ńņš. 42 |

triple and that the category of Eilenberg-Moore algebras (Ļ: A ā’ C, Ļ) is the

ā’

category of left C -objects.

Now come the heavy cannon. By the constructions of Theorem 6 and Corol-

lary 7 of Section 5.3, T factors as the top row of

(d0 )ā— d1

E /C ā’ā’ ā’ā’ E /C1 ā’ā’ ā’ā’ E /C

āā’ā’ā’

ā’ā’ ā’ā’ āā’ā’ā’

ā’ā’ ā’ā’

d0 (d1 )ā—

where the bottom row forms a right adjoint to T . Thus by Theorem 5 of Sec-

tion 3.7, T is adjoint tripleable and the category of left C -objects is the category

of coalgebras for a cotriple whose functor is G = d0 ā—¦ (d1 )ā— . G has a left adjoint,

so gives a left exact cotriple. Hence by Theorem 1 of Section 6.4, we have:

Theorem 2. For any category object C of a topos E , the category of left C -

objects (equivalently the category of split discrete opļ¬brations of C ) is a topos.

It is natural to denote the category of left C -objects by E C .

Theorem 3. Any category object functor f : C ā’ C induces a functor f # : E C

ā’

ā’ E C which is the restriction of the pullback functor f ā— . Moreover, f # has left

ā’

and right adjoints.

Proof. That pulling back a split discrete opļ¬bration produces a split discrete

opļ¬bration is merely the statement that pulling back a pullback gives a pull-

back. The reader may alternatively deļ¬ne f # on left C -objects by stipulating

that f # (A , Ļ , Ļ ) = (A, Ļ, Ļ), where A = [(c, a ) | f c = Ļ a ], Ļ is the ļ¬rst

projection, and Ļ(g, (c, a )) (where necessarily d0 (g) = c and Ļa = f c) must be

d1 (g), Ļ (f (g), a )).

6.6 Categories in a Topos 225

The square

f# E C

C

E E

U

U

c c

E E /C

E /C ā—

f

commutes, where the vertical arrows are the underlying functors from triple alge-

bras. (They are in fact inclusions). Since the bottom arrow has both a left and a

right adjoint, so does the top one by Butlerā™s Theorems (Section 3.7, Theorem 3).

When C is the trivial category object with C = C1 = 1, then E C is E and

the induced map E ā’ E C (where C is some category object) has left and right

ā’

adjoints denoted limā’ C and limā C respectively. When E is Set they are in fact

the left and right Kan extensions. Notice that this says that, given a set function

f : Y ā’ I, then the existence of f ā’1 (i) (which is a completeness propertyā”Y

ā’

and I can both be inļ¬nite) depends only on the fact that in Set we have ļ¬nite

limits and a power object. Thus the existence of P is a powerful hypothesis.

Exercises 6.6

1. A simplicial object in a category E is a functor S: āop ā’ E where ā is

ā’

the category whose objects are the ļ¬nite sets {1, 2, . . . , n} for n = 0, 1, 2, . . . and

whose arrows are the order-preserving maps between these sets. Prove that the

category of category objects and functors between them in a left exact category

E is equivalent to a full subcategory of the category of simplicial objects in E .

2. Let F : D ā’ C be a functor between categories (in Set). If A is an object of

ā’

C , the ļ¬ber DA over C is the subcategory of D consisting of all objects mapping

to A and all arrows mapping to 1A . Its inclusion into D is denoted JA . F is an

opļ¬bration if for each f : A ā’ B in C there is a functor f ā— : DA ā’ DB and a

ā’ ā’

ā—

natural transformation Īøf : JA ā’ JB ā—¦ f with every component lying over f , for

ā’

which for any arrow m: D ā’ E in D lying over f there is a unique arrow f ā— (D)

ā’

ā’ E making

ā’

m EE

D

d Ā

(Īøf )Dd Ā

d Ā

dĀ

Ā‚Ā©

f ā— (D)

226 6 Permanence Properties of Toposes

commute. The opļ¬bration is split if (1A )ā— = 1(DA ) and g ā— ā—¦ f ā— = (g ā—¦ f )ā— whenever

g ā—¦ f is deļ¬ned. It is discrete if the ļ¬bers DA are all setsā”i.e., their only arrows

are identity arrows. (āSplitā is also called āsplit normal,ā the normal referring to

preservation of identities). A functor between opļ¬brations over C is a functor in

Cat/C (it does not have to preserve ā— ).

