’

f E UA

B0

T

©

·B0 ·T U A (10)

B UA

gd

dc

‚ c

E TUA

T B0

Tf

6.5 Left exact triples 219

Here, u is given by Lemma 5 and U A —¦ ·U A = id by Exercise 15 of Section 1.9.

It follows that U A —¦ T f —¦ u is the required arrow.

We now have a factorization of the given geometric morphism through Ej into

two geometric morphisms.

Ej

d s

d

dd

— V

V — F d dincl

dd

©

d

‚

d

U— E

E' E

—

U

Here, V— is U— regarded as going into Ej and V — is U — composed with inclusion.

V — is cotripleable.

Theorem 8.

Proof. V — is left exact by assumption, and so preserves all equalizers (and they

exist because Ej is a topos). So all we need to show is that it re¬‚ects isomorphisms.

Suppose f : A ’ B in Ej is such that V — (f ) is an isomorphism. In the

’

following diagram, ∆ is the diagonal map, T = V— —¦ V — , and d0 and d1 are the

projections from the ¬ber product. All the vertical maps are components of the

unit · corresponding to the adjunction of V — and V— .

d0 E fE

∆

E E A —B A

A EA B

d1

c c T d0E c c

E T A —T ETB

TA TA ETA

B

Tf

T (∆) 1

Td

The composite across the top is f and T f is an isomorphism by assumption,

so T d0 = T d1 . This means T (∆) is an isomorphism. But A is a j-sheaf, so it

is j-separated, meaning the left square is a pullback. (Note that as far as objects

of Ej are concerned, T is the triple determined by U and its left adjoint). That

means that ∆ is an isomorphism, so f is monic. That means that by the same

argument the right square is a pullback, so f is an isomorphism.

Thus every sheaf category in a topos is the category of coalgebras for a left

exact cotriple, and every geometric morphism is the composite of the cofree map

for a left exact cotriple followed by the inclusion of its category of coalgebras as

a sheaf category in the codomain topos.

220 6 Permanence Properties of Toposes

Exercises 6.5

1. Use Lemma 3 to give a direct proof that any object of the form T A is separated.

2. If j is a topology on a topos E , the inclusion of the category of sheaves in E

and its left adjoint the shea¬¬cation functor F given by Theorem 9 of Section 6.2

produce a triple T = (T, ·, µ) in E .

(a) Show that µ is an isomorphism. (A triple for which µ is an isomorphism

is said to be idempotent ).

(b) Show that the topology induced by that triple is j.

3. Show that the Kleisli and Eilenberg-Moore categories of an idempotent triple

are equivalent.

6.6 Categories in a Topos

We will de¬ne category objects in a category E with ¬nite limits by commuta-

tive diagrams as in Section 1.1. Functors between such category objects have

a straightforward de¬nition. What is more interesting is that functor from a

category object C in E to E itself may be de¬ned even though E is not itself a

category object in E . It turns out (as in Set) that the category of such E -valued

functors is a topos when E is a topos.

Category objects

A category object in E is C = (C, C1 , d0 , d1 , u, c), where C is the object of

objects, C1 is the object of morphisms, d0 : C1 ’ C is the domain map, d1 : C1

’

’ C is the codomain map, u: C ’ C1 the unit map, and c: C2 ’ C1 the

’ ’ ’

0 1

composition. Here, C2 is the ¬ber product [(f, g) | d (f ) = d (g)]. In general,

Cn = [(f1 , . . . , fn ) | d1 (fi ) = d0 (fi+1 ), i = 1, . . . , n ’ 1], the object of composable

n-tuples of maps of C . These objects and maps must satisfy the following laws:

(i) d0 —¦ u = d1 —¦ u = idC .

