ISBN 82-553-0265-4 Mathematics No 5 - April 7

UNIFOR1'1 SPACES IN TOPOI

by

Hans Engenes Oslo

'1976

PREPRINT SERIES - Matematisk institutt, Universitetet i Oslo

ABSTRACT

The present dissertation is part of the theory of categories, or - more specifically - the theory of those categories that have become known as elementary topoi. As was realized by F.W. Lawvere and M. Tierney around 1969, these are precisely the categories in which the usual interpretation of equational theories can be extended to an internal interpretation of any - not only first order, but arbitrarily higher order - formal 'theory. This inter­pretation will in general not comply with the rules of classical logic, but rather with those of intuitionistic systems. This ties up with earlier uses of topological spaces in interpretations of intuitionistic theories, for the category of set-valued sheaves on a topological space is the prime example of an elementary topes.

Thus one can equip the objects of a topes with the structure of local rings, fields, well-orderings or topologies. Or unifor­mities, which are the subject of this dissertation.

A fairly long introduction is devoted to a detailed explana­tion of how one goes about interpreting a higher-order theory in a topes. Then a particular version of the theory of uniform spaces is discussed and interpreted, and basic properties established. This is used for the construction, following Bourbaki, of the completion of a given "uniform space object", and for proofs that the completion in fact has all the properties characteristic of the ordinary theory of uniform spaces.

A short appendix takes up the external approach to structured objects, which identifies a T-structure on an object A with a factorization of the functor Hom(-,A) through the underlying­set-functor T-models +Sets. This approach can be used for fairly general theories T, if only the factorizations of Hom(-,A) are required to be continuous.

0. PREFACE.

A topes is a category in which one can interpret finitary

higher-order theories. This can be taken as a definition of a

topes, together with a suitable specification of what it is to

"interpret" • However, there is another way to define this class

of categories~ as was realized by Lawvere and Tierney around 1970,

namely by extracting the elementary properties of categories of

sAt-valued sheaves on sites (= small categories equipped with

Grothendieck topologies, see [De]). The resulting set of axioms

can be made extremely simple (see Ch. 1), and it must be considered

a remarkable fact that one can get to "interpretability·· of

finitary higher-order theories" in such a simple fashion.

Interpretations in topoi of various concepts of ring theory

were studied by Christopher Mulvey in [Mu], published in 1974. It

was also in that year that Lawrenee N. Stout finished his docto­

ral dissertion on General Topology in Topoi ([St]). The present

dissertation takes up the theory of uniform spaces in the same

spirit, by interpreting the basic concepts and constructing the

completion of an arbitrary "uniform space object". We have not

discussed the close connections, which fairly obviously must exist,

between the present work and earlier works in intuitionistic and

non-standard analysis. Examples of the latter are [FN], [MH] and

[Lu]. Also, we have not discussed the problem of equipping uniform

space objects with algebraic structure.

Finally, I hereby express gratitude to a number of people -

to Fred Linton, who taught me category theory, suggested the subject

of this paper, and guided the work on it with limitless patience -

to Wesleyan University, of Middleton, Conn., for supporting me and

my family through five years of study and research - to all my

former teachers and colleagues at Wesleyan, for innumerable di$­

cussions, suggestions and encouragements - to the Serninaire de

Mathematique Superieure, of the Universite de Montreal, for

supporting my participation of their me0ting in June 1974, an

occation which concentrated my interest at topes theory, and

finally to Agnes Michelet, for typing this manuscript so flawlessly.

Oslo, April 1976

Hans Engenes

,.. 1 -

1 INTRODUCTION

1.1. The problem of structured objects. The privileged

status of set theory in the mathematics of the 20th century

is universally accepted, if not as desirable then at least as

factual. This is so,of course, because most branches of modern

mathematics are formulated as studies of structured sets. It

is natural for a category theorist to view this as studies of

structured objects in the category Set , of all sets and all

functions between them. From that point of view it is also

natural to seek an understanding of structured objects in

other categories, and indeed in abstract, unspecified catego­

ries satisfying certain general conditions that may be needed.

It was early realized that in a category with finite

powers (i.e. for each ob'ject A and each non-negative integer

n there is a product of n copies of A, denoted An) one

can equip any object with finitary operations satisfYing given

equations (axioms). The n-ary operations on A are then

interpreted as morphisms An ~ A, and satisfaction of a certain

equation is interpreted as commutativity of a certain diagram.

Notice that "constants" can be incorporated into this setting

as 0-ary operations, and that A0 is always a terminal object,

usually denoted 1.

Thus one can study, for example, "group-objects" in such

categories, as was done by Ekmann and Hilton in 1962 (see [EH]).

It turns out, not surprisingly, that group-objects in the cate­

gory of topological spaces are nothing more nor less than

- 2 -

topological groups, and a group-object in the category of set­

valued functors on a small category ~ is just a group-valued

functor on <& •

Objects structured in the manner described above, by a

finitary equational theory, can also (by a simple application

of the Yoneda lemma) be described as follows : First of all

there is a category, ~, of sets witn structure of the given

type and the structure-preserving functions (homomorphisms)

between them, and thet'e is a natural faithful functor (the

"forgetful" functor) UJ : J + Set. Then, with A being an

object in a category C& with finite powers, the various ways

of equipping A with a structure of the given type correspond

to the liftings, over U~ , of the contravariant Set-valued

functor represented by A, as in the following diagram :

This approach suggests an obvious way of defining an

" ~-structure" on an object in any category a, even if it

does not have finite powers, and everything works as it should

as long as the structure we are dealing with is equational.