(a) Show that, for Set, a split discrete opļ¬bration as deļ¬ned in the text is

the same as that deļ¬ned here.

(b) Show that the subcategory of split opļ¬brations over C and functors which

commute with ā— is equivalent to the category of functors from C to Cat and natural

transformations.

(c) Show that opļ¬brations have the āhomotopy lifting propertyā: If G and H

are functors from A to C , F : D ā’ C is an opļ¬bration, Ī»: G ā’ H is a natural

ā’ ā’

transformation, and G = F ā—¦ G for some functor G : A ā’ D, then there is a

ā’

functor H : A ā’ D such that H = F ā—¦ H and a natural transformation Ī» : G

ā’

ā’ H.

ā’

3. (Categorical deļ¬nition of opļ¬bration). Let F : D ā’ C be a functor. Let 2

ā’

denote the category with two objects 0 and 1 whose only nonidentity arrow u

goes from 0 to 1. Let S be the functor from Hom(2, D) to the comma category

(F, 1C ) which takes M : 2 ā’ D to (M (0), F (M (u)), F (M (1))) (you can ļ¬gure

ā’

out what it does to arrows of Hom(2, D), which are natural transformations).

Show that F is an opļ¬bration if and only if S has a left adjoint which is also a

right inverse of S.

6.7 Grothendieck Topologies

A Grothendieck topology on a category is a generalization of the concept of all

open covers of all open sets in a topological space.

A sieve (called ācribleā by some authorsā”ācribleā is the French word for

āsieveā) on an object A is a family of arrows with codomain A. We will use the

notation {Ai ā’ A} for a sieve, the i varying over an unspeciļ¬ed index set. We

ā’

will follow the convention that diļ¬erent sieves have possibly diļ¬erent index sets

even if the same letter i is used, unless speciļ¬cally stated otherwise.

The set of ļ¬ber products {Ai Ć—A Aj }, where i and j both run over the index

set of the sieve, will be used repeatedly in the sequel. If f : A ā’ B and {Ai

ā’

ā’ A} is a sieve on A, we write f |Ai for the composite of the projection Ai ā’ A

ā’ ā’

followed by f , and f |Ai Ć—A Aj for the composite of Ai Ć—A Aj ā’ A followed by f .

ā’

A sieve {Ai ā’ A} reļ¬nes a sieve {Bj ā’ A} if every arrow in the ļ¬rst

ā’ ā’

factors through at least one arrow in the second.

6.7 Grothendieck Topologies 227

A Grothendieck topology in a left exact category A is a family of sieves,

called covers, with the following properties.

(i) For each object A of A , idA is a sieve.

(ii) (Stability) If {Ai ā’ A} is a cover and B ā’ A is an arrow, then {B Ć—A Ai

ā’ ā’

ā’ B} is a cover.

ā’

(iii) (Composability) If {Ai ā’ A} is a cover, and for each i, {Aij ā’ Ai } is

ā’ ā’

a cover, then {Aij ā’ A} is a cover.

ā’

A Grothendieck topology is saturated if whenever {Ai ā’ A} is a sieve and

ā’

for each i, {Aij ā’ Ai } is a sieve for which {Aij ā’ A} (doubly indexed!) is a

ā’ ā’

cover, then {Ai ā’ A} is a cover. It is clear that each Grothendieck topology

ā’

is contained in a unique saturated Grothendieck topology. It follows from (ii)

(stability) that the saturation contains all the sieves which have a reļ¬nement in

the original topology.

Reļ¬nement is also deļ¬ned for topologies: one topology reļ¬nes another if every

cover in the saturation of the second is in the saturation of the ļ¬rst.

A site is a left exact category together with a speciļ¬c saturated Grothendieck

topology. A morphism of sites is a left exact functor between sites which takes

covers to covers.

Some examples of sites:

(a) The category of open sets of a ļ¬xed topological space together with all

the open covers of all open sets.

(b) Any left exact category together with all the universal regular epimor-

phisms, each regarded as a sieve containing a single arrow. (āUniversalā or

āstableā means preserved under pullbacks.)

(c) A sieve {fi : Ai ā’ A} is an epimorphic family if and only if whenever

ā’

g = h are two maps from A to B, there is at least one index i for which g ā—¦fi = hā—¦fi .

