phism making the triangle in (3) below commute. There is one, since LPB and

P LB represent the same functor.

L E Hom(LA, LPB) Hom(LA, ±B) Hom(LA, P LB)

E

Hom(A, PB)

d

d

(3)

βBd φ (LA, LB)

φ(A, B)

d

d

‚©

d

c

E Sub(LA — LB)

Sub(A — B)

The arrow along the top of (3) is γB, so (2) commutes.

Conversely, given the arrows ±B, de¬ne β by requiring that the triangle in

(3) commute; then (1) commutes as required. β is natural in X because both

small squares in (4) below commute for any f : X ’ Y :

’

Hom(X, LPB) ' Hom(Y, LPB)

Hom(X, ±B) Hom(Y, ±B)

c c

Hom(X, P LB) ' Hom(Y, P LB) (4)

φ(X, LB) φ(Y, LB)

c c

Sub(X — LB) ' Sub(Y — LB)

180 5 Properties of Toposes

If L: E ’ E is logical, then P

Proposition 2. ’ L is naturally isomorphic to

—¦

Lop —¦ P.

Proof. In fact, the ± of Proposition 1 is a natural isomorphism. In view of the

way that ± is de¬ned in terms of γ, the naturality follows from the fact that the

top face in diagram (5) below commutes for any g: B ’ C. This in turn follows

’

from the fact that all the other faces commute and φ and φ are isomorphisms.

The left and right faces commute by de¬nition of P and P . The bottom face

commutes because L preserves monos and pullbacks.

γC E Hom(LA, P LC)

Hom(A, PC)

d

d

Hom(A, Pg) Hom(LA, P Lg)

d

d

‚

d

©

γBE

Hom(A, PB) Hom(LA, P LB)

(5)

φ(A, C) φ(A, B) φ (LA, LB) φ (LA, LC)

c c

E Sub(LA — LB)

Sub(A — B)

d

s

d

Sub(A — g) Sub(LA — Lg)d

d

c dc

E Sub(LA — LC)

Sub(A — C)

Proposition 3. Logical functors preserve the subobject classi¬er.

Proof. A logical functor preserves the terminal object since it preserves ¬nite

limits, and the subobject classi¬er is P1.

Proposition 4. Logical functors preserve ¬nite colimits.

Proof. Let L: E ’ E be a logical functor. Since P has a left adjoint, it preserves

’

¬nite limits, so L —¦ P preserves ¬nite limits. Hence by Proposition 2, P —¦ Lop

preserves ¬nite limits. Since P is tripleable, it re¬‚ects limits (Proposition 1 and

Exercise 1 of Section 3.3); hence Lop : E op ’ E op must preserve ¬nite limits.

’

Thus L preserves ¬nite colimits.

5.3 Logical Functors 181

Proposition 5. A logical functor L has a right adjoint if and only if it has a

left adjoint.

Proof. If L has a right adjoint, then apply Butler™s Theorem 3(a) of Section 3.7

to this diagram to conclude that Lop has a right adjoint, whence L has a left

adjoint.

LopE op

op

E E

T T

op

op (6)

P P P

P

c c

EE

E

L

A similar argument using Butler™s Theorem 3(b) of Section 3.7 yields the other

implication.

Theorem 6. Let A— be the functor which takes an object B of E to the object

B — A ’ A (the arrow is projection) of E /A, and an arrow f : B ’ C to

’ ’

f — idA : B — A ’ C — A. Then A—

’

(i) is right adjoint to the forgetful functor,

(ii) is logical,

(iii) has a right adjoint A— , and

(iv) is faithful if and only if A ’ 1 is epi.

’

Proof. (i) is Exercise 13 of Section 1.9 and (iii) follows from (ii) by Proposition 5.

To prove (ii), observe ¬rst that A— clearly preserves limits. For a given object C

’ A of E /A and an object B of E , diagram (1) becomes the diagram below.

’

A— E Hom(C — A ’ A, PB — A ’ A)

Hom(C, PB) ’ ’

(7)

φ(C, B) β(C, B)

c c

E Sub(C — B — A ’ A)

Sub(C — B) ’

The lower right corner really is the set of subobjects of the product of C — A

’ A and B — A ’ A in E /A, since that product is the pullback [(c, a, b, a) |

’ ’

a = a] = C — B — A. The bottom arrow takes a subobject u to (u, p2 ).

