<<

. 33
( 60 .)



>>

c c
·A ·B (1)
c c
PPA ' PPB
PPf
the vertical arrows are monic by Lemma 3, so f is monic. Since for any object
C of E , HomE (C, Pf ) is essentially the same (via the natural isomorphism φ
of Section 2.1) as Sub(C — f ), the fact that Pf is an isomorphism means that
Sub f : Sub A ’ Sub B is an isomorphism in Set, i.e., a bijection.

Now A is a subobject of A via the identity and B is a subobject of A via f ,
which we now know is monic. It is easy to see that Sub f (A) = Sub f (B) is the
subobject idB : B ’ B, so since Sub f is a bijection, idA : A ’ A and f : B ’ A
’ ’ ’
determine the same subobject of A. It follows easily that f is an isomorphism.

Lemma 5. P preserves the coequalizers of re¬‚exive pairs. In fact, it takes them
to contractible coequalizers.
Proof. This elegant proof is due to Par´. A coequalizer diagram of a re¬‚exive
e
pair in E op is an equalizer diagram
g
f
A ’’ B ’’ C

’ (2)
’’

h
in E in which g and h are split monos. All these diagrams are pullbacks:
fE
1E 1E
A A A B B B

g
f f (3)
1 1 h
c c c c c c
EB EC EC
A B B
h h
f
5.2 Slices of Toposes 175
Hence, by the Beck condition and the fact that ∃idA = idPA , these diagrams
commute:
∃fE
1E 1E
PA PA PA PB PB PB
T T T T T T
Pg
Pf Pf (4)
1 1 Ph

E PB E PC E PC
PA PB PB
∃h ∃h
∃f
It follows from this that
Pg
’’ PB ’Pf’ PA
PC ’’ ’’
’ (5)
’’’’
Ph
is a contractible coequalizer, with its contraction given by ∃f and ∃h.

Exercises 5.1

1. Show that a topos has colimits corresponding to whatever class of limits it
has and conversely. (Hint: To have a limit (resp. colimit) for all diagrams based
on a graph I is to have a left (resp. right) adjoint to the diagonal”or constant
functor”functor E ’ Func(I , E ). Use the Butler Theorem 3 of 3.7 and the

tripleability of E op ’ E to derive the two directions.)


2. Prove that (5) is a contractible coequalizer diagram.


5.2 Slices of Toposes
Recall from Section 1.1 that if C is a category and A an object of C , the category
C /A, called the slice of C by A, has as objects arrows C ’ A and morphisms

commutative triangles.
The following theorem is heavily used in proving the embedding theorems of
Chapter 7.
If E is a topos and A an object of E , then E /A is a topos.
Theorem 1.
Proof. E /A has a terminal object, namely the identity on A, and the map E /A
’ E creates pullbacks (Exercise 5 of Section 1.7), so E /A has ¬nite limits.

We must construct a power object for each object of E /A.
The product of objects B ’ A and C ’ A in E /A is the pullback B —A C,
’ ’
which is an object over A, and for any object X ’ A of E /A, Sub(X ’ A)
’ ’
176 5 Properties of Toposes
is the same as Sub(X) in E . Thus given an object f : B ’ A of E /A, we must

construct an object P(B ’ A) which represents Sub(B —A C) regarded as a

functor of objects g: C ’ A of E /A. The key to the proof lies in representing

the pullback B —A C as the equalizer [(b, c) | (b, f b, c) = (b, gc, c)], thus given as
the equalizer:
d
d ’0
’’ B — A — C
’’
B —A C ’ ’ B — C ’ ’
’ ’
d1
Now suppose we are given the arrow
EC
C
d  
d  
d  
d
‚  
©
A
in E /A. We have a serially commutative diagram in which both rows are equal-
izers of re¬‚exive pairs:
d0 E
d E B—C E B—A—C
B —A C
d1


c c c
d0 E
E B—C E B—A—C
B —A C
d d1
Applying P to this diagram gives rise to the diagram in which by Lemma 5
of the preceding section the rows are contractible coequalizers. After Hom(1, ’)
is applied, we get