Going beyond this limitation, however, we immediately

encounter serious problems. As an example consider the theory

of fields. The axioms can be expressed in various ways, but

not as a set of equations in certain operations, because the

- 3 -

operation of inversion is not everywhere defined. And, indeed,

it is clear that a field structure on a set A does not in

general correspond to a lifting of the functor

Set(-,A) : Set 0 P ~Set over UFields : Fields~ Set. In fact,

if A has more than one element then there are no such liftings

at all.

As a second example, consider tr.e theory of partial

orderings, which is also a relational, not equational, theory.

Let Pos be the category of partially ordered sets and order­

preserving functions. Any partial ordering on a set A clearly

yields a lifting of Set(-,A) over UPos' by the pointwise

ordering of functions, but generally there will be more liftings

than those obtained this way. To see this, let A be a two­

element set, so that Set(-,A) is naturally equivalent to the

contravariant power set functor p* : Set 0 P ~Set. There are

precisely four different orderings on A, corresponding to

liftings of P* obtained by ordering each P*(B) discretely,

indiscretely, by the inclusion relation for subsets and by the

reversed inclusion relation. But there are other ways of

ordering each P*(B) functorially. One is to let ¢ ~ B' for

all B' c: B, and B' < B" for non-empty B', B" if and only

if B' = B". This is an ordering of each p*(B), different from

those already mentioned, and it is functorial, for if f: C + B

is a function between sets, and B' < B" in p* (B), then

B' = ¢, so p*(f)(B') = f- 1 (B') = f- 1 (¢) ~ p*(f)(B") in

P*(c). So this orderig defines a fifth lifting of P* over

UPos •

- 4 -

As a third example, consider the theory of topological

spaces, and let Top be the category of all topological spaces

and all continuous maps. Although this is a second order

theory, each topology on a given set A induces a lifting of

Set(-,A) over UTop' by giving each hom-set ~(B,A) the

topology of pointwise convergence. But we can also give each

Set(B,A) the discrete topology, thereby defining a lifting of

Set(-,A) over UTop not induced by any topology on A (for,

if B is infinite and A has at least two points, then the

topology of pointwise convergence on Set(B,A) is not discrete).

In Appendix A we will give some general conditions under which this external method for structuring objects "works".

1.2 Internal languages in categories. The reason why we

can equip sets with so many kinds of structure is that in the

formal language of set theory one can express not only equational

theories, but also relational and higher-order theories.

This insight makes it natural to try and interpret formal

languages in other categories, in a way which makes it' possible

to describe models, in a given category, for theories expressed

in a given formal language.

We will now describe a rudimentary beginning of such an

interpretation, possible in any category. Let C& be a given

(arbitrary) category, and let A be an object in tt. The class

of all Glrmonomorphisms with A as codomain is denoted

L((Z)/A. We call the elements of this formulas of~ A. The

natural pre-ordering of L(ll)/A is viewed as a logical structure:

For tP,111 e: L(()){A we say "tP implies 111" or "from q> we can

infer w", if (f) factors through 111

- 5 -

, ------ --·

~~ A

and ~ and $ are equivalent if each implies the other. The

equivalence-classes in L(O,) I A are called subobj.ects of A.

There is at least one formula of type A which can be inferred

from every other, and that is idA' the identity map on A. The

formulas equivalent to idA are said to be valid on A. These

are precisely the isomorphisms in L(aJ/A.

The class L(Q) = U{L(~/AIA € IQJl is the beginning of

what we will call the internal language of CL. In general

this language will not have much expressive power, but under

certain conditions on Ci, to be listed in the following, it

will.

The first condition is that CL have pull-backs. Then,

for ~,$ € L(G0/A, let ~ A $ € L(CD/A be the formula ob­

tained by choosing a pull-back of ~ and $ :

pull-back -----> ~

v v

• ------:>A ~

(recall that pull-backs of monies are again monic). It is

easy to see that this operation (once well defined) yields

binary infima in L(Q)/A -- i.e. from ~ A $ we can infer

- 6 -

both ~ and ~' and if from ~ € L(~/A we can infer both

q, and 1/J then from ~ we can also infer ~ A ~.

Pull-backs in ~ also make available an operation

f- 1 : L(a)/B + L(UP/A, induced by a given f E Q(A,B), by

simply pulling formulas back along f :

pull-back

lr-t(<P)

----------------:>

f !"' A ---------------------> B

Notice that again we must choose pull-backs to make

well defined operation. This is a frequently occurring

phenomenon, but most often it is of little importance to

mention the choosing explicitly, so we will usually simply

a

-1 suppress it. The operation f is called substitution along

f. It is order-preserving and validity-preserving (i.e.

f- 1(id8 ) is valid on A), but that is about all we can say

about it without further assumptions on ~.