If epimorphic families are stable under pullbacks, they form a site.

op

If C is any category, an object of SetC is called a presheaf, a terminology

used particularly when C is a site. If f : A1 ā’ A2 in A and a ā A2 , then,

ā’

motivated by the discussion of sheaves on a topological space in Section 2.2, we

write a|A1 for F f (a). (We recommend this notationā”it makes the theory much

more manageable.)

Let S = {Ai ā’ A} be a sieve in a category C . If F : C op ā’ Set is a presheaf,

ā’ ā’

we say that F is S-separated if F A ā’ ā’ F Ai is injective and that F is an

S-sheaf if

ā’ F Ai ā’ ā’ F (Ai Ć—A Aj )

FA ā’ ā’ā’

228 6 Permanence Properties of Toposes

is an equalizer. This generalizes the sheaf condition for topological spaces given

by Proposition 1, Section 2.2. Stated in terms of restrictions, F is S-separated

if whenever a, a ā F A with the property that for every i, a|Ai = a |Ai , then

a = a . F is an S-sheaf if in addition for every tuple of elements ai ā Ai with

the property that ai |Ai Ć—A Aj = aj |Ai Ć—A Aj , there is a (unique) element a ā F A

such that a|Ai = ai .

It is straightforward to see that a sieve S is an epimorphic family if every

representable functor Hom(ā’, B) is S-separated, and we say that S is a regular

or eļ¬ective epimorphic family if every representable functor is an S-sheaf.

Both epimorphic and regular epimorphic families are called stable or universal

if they remain epimorphic (respectively regular epimorphic) when pulled back.

In any category, stable epimorphic families, and also stable regular epimorphic

families, form a Grothendieck topology.

Note that since every epimorphism in a topos is universal and regular by

Corollary 8 of Section 5.3 and Proposition 3 of Section 5.5, the class of all epi-

morphisms in a topos forms a Grothendieck topology on the topos. However,

epimorphic families in a topos need not be regular unless the topos has arbitrary

sums.

op

A site A determines a topology j on SetA as follows. If F is a presheaf in

op

SetA and F0 is a subpresheaf, then jF0 = jF (F0 ) is the functor whose value

at A consists of all a ā F A for which there is a cover {Ai ā’ A} such that

ā’

a|Ai ā F0 Ai for all i.

op

j as constructed above is a topology on SetA .

Proposition 1.

Proof.

(i) Naturality translates into showing that if the left square below is a pullback

then so is the right one.

FE EF jFE EF

0

0

c c c c

GE EG jGE EG

0

0

Suppose a ā jG0 A, b ā F A have the same image in GA. We must show

b ā jF0 A. Let {Ai ā’ A} witness that a ā jG0 (A). In other words, a|Ai ā G0 Ai

ā’

6.7 Grothendieck Topologies 229

for all i. Now consider this pullback diagram:

E F Ai

F0 Ai

c c

E GAi

G0 Ai

We know a|Ai ā G0 Ai and that b|Ai ā F Ai for all i. They must have the

same image in GAi , so b|Ai ā F0 Ai . Therefore b ā jF0 (A).

(ii) The inļ¬‚ationary property follows from the fact that idA is a cover.

(iii) The monotone property is a trivial consequence of the deļ¬nition.

(iv) Idempotence follows from the composition property for covers.

op

Conversely, given a topology j on SetA , we can construct a Grothendieck

topology on A as follows. Any sieve {Ai ā’ A} determines a subfunctor R of

ā’

Hom(ā’, A) deļ¬ned for an object B by letting RB be the set of f : B ā’ A for

ā’

which for some i, there is a factorization B ā’ Ai ā’ A of f . This is the same

ā’ ā’

as saying R is the union of the images of Hom(ā’, Ai ) in Hom(ā’, A). Then we

say {Ai ā’ A} is a covering sieve for the Grothendieck topology if R is j-dense.

ā’

Proposition 2. For any topology j, the deļ¬nition just given produces a Grothen-

dieck topology on A .

The proof is straightforward and will be omitted.

The two constructions given above produce a one to one correspondence be-

op

tween saturated Grothendieck topologies on A and topologies on SetA . We

make no use of this fact and the proof is uninteresting, so we omit it.

op

ńņš. 42 |