d0 E
Sub(B — A — C ) ' E Sub(B — C ) E Sub(B —A C )
d1
(—)
c c c
d0 E
' E Sub(B —A C)
Sub(B — A — C) E Sub(B — C)
d1
5.2 Slices of Toposes 177
in which by the external Beck condition the rows are contractible coequalizers
with contractions d0 —¦ ’, so this diagram commutes serially too. Thus
d0 E
Hom(C , P(B — A)) ' E Hom(C , PB)
E
1
d

c c
d0 E
'
Hom(C, P(B — A)) Hom(C, PB)
is serially commutative and has contractible pairs as rows. By the adjunction
between E and E /A (see Exercise 13 of Section 1.9), the following is a serially
commutative diagram of contractible pairs:
d0 E
Hom(C ’ A, P(B — A) — A ’ A) ' E Hom(C ’ A, PB — A ’ A)
’ ’ ’ ’
1
d

c c
d0 E
'
Hom(C ’ A, P(B — A) — A ’ A)
’ ’ E Hom(C ’ A, PB — A ’ A)
’ ’
d1
It follows from the Yoneda lemma that we have a contractible pair
P(B — A) — A ’ A ’’ ’’ PB — A ’ A
’ ’ ’’’
←’ ’’
’’ ’
’ ’’’
in E /A. Coequalizers exist in E /A; they are created by the underlying functor to
E . Using the facts that Hom functors, like all functors, preserve the coequalizers
of contractible pairs, and subobjects in E /A are identical to subobjects in E , it
is easy to see from diagram (*) above that the coequalizer of the above parallel
pair represents Sub(B —A ’).

Exercise 5.2

1. Assuming, as we establish in the next section, that pullbacks of regular epis
are regular epis, show that E ’ E /A is faithful if and only if A ’ 1 is epi.
’ ’
(Hint: consider, for
B’ C

’’
the diagram
E
E A—C
A—B


c Ec
EC
B
178 5 Properties of Toposes
and use regularity.)


5.3 Logical Functors
Two sorts of functors between toposes have proved to be important. Logical
functors are those which preserve the structure given in the de¬nition of a topos.
They will be discussed in this section. The other kind is geometric functors,
which arise as an abstraction of the map induced by a continuous map between
topological spaces on the corresponding categories of sheaves of sets. They will
be discussed in Section 6.5.
A functor L which preserves the structure of a topos must be left exact (pre-
serve ¬nite limits) and preserve power objects in the sense that L applied to each
power object must represent in a strong sense speci¬ed below the corresponding
subobject functor in the codomain category. To make sense of this, note that
any functor L: E ’ E induces by restriction a function (also denoted L) from

HomE (A, B) to HomE (LA, LB) for any objects A and B of E . Furthermore, if
L is left exact (hence preserves monos and products in particular), it takes any
subobject U )’ A — B to a subobject LU )’ LA — LB.
’ ’
In the following de¬nition, we put a prime on P or φ to indicate that it is
part of the structure of E .
A functor L: E ’ E between toposes is called logical if it preserves ¬nite

limits and if for each object B, there is an isomorphism

βB: Hom(’, LPB) ’ Sub(’ — LB)


such that the following diagram commutes (φ is the natural transformation in
diagram (1) of Section 2.1).

L E Hom(LA, LPB)
Hom(A, PB)

(1)
φ(A, B) (βB)A
c c
E Sub(LA — LB)
Sub(A — B)

Of course, the de¬nition implies that for every object B of E, LPB is isomor-
phic to P LB. We will see below that in fact the induced isomorphism is natural
in B.
5.3 Logical Functors 179
Proposition 1. A functor L: E ’ E is logical if and only if for each object

B of E there is an isomorphism ±B: LPB ’ P LB for which the induced map

γB: Hom(A, PB) ’ Hom(LA, P LB) de¬ned by γB(f ) = ±B —¦ Lf for f : A

’ PB preserves φ in the sense that

γB E
Hom(A, PB) Hom(LA, LPB)

(2)
φ(A, B) φ (LA, LB)
c c
E Sub(LA — LB)
Sub(A — B)
commutes.

<<

. 33
( 60 .)



>>