representable, say F = Hom(’, A). We have

Nat(F, PE) = Nat(Hom(’, A), PE) PE(A)

= Sub(Hom(’, A) — E) = Sub(F — E)

The case for general F follows from Proposition 3 and the fact that F is a colimit

of representable functors (Exercise 7, Section 1.8).

Exercises 2.1

1. Prove that the product of two toposes is a topos.

2. Let B = PA in Diagram (1); the subobject of PA—A corresponding to idPA is

denoted ∈A (it is the “element of” relation in Set). Prove that for any subobject

78 2 Toposes

U ’ A — B there is a unique arrow ¦U : A ’ PB which makes the following

’ ’

diagram a pullback.

E A—B

U

¦U — idB

c c

E PB — B

∈B

3. An object B of a category is injective if for any subobject A0 ’ A and arrow

’

f : A0 ’ B there is an arrow f : A ’ B extending f . Prove that in a topos any

’ ’

power object is injective. (Hint: Every subobject of A0 — B is a subobject of

A — B. Now use Exercise 2.)

4. Show that in diagram (2) in Set, if X is an equivalence relation and c is the

class map, then Sub c in diagram (3) takes a set of equivalence classes to its union.

What are Sub p1 and Sub p2 ?

5. Complete the proof of Proposition 3.

6. Show that if X is a set, the functor ’ — X: Set ’ Set commutes with

’

colimits. (Hint: Show that ’ — X is left adjoint to (’)X and use Corollary 2 of

Section 1.9).

2.2 Sheaves on a Space

Categories of sheaves were the original examples of toposes. In this section we

will consider sheaves over topological spaces in some detail and prove that the

category of sheaves over a ¬xed space is a topos. In Section 5.1, we give Grothen-

dieck™s generalization of the concept of sheaf. He invented it for use in algebraic

geometry, but we will use it as the fundamental tool in building the connection

between toposes and theories.

Let X be a topological space and O(X) the category of open sets of X and

inclusions. As we have seen, the category Func(O(X)op , Set) is a topos. An object

of this category is called a presheaf on X, and the topos is denoted Psh(X). If

the open set V is contained in the open set U , the induced map from F U to F V

is denoted F (U, V ) and is called a restriction map. In fact, we often write x|V

instead of F (U, V )x for x ∈ F U . This terminology is motivated by the example

of rings of continuous functions mentioned on page 24.

A presheaf is called a sheaf if it satis¬es the following “local character” con-

dition: If {Ui } is an open cover of U and xi ∈ F Ui is given for each i in such a

2.2 Sheaves on a Space 79

way that xi |Ui © Uj = xj |Ui © Uj for all i and j, then there is a unique x ∈ F U

such that x|Ui = xi .

The full subcategory of Psh(X) whose objects are the sheaves on X is denoted

Sh(X).

Examples

(i) For each topological space Y , the functor which assigns to each open set

U of X the set of continuous functions from U to Y can easily be seen to be a

sheaf.

(ii) Given a topological space Y and continuous map p: Y ’ X, for each

’

open U in X let “(U, Y ) denote the set of all continuous maps s: U ’ Y such

’

that p —¦ s is the inclusion of U in X. These are called sections of p. Then

“(’, Y ): O(X)op ’ Set is a sheaf, called the sheaf of sections of p. We will

’

see below (Theorem 3) that every sheaf arises this way.

The de¬nition of sheaf is expressible by an exactness condition:

Proposition 1. F : O(X)op ’ Set is a sheaf if and only if for every open set

’

U and for every open cover {Ui } of U , the following diagram is an equalizer.

d

’0

(F Ui ) ’ ’

’’

FU ’’ F (Ui © Uj )

’’ ’

d1

In this diagram, the left arrow is induced by restrictions. As for d0 and d1 ,

they are the unique arrows for which the diagrams

d0 E

(F Ui ) F (Ui © Uj )

pij

pi

c c

E F (Ui © Uj )

F Ui

d1 E

(F Ui ) F (Ui © Uj )

pj pij

c c

E F (Ui © Uj )

F Ui

commute. The bottom arrows are the restriction maps.

Proof. Exercise.

80 2 Toposes

Sheaf categories are toposes

Theorem 2. Sh(X) is a topos.

Proof. We know Func(O(X)op , Set) has limits, so to see that Sh(X) has ¬nite

limits, it is su¬cient to show that the limit of a diagram of sheaves is a sheaf.

This is an easy consequence of Proposition 1 and is omitted.

The method by which we proved that Set-valued functor categories are toposes

suggests that we de¬ne P(F ) to be the functor whose values at U is the set of

subsheaves (i.e., subobjects in Sh(X)) of F —Hom(’, U ). Since O(X) is a partially

ordered set, the sheaf G = F —Hom(’, U ) has a particularly simple form, namely

G(V ) = F (V ) if V ⊆ U and G(V ) is empty otherwise. Thus we write F |U for

F — Hom(’, U ). Hence (PF )U is the set of subsheaves of F |U . It is necessary

to show that this de¬nes a sheaf (it is clearly a presheaf).

Let {Ui } be a cover of U . Suppose for each i we have a subsheaf Gi of F |Ui

such that Gi |Ui © Uj = Gj |Ui © Uj . De¬ne G so that for all V ⊆ U ,

E FV

GV

(1)

c c

E

Gi (Ui © V ) F (Ui © V )

is a pullback. For other open sets V , GV is of course empty. Restriction maps

are induced by the pullback property. It is clear that G is a subfunctor of F .

