·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.