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Proof. If L is logical, so that (1) commutes, de¬ne ±B to be the unique isomor-
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)
Hom(A, PB)
βBd  φ (LA, LB)
φ(A, B)

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)
Hom(A, Pg) Hom(LA, P Lg) 

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

φ(A, C) φ(A, B) φ (LA, LB) φ (LA, LC)

c c
E Sub(LA — LB)
Sub(A — B)

 Sub(A — g) Sub(LA — Lg)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
LopE op
op (6)
c c
A similar argument using Butler™s Theorem 3(b) of Section 3.7 yields the other
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) ’ ’

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


c c
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,


f id
c c
must be a pushout. Applying the functor B —B ’, which preserves pushouts, to
that diagram gives
B —B A

c c
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:


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