. 16
( 60 .)


spondence with natural transformations from F to PE. We show this ¬rst for F
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.

¦U — idB
c c
E PB — 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


(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
(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.
(F Ui ) ’ ’
FU ’’ F (Ui © Uj )
’’ ’
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 )

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 ,

c c
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
Gj V

c c
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
E GVk E G(Vk © Vl )

c c c
FV F Vk F (Vk © Vl )
In the following commutative cube,
‚ ©
Gi (Ui © V ) Gi (Ui © Vk )


c c
F (Ui © V ) F (Ui © Vk )
c   dc

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

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
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


. 16
( 60 .)