(ii) The following diagrams commute:

p2 E p1 E

C2 C1 C2 C1

c c

d1 d0

c c c c

EC EC

C1 C1

d0 d1

6.6 Categories in a Topos 221

(in other words, d0 (c(f, g)) = d0 (g) and d1 (c(f, g)) = d1 (f ) for all elements (f, g)

of C1 ),

(iii) c(c — id) = c(id — c): C3 ’ C1 , and

’

(iv) c(ud1 , id) = c(id, ud0 ) = id: C1 ’ C1 .

’

In general, we will use the notation that when a letter denotes a category

object, that letter with subscript 1 denotes its arrows. The maps d0 , d1 and u

will always be called by the same name.

An internal functor F : C ’ D between category objects of E is a pair

’

of maps F : C ’ D and F1 : C1 ’ D1 which commutes with all the structure

’ ’

maps: F —¦ d0 = d0 —¦ F1 , F —¦ d1 = d1 —¦ F1 , F1 —¦ u = u —¦ F , and F1 —¦ c = c —¦ (F1 — F1 ).

It is straightforward to show that the category of category objects and functors

between them form a category Cat(E ).

E -valued functors

There are two approaches to the problem of de¬ning the notion of an E -

valued functor from a category object C in a topos E to E . They turn out to be

equivalent. One is a generalization of the algebraic notion of monoid action and

the other is analogous to the topological concept of ¬bration. We describe each

construction in Set ¬rst and then give the general de¬nition.

The algebraic approach is to regard C as a generalized monoid. Then a Set-

valued functor is a generalization of the notion of monoid action. Thus if F : C

’ Set and f : A ’ B in C , one writes f x for F f (x) when x ∈ F A and shows

’ ’

that the map (f, x) ’ f x satis¬es laws generalizing those of a monoid action:

’

(1A )x = x and (f g)x = f (gx) whenever f g and gx are de¬ned. This map (f, x)

’ f x has a ¬ber product as domain: x must be an element of d0 (f ). Moreover,

’

it is a map over C.

Guided by this, we say a left C -object is a structure (A, φ, ψ) where φ: A

’ C, and ψ: C1 —C A ’ A, where C1 —C A = [(g, a) | a ∈ A, g ∈ C1 and

’ ’

φ(a) = d0 (g)], for which

(i) ψ(u(φ(a)), a) = a for all elements a of A,

(ii) φψ(g, a) = d1 (g) whenever φ(a) = d0 (g), and

(iii) ψ(c(f, g), a) = ψ(f, ψ(g, a)) for all (f, g, a) for which a ∈ A, f , g ∈ C1

and φ(a) = d0 (g), d1 (g) = d0 (f ).

A morphism of left C -objects is a map over C which commutes with φ and ψ

in the obvious way. In Set, given a functor G: C ’ Set, A would be the disjoint

’

union of all the values of G for all objects of C , φ(x) would denote the object C

for which x ∈ GC, and ψ(x, g) would denote G(g)(x).

222 6 Permanence Properties of Toposes

Contravariant functors can be handled by considering right C -objects.

The other approach, via ¬brations, takes the values of a functor F : C ’ ’

E and joins them together in a category over C . The result has a property

analogous to the homotopy lifting property (Exercise ??) and is a particular

type of “op¬bration”. The general notion of op¬bration (for Set) is given in

Exercise 2 and will not be used in this book (see Gray [1974]).

(The corresponding object for contravariant functors to E ”i.e. presheaves”

is a “¬bration.” These ideas were discovered by Grothendieck, who was primarily

interested in presheaves. What we call op¬brations he called co¬brations.)

The way this construction works in Set is this: Given F : C ’ Set, construct

’

the category D whose objects are the elements of the (disjoint) sets F C for objects

C of C . If you were explaining to someone the way F works, you might draw, for

each element x of F C and f : C ’ C , an arrow from x to F f (x). These arrows

’

are the arrows of D. They compose in the obvious way, and there is an obvious

map from D to C . Then for C ∈ Ob(C ), F C is the inverse image of C under

that map.

