

of Section 1.9, so that f — is left adjoint to f— . The detailed veri¬cations, which

get a bit intricate, are left to the reader. This is a special case of adjoints of

functors induced by maps of theories. The general situation will be dealt with in

Chapter 8.

A geometric morphism between toposes is a functor f : E ’ E with a

’

—

left adjoint f which preserves ¬nite limits. The functor f is usually written

f— , and f — is called its “inverse image”. Thus a continuous map f : X ’ Y of

’

topological spaces induces a geometric morphism from Sh(X) to Sh(Y ). We will

study geometric morphisms in detail in Chapter 6.

Exercises 2.2

1. (a) Verify that Examples (i) and (ii) really are sheaves.

2.2 Sheaves on a Space 85

(b) Show that Example (i) is a special case of Example (ii). (Hint: Consider

the projection from Y — X to X.)

2. Prove Proposition 1.

3. Show that L is a functor.

4. Show that a subfunctor of a sheaf need not be a sheaf.

5. Let F be a presheaf on a topological space X. Show that the images s(U ) for

all open sets U in X and all sections s ∈ F U form a basis for a topology on LF .

6. Carry out the veri¬cations that prove Theorem 3. (You have to prove that for

each F , ·F is a natural transformation, and for each p, p is a continuous map;

that · and are natural transformations; and that they satisfy the requirements

in (a) of Exercise 15 of Section 1.9.)

7. Using the notation of the preceding exercise, prove that E is an isomorphism

if and only if p: E ’ X is a local homeomorphism, and ·F is an isomorphism if

’

and only if F is a sheaf.

8. Show that two sheaves over the same space can have the same stalks at every

point without being the same sheaf. (Hint: Look at a double covering of a circle

versus two single circles lying over a circle.)

9. Prove that the pullback of a local homeomorphism is a local homeomorphism.

10. Show that every point of a topological space X induces a geometric morphism

from Set to Sh(X) which takes a sheaf over X to its stalk over the point.

11. Consider the diagram

E

EB EC

A

g h

c c Ec

EB

A˜ EC

in which we assume the left square commutes, and h —¦ di = di —¦ g, i = 1, 2. (We

often say that such a square commutes serially, a notion which we will use a

lot in later chapters.)

86 2 Toposes

(a) Show that if the bottom row is an equalizer and the left square is a

pullback, then the top row is an equalizer.

(b) Show that if the top row is an equalizer, the bottom row has both com-

posites the same, and f is monic, then the left square is pullback.

12. Consider the diagram

fE

EB

A C

g

c c c

EB EC

A

(a) Show that if both squares are pullbacks then so is the outer rectangle.

(b) Show that if the outer rectangle is a pullback and f and g are jointly

monic, then the left square is a pullback. (f and g are jointly monic if f (x) = f (y)

and g(x) = g(y) implies that x = y. Such a square is called a mono square.)

2.3 Properties of Toposes

In this section and the next, we will state and prove those basic properties of

toposes which can conveniently be proved without using triple theory.

In the following, E is a topos with power-object function P.

Functoriality of P

Proposition 1. Let A and B be categories and ¦: A op —B ’ Set be a functor.

’

Let F : ObA ’ ObB be a function such that for each object A of A there is a

’

natural (in B) equivalence

¦(A, B) Hom(F A, B)

Then there is a unique way of extending F to a functor (also denoted F ) A ’ B

’

in such a way that the above equivalence is a natural equivalence in both A and

B.

2.3 Properties of Toposes 87

Proof. Fix a morphism f : A ’ A. We have a diagram of functors on B

’

∼

= E Hom(F A, ’)

¦(A, ’)

¦(f, ’)

c c

E Hom(F A , ’)

¦(A ) ∼

=

in which the right arrow is de¬ned by the indicated isomorphisms to make the dia-

gram commute. The result is a natural (in B) transformation from Hom(F A, ’)

to Hom(F A , ’) which is induced by a morphism we denote F f : F A ’ F A.’

The naturality in A and the functoriality of F are clear.

Proposition 2. P has a unique extension to a functor from E op to E with the

property that for any arrow g: A ’ A, the following diagram commutes. Here,

’

φ is the natural isomorphism of Section 2.1.

φ(A , B)

E

HomE (B, PA ) Sub(B — A )

T T

Hom(B, Pg) Sub(idB — g) (1)

E Sub(B — A)

HomE (B, PA)

φ(A, B)

Proof. Apply Proposition 1 above with A = E , B = E op , and F = Pop .

It is worthwhile to restate what we now know about P in view of the de¬nition

of Sub. If s: U ’ B — A is a subobject and [s] = ¦’1 (s): B ’ PA is the

’ ’

corresponding element of PA, then diagram (1) of Section 2.1 says that for an

arrow f : B ’ B, the element [s]f of PA de¬ned on B corresponds via the

’

adjunction to Sub(f — idB )(s) which is the pullback of s along f — idB . On the

other hand, given g: A ’ A, then according to diagram (1) of this section, the

’

element Pg(s) of PA corresponds to Sub(idB — g)(s), which is the pullback of s

along idB — g.

Pop : E ’ E op is left adjoint to P: E op ’ E .

Proposition 3. ’ ’

Proof. The arrow A — B ’ B — A which switches coordinates induces a natural

’

isomorphism from the bifunctor whose value at (B, A) is Sub(B — A) to the

bifunctor whose value at (B, A) is Sub(A — B). This then induces a natural

isomorphism

HomE (B, PA) HomE (A, PB) = HomE op (PB, A)

88 2 Toposes

which proves the Proposition.

We sometimes say, “P is adjoint to itself on the left.”

The subobject classi¬er

Since A A — 1, the subobject functor is represented by P(1). This object

is so important in a topos that it deserves its own name, which is traditionally

„¦. It follows from the Yoneda Lemma that „¦ has a representative subobject

true: „¦0 ’ „¦ with the property that for any object A and any subobject a: A0

’

’ A there is a unique map χa: A ’ „¦ such that a is the pullback of true along

’ ’

χa. This means that there is a map A0 ’ „¦0 (whose nature will be clari¬ed by

’

Proposition 4) for which the following diagram is a pullback:

E „¦0

A0

a true

c c

E„¦

A χa

Proposition 4. „¦0 is the terminal object.

Proof. For a given object A, there is at least one map from A to „¦0 , namely the

map u given by the following pullback:

u E„¦

A 0

idA true (2)

c c

E„¦

A

χ(idA )

To see that u is the only map, suppose v: A ’ „¦0 is another map. Then this

’

diagram is a pullback (see Exercise 3 of Chapter 1.6):

v E„¦

A 0

idA true

c c

E„¦

A true —¦ v

The uniqueness part of the universal mapping property of „¦ says that true —¦v =

χ(idA ), which is true —¦ u by diagram (2). Since true is mono, this means u = v.

It follows that every object has exactly one map to „¦0 .

2.3 Properties of Toposes 89

„¦ is called the subobject classi¬er. In Set, any two element set is a subob-

ject classi¬er in two di¬erent ways, depending on which of the two elements you

take to be true. If for each set A you take PA to be the actual set of subsets

of A, then the preceding construction makes the subobject classi¬er the set of