We ¬rst show that G|Uj = Gj . For V ⊆ Uj , the fact that Gi (Ui © Uj ) =

Gj (Ui ©Uj ) implies that Gi (V ©Uj ) = Gj (V ©Uj ). Thus we have the commutative

diagram

E FV

Gj V

c c

E

Gi (Ui © V ) = Gi (Uj © V ) F (Ui © V )

c c

E F (Uj © V ) = F V

Gj (Uj © V ) = Gj V

in which the top middle node is also Gi (Uj © V ), the outer rectangle is a

pullback and the middle arrow is a mono as Gi is a subfunctor of F . It follows

2.2 Sheaves on a Space 81

from Exercise 12 that the left square is a pullback too. But that pullback is G(V )

by de¬nition.

To see that G is a subsheaf, let {Vk } be a cover of V . By Proposition 1

we need to show that the top row of the following diagram is an equalizer. By

Exercise 11(a) it is su¬cient to show that the left square in the diagram is a

pullback.

E GVk E G(Vk © Vl )

E

GV

(2)

c c c

E

E E

FV F Vk F (Vk © Vl )

In the following commutative cube,

E

GV GVk

d

d

d

d

‚ ©

E

Gi (Ui © V ) Gi (Ui © Vk )

I II III

c c

E

F (Ui © V ) F (Ui © Vk )

s

d

d

d

c dc

E

FV F Vk

the square labelled I is a pullback by the de¬nition of G and because G|Ui = Gi .

Number III is a product of squares which are pullbacks from the de¬nition of

G. Finally II is a product of squares, each of which is the left hand square in a

diagram of type (2) above with Gi replacing G and is a pullback because Gi is

a sheaf (see Exercise 11(b)). It follows from Exercise 8 of Section 1.6 that the

outer square is a pullback, which completes the proof.

82 2 Toposes

Shea¬¬cation

In the rest of this section, we outline some functorial properties of toposes of

sheaves. These results follow from more general results to be proved later when

we discuss Grothendieck topologies, and so may be skipped. However, you may

¬nd considering this special case ¬rst helpful in understanding and motivating

the ideas introduced later. We give only the constructions involved in the proofs;

the veri¬cations are left as exercises.

We saw in example (ii) that any space over X de¬nes a sheaf on X. Given a

presheaf F on X we will construct a space LF and a continuous map (in fact local

homeomorphism) p: LF ’ X for which, for an open set U of X, the elements

’

of F U are sections of p (when F is a sheaf). Thus we need somehow to ¬nd the

points of LF that lie over a particular point x of X. We construct the set of such

points by using colimits.

Form the diagram of sets and functions consisting of all those sets F U for all

open U which contain x and all the restriction maps between them. This diagram

is actually a directed system. The colimit in Set of this diagram is denoted Fx

and called the stalk or ¬ber of F at x. An element of Fx is an equivalence class

of pairs (U, s) where x ∈ U and s ∈ F U . The equivalence relation is de¬ned

by requiring that (U, s) ∼ (V, t) if and only if there is an open set W such that

=

x ∈ W ⊆ U © V and s|W = t|W . The equivalence class determined by (U, s) is

denoted sx and is called the germ determined by s at x. Thus s determines a

map s from U to Fx which takes x to sx .

Now let LF be the disjoint union of all the stalks of F . We topologize LF

with the topology generated by the images of all the maps s for all open U in

X and all sections s ∈ F U . These images actually form a basis (Exercise 5).

LF comes equipped with a projection p: LF ’ X which takes sx to x. This

’

projection p is continuous and is in fact a local homeomorphism, meaning that

for any y in LF there is an open set V of LF containing y for which the image

of p|V is an open set in X and p|V is a homeomorphism from V to p(V ). To see

this, let U be an open set containing p(y) and de¬ne V = s(U ), where y is the

equivalence class containing (U, s). Note that LF is not usually Hausdor¬, even

when X is.

If f : F ’ G is a map of presheaves, then f induces (by the universal property

’

of colimits) a map fx : Fx ’ Gx for each x ∈ X, and so a map Lf : LF ’ LG.

’ ’

It is a nice exercise to see that Lf is continuous and that this makes L a functor

from Psh(X) to T op/X.

LF is called the total space of F (or just the “space” of F ). (Many people

follow the French in calling the total space the “espace ´tal´”. It is wrong to call

ee

it the “´tale space” since “´tale” is a di¬erent and also mathematically signi¬cant

e e

2.2 Sheaves on a Space 83

word.) We will denote by LH /X the category of spaces (E, p) over X with p a

local homeomorphism.

The function “ de¬ned in Example (ii) above also induces a functor (also

called “) from the category T op/X of spaces over X to Psh(X): If E and E are

spaces over X and u: E ’ E is a map over X, then “(u) is de¬ned to take a

’

section s of the structure map of E to u —¦ s. It is easy to see that this makes “ a

functor. Note that for E over X, “(E) is a presheaf, hence a functor from O(X)op

to Set; its value “(E)(U ) at an open set U is customarily written “(U, E).

Theorem 3. L is left adjoint to “. Moreover, L is a natural equivalence between

Sh(X) and LH /X.

Proof. We will construct natural transformations ·: id ’ “ —¦ L and : L —¦ “ ’ id