This has to be approached more indirectly in a topos. Given a topos E and

a category object C of E , a morphism φ: D ’ C of category objects is a split

’

discrete op¬bration if

d0 E D

D1

φ1 φ

c c

EC

C1

d0

is a pullback. This says that the set of arrows of D is exactly the set

[(g, d) | g ∈ C1 , d ∈ D, and d0 (g) = φ(d)]

Furthermore, the identi¬cation as a pullback square means that d0 (g, d) = d

(because the top arrow must be the second projection) and similarly φ1 (g, d) = g

(hence φ(d1 (g, d)) = d1 (g)).

It follows that for each object d of D and each arrow g out of φ(d) in C , there

is exactly one arrow of D over g with domain d; we denote this arrow (g, d). This

property will be referred to as the unique lifting property of op¬brations.

A split discrete op¬bration over C is thus an object in E /C; we de¬ne a

morphism of split discrete op¬brations over C to be just a morphism in E /C.

Proposition 1. Let E be a left exact category and C a category object of E .

Then the category of split discrete op¬brations over C is equivalent to the category

of left C objects. When E = Set, they are equivalent to the functor category SetC .

6.6 Categories in a Topos 223

Proof. Suppose φ: D ’ C is a split discrete op¬bration. Then D1 = C1 —C D

’

and d1 : D1 ’ D. We claim that (D, φ, d1 ) is a left C action. All the veri¬cations,

’

including that morphisms of split discrete op¬brations are taken to morphisms

of left C actions, make use of the unique lifting property. We show two of the

required properties and leave the others to you.

We show ¬rst that d1 (u(φd), d) = d. Observe that φ1 (u(φd), d) = u(φd) =

φ1 (ud) and d0 (u(φd), d) = d = d0 (ud). Therefore by the unique lifting property,

ud = (u(φd), d). The result follows from the fact that d1 (ud) = d.

We also need

d1 (c(f, g), d) = d1 (f, d1 (g, d))

where d0 (f ) = d1 (g) and d0 (g) = φd. Now the arrows (c(f, g), d) and c((f, d1 (g, d)), (g, d))

(composition in D) both have domain d and lie over c(f, g). Therefore they are

the same arrow, so d1 of them is the same.

Going the other way, suppose that (A, φ, ψ) is a left C -object. Let D =

C1 —C A = [(g, a) | d0 (g) = φ(a)]. Then the top part of the following serially

commutative diagram

(uφ , idA )

© p2 EA

D ψ

p1 φ

u

c ‚c

d0 EC

C1 1

d

s

is a category object with composition taking ((g, a), (g , a )) to (c(g, g ), a), and φ

is a morphism of category objects. The veri¬cation of all the laws is tedious but

straightforward. It is then immediate from the de¬nition of D that (φ, p1 ) is an

split discrete op¬bration.

Suppose (A, φ, ψ) is a left C -object in Set. Then de¬ne a functor F : C ’ Set’

by requiring that for an object C of C , F C = φ’1 (C). If x ∈ F C and g: C

’ D, set F g(x) = ψ(g, x). If (A , φ , ψ ) is another left C -object and »: A

’

’ A is a morphism, then » corresponds to a natural transformation between

’

the corresponding functors whose component at C is the restriction of » to F C

(which is a subset of A). That this construction gives an equivalence between the

category of left C -objects and SetC follows directly from (i)-(iii) in the de¬nition

of left C -object and the de¬nition of morphism.

224 6 Permanence Properties of Toposes

Left C -objects form a topos

We will now bring in heavy artillery from several preceding sections to show

that the category of left C -objects and their morphisms form a topos.

Let C be a category object in the topos E and let T : E /C ’ E /C be the

’

functor which takes A ’ C to C1 —C A ’ C and f : A ’ B over C to

’ ’ ’

C1 —C f . Let · be the natural transformation from idE to T whose component at

an object s: A ’ C is ·(s: A ’ C) = (us, idA ). Let µ: T 2 ’ T be the natural

’ ’ ’

transformation whose component at A ’ C is (c, idA ). It then follows directly

’