An arrow in the upper right corner must be of the form (u, p2 ) for some u: C—A

’ PB). We de¬ne β by requiring that β(C, B)(u, p2 ) to be the subobject

’

(φ(C — A, B)(u)) of Sub(C — B — A). Then β is natural in B: given g: B ’ B, ’

the commutativity condition requires that

P(idC , g, p2 )(φ(u), p2 ) = (φ(Pg(u)), p2 )

182 5 Properties of Toposes

which follows because φ is a natural isomorphism and so commutes with the

functor C — ’ — A.

We must show that (7) commutes. If f : C ’ PB, the northern route around

’

the diagram takes f to φ(C — A, B)(f, p2 ), whereas the southern route takes it

to (φ(C, B)f, p2 ). These are the same since φ commutes with the functor ’ — A.

This completes the proof of (ii). (iii) is immediate from (i), (ii) and Proposition 5.

Finally, we must prove (iv). If A ’ 1 is epi, then since B — ’ has a right

’

adjoint, it preserves epis and we conclude that A — B ’ B is also epi. Now we

’

must show that if f = g then A — f = A — g. This follows from the fact just

noted and the fact that this diagram commutes:

A—f

E

E A—C

A—B

A—g

(8)

proj proj

c Ec

f

EC

B

g

Conversely, if f : A ’ 1 is not epi, consider g, h: 1 ’ B with g —¦ f = h —¦ f .

’ ’

Then

A — g = A — h: A ’ A — 1 ’ A — B

’ ’

which contradicts faithfulness.

xThe right adjoint A— is often called A , and the forgetful functor, which is

left adjoint to A— , is called ΣA . This is because an object B ’ A of E /A can be

’

thought of as an indexed family {Ba : a ∈ A} of sets. In Set, ΣA (B ’ A), which

’

of course is B, is the union of the family and A (B ’ A) is the product of

’

the family. The two notations are both useful and suggest di¬erent, but equally

correct, aspects of the story.

An object A for which A ’ 1 is epi is said to have global support. (Think

’

of a sheaf of continuous functions to see why).

Corollary 7. If f : A ’ B in E , then the pullback functor f — : E /B ’ E /A

’ ’

which takes g: X ’ B to the pullback

’

EX

P

g

c c

EB

A

f

has left and right adjoints.

5.4 Toposes are Cartesian Closed 183

Proof. This follows from Theorem 6 by observing that for an object A ’ B of

’

E /B, (E /B)/(A ’ B) = E /A.

’

Corollary 8. In a topos, pullbacks commute with colimits. In particular, the

pullback of an epimorphism is an epimorphism.

Proof. The ¬rst sentence follows from Corollary 7. Given f : A ’ B epi,

’

fE

A B

f id

c c

EB

B

id

must be a pushout. Applying the functor B —B ’, which preserves pushouts, to

that diagram gives

EB

B —B A

id

c c

EB

B

id

which must therefore be a pushout. Hence the map B — A ’ B must be epi.

’

Exercises 5.3

1. Show that in Set, A— (g: C ’ A) can be taken to be the set of all functions

’

from A to C which split g, or alternatively as the product of the ¬bers of g.

5.4 Toposes are Cartesian Closed

A category is cartesian closed if the functor Hom(’ — A, B) is representable.

The representing object is denoted B A , so that

Hom(C, B A )

Hom(C — A, B)

for all objects C of E . The object B A is called the exponential of B by A. The

notation B A is used because the global elements of B A , by the adjunction, are

just the elements of Hom(A, B).

184 5 Properties of Toposes

The general elements of B A can also be thought of as functions. If f ∈T B A ,

y ∈T A, de¬ne f (y) = ev A(f, y), where ev A: B A — A ’ B is the counit of

’

the adjunction (see Exercise 2). This notation has been developed extensively,

for example in Kock [1981]. It can con¬‚ict with our notation f (y) = f —¦ y if T

happens to be the same as A; Kock™s treatment shows how to handle this con¬‚ict.

Theorem 1. A topos is cartesian closed.

We will give two constructions of the exponential. One is an easy consequence

of the existence of a right adjoint to the functor A— (= A — ’) of Section 5.3; the

other constructs B A as an equalizer, from which it follows that logical functors

preserve the construction.

The ¬rst construction follows from the observation that in Set, in the notation

of Section 5.3, A— (B — A ’ A) is the set of functions from A to B — A which

’

split the structure map of B — A ’ A and this is just the set of all maps A to

’

B. This suggests trying A— (B — A ’ A) as a candidate for B A , which works

’

because of the following sequence of calculations: