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Proof. If L is logical, so that (1) commutes, deп¬Ѓne О±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, О±B) Hom(LA, P LB)
E
Hom(A, PB)
d В
d В
(3)
ОІBd В П† (LA, LB)
П†(A, B)
d В
d В
dВ
c
E Sub(LA Г— LB)
Sub(A Г— B)

The arrow along the top of (3) is ОіB, so (2) commutes.
Conversely, given the arrows О±B, deп¬Ѓne ОІ by requiring that the triangle in
(3) commute; then (1) commutes as required. ОІ is natural in X because both
small squares in (4) below commute for any f : X в€’ Y :
в†’

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

Hom(X, О±B) Hom(Y, О±B)
c c
Hom(X, P LB) ' Hom(Y, P LB) (4)

П†(X, LB) П†(Y, LB)
c c
Sub(X Г— LB) ' Sub(Y Г— LB)
180 5 Properties of Toposes
If L: E в€’ E is logical, then P
Proposition 2. в†’ L is naturally isomorphic to
в—¦
Lop в—¦ P.
Proof. In fact, the О± of Proposition 1 is a natural isomorphism. In view of the
way that О± is deп¬Ѓned in terms of Оі, the naturality follows from the fact that the
top face in diagram (5) below commutes for any g: B в€’ C. This in turn follows
в†’
from the fact that all the other faces commute and П† and П† are isomorphisms.
The left and right faces commute by deп¬Ѓnition of P and P . The bottom face
commutes because L preserves monos and pullbacks.
ОіC E Hom(LA, P LC)
Hom(A, PC)
d В
d В
Hom(A, Pg) Hom(LA, P Lg)В
d
d В
В‚
d В
ОіBE
Hom(A, PB) Hom(LA, P LB)

(5)
П†(A, C) П†(A, B) П† (LA, LB) П† (LA, LC)

c c
E Sub(LA Г— LB)
Sub(A Г— B)

В  d
s
В  d
В Sub(A Г— g) Sub(LA Г— Lg)d
В  d
cВ  dc
E Sub(LA Г— LC)
Sub(A Г— C)

Proposition 3. Logical functors preserve the subobject classiп¬Ѓer.
Proof. A logical functor preserves the terminal object since it preserves п¬Ѓnite
limits, and the subobject classiп¬Ѓer is P1.
Proposition 4. Logical functors preserve п¬Ѓnite colimits.
Proof. Let L: E в€’ E be a logical functor. Since P has a left adjoint, it preserves
в†’
п¬Ѓnite limits, so L в—¦ P preserves п¬Ѓnite limits. Hence by Proposition 2, P в—¦ Lop
preserves п¬Ѓnite limits. Since P is tripleable, it reп¬‚ects limits (Proposition 1 and
Exercise 1 of Section 3.3); hence Lop : E op в€’ E op must preserve п¬Ѓnite limits.
в†’
Thus L preserves п¬Ѓnite colimits.
5.3 Logical Functors 181
Proposition 5. A logical functor L has a right adjoint if and only if it has a
Proof. If L has a right adjoint, then apply ButlerвЂ™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
E E
T T
op
op (6)
P P P
P
c c
EE
E
L
A similar argument using ButlerвЂ™s Theorem 3(b) of Section 3.7 yields the other
implication.
Theorem 6. Let Aв€— be the functor which takes an object B of E to the object
B Г— A в€’ A (the arrow is projection) of E /A, and an arrow f : B в€’ C to
в†’ в†’
f Г— idA : B Г— A в€’ C Г— A. Then Aв€—
в†’
(i) is right adjoint to the forgetful functor,
(ii) is logical,
(iii) has a right adjoint Aв€— , and
(iv) is faithful if and only if A в€’ 1 is epi.
в†’
Proof. (i) is Exercise 13 of Section 1.9 and (iii) follows from (ii) by Proposition 5.
To prove (ii), observe п¬Ѓrst that Aв€— clearly preserves limits. For a given object C
в€’ A of E /A and an object B of E , diagram (1) becomes the diagram below.
в†’
Aв€— E Hom(C Г— A в€’ A, PB Г— A в€’ A)
Hom(C, PB) в†’ в†’

(7)
П†(C, B) ОІ(C, B)
c c
E Sub(C Г— B Г— A в€’ A)
Sub(C Г— B) в†’
The lower right corner really is the set of subobjects of the product of C Г— A
в€’ A and B Г— A в€’ A in E /A, since that product is the pullback [(c, a, b, a) |
в†’ в†’
a = a] = C Г— B Г— 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: CГ—A
в€’ PB). We deп¬Ѓne ОІ by requiring that ОІ(C, B)(u, p2 ) to be the subobject
в†’
(П†(C Г— A, B)(u)) of Sub(C Г— B Г— A). Then ОІ is natural in B: given g: B в€’ B, в†’
the commutativity condition requires that
P(idC , g, p2 )(П†(u), p2 ) = (П†(Pg(u)), p2 )
182 5 Properties of Toposes
which follows because П† is a natural isomorphism and so commutes with the
functor C Г— в€’ Г— A.
We must show that (7) commutes. If f : C в€’ PB, the northern route around
в†’
the diagram takes f to П†(C Г— A, B)(f, p2 ), whereas the southern route takes it
to (П†(C, B)f, p2 ). These are the same since П† commutes with the functor в€’ Г— A.
This completes the proof of (ii). (iii) is immediate from (i), (ii) and Proposition 5.
Finally, we must prove (iv). If A в€’ 1 is epi, then since B Г— в€’ has a right
в†’
adjoint, it preserves epis and we conclude that A Г— B в€’ B is also epi. Now we
в†’
must show that if f = g then A Г— f = A Г— g. This follows from the fact just
noted and the fact that this diagram commutes:
AГ—f
E
E AГ—C
AГ—B
AГ—g
(8)
proj proj
c Ec
f
EC
B
g
Conversely, if f : A в€’ 1 is not epi, consider g, h: 1 в€’ B with g в—¦ f = h в—¦ f .
в†’ в†’
Then
A Г— g = A Г— h: A в€’ A Г— 1 в€’ A Г— B
в†’ в†’
xThe right adjoint Aв€— is often called A , and the forgetful functor, which is
left adjoint to Aв€— , is called ОЈA . This is because an object B в€’ A of E /A can be
в†’
thought of as an indexed family {Ba : a в€€ A} of sets. In Set, ОЈA (B в€’ A), which
в†’
of course is B, is the union of the family and A (B в€’ A) is the product of
в†’
the family. The two notations are both useful and suggest diп¬Ђerent, but equally
correct, aspects of the story.
An object A for which A в€’ 1 is epi is said to have global support. (Think
в†’
of a sheaf of continuous functions to see why).
Corollary 7. If f : A в€’ B in E , then the pullback functor f в€— : E /B в€’ E /A
в†’ в†’
which takes g: X в€’ B to the pullback
в†’
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 в€’ B of
в†’
E /B, (E /B)/(A в€’ B) = E /A.
в†’
Corollary 8. In a topos, pullbacks commute with colimits. In particular, the
pullback of an epimorphism is an epimorphism.
Proof. The п¬Ѓrst sentence follows from Corollary 7. Given f : A в€’ B epi,
в†’
fE
A B

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

id
c c
EB
B
id
which must therefore be a pushout. Hence the map B Г— A в€’ B must be epi.
в†’

Exercises 5.3

1. Show that in Set, Aв€— (g: C в€’ A) can be taken to be the set of all functions
в†’
from A to C which split g, or alternatively as the product of the п¬Ѓbers of g.

5.4 Toposes are Cartesian Closed
A category is cartesian closed if the functor Hom(в€’ Г— A, B) is representable.
The representing object is denoted B A , so that

Hom(C, B A )
Hom(C Г— 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 в€€T B A ,
y в€€T A, deп¬Ѓne f (y) = ev A(f, y), where ev A: B A Г— A в€’ B is the counit of
в†’
the adjunction (see Exercise 2). This notation has been developed extensively,
for example in Kock . It can conп¬‚ict with our notation f (y) = f в—¦ y if T
happens to be the same as A; KockвЂ™s treatment shows how to handle this conп¬‚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 Aв€— (= A Г— в€’) of Section 5.3; the
other constructs B A as an equalizer, from which it follows that logical functors
preserve the construction.
The п¬Ѓrst construction follows from the observation that in Set, in the notation
of Section 5.3, Aв€— (B Г— A в€’ A) is the set of functions from A to B Г— A which
в†’
split the structure map of B Г— A в€’ A and this is just the set of all maps A to
в†’
B. This suggests trying Aв€— (B Г— A в€’ A) as a candidate for B A , which works
в†’
because of the following sequence of calculations:

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