Now suppose that ~ has unique epic-monic factorizations.

That is, for each CL-morphism f there is a monomorphism f 1

and an epimorphism f 2 such that f1 0 f2 = f ' and if f' 1

is also a monomorphism, f 2 ' an epimorphism, with f 1' o f 2' = f,

then there is a unique morphism g with g o f 2 = f 1 ' and

f 1 ' o g = f 1, and this unique g is an isomorphism

f

- 7 -

Thus each morphism has a monic part. Given f E ~(A,B) and

~ E L(~)/A, let 3f(~) be the monic part of f o ~. Then

3f(~) is a formula of type B, and we call it the existential

quantification of ~ along f.

It is now an easy exercise to prove that

3f : L(a) /A -+ L(a.) /B is a left adjoint to -1 f , in the sense

that 3f(~) implies . 1/1 if and only if ~ implies f- 1(111),

whenever ~ E L( a) /A and 111 E L(£Z,) /B.

To see how these notions are connected with the logical

ideas their names suggest, take ~ to be Set, where the for-

mulas can simply be considered as subsets of their types. Then,

given f A-+ B, A' c A and B' c B, f- 1(B') is the set of

all a E A for which f(a) E B', as the notation suggests, and

3f(A') is just the image of A' under f, i.e. the set of all

b E B for which there exists a E A' with f(a) = b.

In Set, with f as above, there is also an operation

vf L(Set)/A + L(Set)/B, which is right adjoint to f-1 :

ljJ implies Vf(~) if and only if f- 1 (111) implies ~, or

B' c vf(A') if and only if f- 1 (B' ) c A' • This makes it clear

how V f (A') can be defined, namely as the set

{b E B I (Va E A)(f(a) = b •aEA')}= {b E B I f- 1({b}) c: A'}.

In other categories, with pull-backs and unique epic-monic

factorization there will generally not be a right adjoint to

f- 1, for each morphism f. But whenever there is one, we will

denote it by vf and call it universal quantification along f.

Asruming existence of vf for each f makes the language

L(~ fairly powerful, but to ensure it as much expressive

- 8

power as the first-order part of L(Set) we must have available

some further propositional connectives. We first assume

existence of a minimal formula of each type For each object

A there shall be ~A E L(aQIA from which every other formula

of type A can be inferred. Furthermore, we assume existence

of a binary operation - + - on L(a) I A, for each A E 1a1, with the property that ~ + - L(Q41A + L(CDIA is order-

preserving and is right adjoint to - A ~: L(~)IA + L(a)/A, for

each ~ € L(a)IA. That is, w A ~ implies ~ if and only if

~ implies ~ + ~. Such an operation on L(a)IA is essentially

unique. Having it available, we define a unary operation, 1 ,

on L(a.) I A, for each A € Ill!' by ! ~ = ~ .... erA. Finally'

we will assume that each L(CDIA, besides having binary infima,

given by the operation - A - , also has binary suprema, given

by a binary operation - v - on L(a)IA. That is, for ~ and

~ in L(COIA we assume there is a formula ~ v ~ in L(GDIA,

which can be inferred from both ~ and ~' with the property

that any ~ E L(GOIA which can be inferred from both ~ and

$ can also be inferred from ~ v $ •

Assuming existence of OA_, A, v and + on L((J)IA means

precisely that the partial ordering arising from the pre-ordering

L(a)IA is a Heyting algebra, or a pseudo-Bodean algebra (see

[Fr1 ] or [RS]).

If the product A x A, exists, let A2 be a chosen such,

with a chosen pair of projections. Then formulas of type A2

will also be called binary formulas of type A. Similarly,

given ar1y non-negative integer n, formulas of type An will

- 9 -

also be called n-ary formulas of type A, where An is a

chosen nth power of A in GL. As an example take the binary

formula AA : A ~ A2 on A, i.e. the diagonal map, which is

always a monomorphism. This particular binary formula on A

is called the equality predicate on A.

Since we .want to be able to interpret in each L(CD/A any

finitary predicate symbol, we now mak~ the cssumption that

has finite powers. Then, in particular, it has Oth powers, i.e.

it has a terminal object, and since we have already assumed it

has pull-backs it follows that it has all finite limits, i.e.

is finitely complete.

We say a is a pre-logos (see [Fr2 ]) if it has the pro­

perties we have assumed so far, i.e. if it is finitely complete,

has unique epic-monic factorizations, has universal quantifi­

cation along any morphism, and if each L(a)/A has ~A' v and + •

1.3 Interpreting a formal language in a pre-logos.

In this section a, will be a pre-logos. Let ~ be a formal

first-order language, as described for example in [Sh] and

!CK], with logical symbols A, v, +, ~, 3 and V , variable-

symbols x 0 ,x1 ,•••, and with only relation symbols as non­

logical symbols, so for each non-negative integer, n, ft has

a class Rn of n-ary relation symbols.

An interpretation of ~ in a. consists of a choice of

a-object, A, a choice of powers An and projection maps

n~ An+ A (i E {O,•••,n-1}), and for each n a map

In Rn ~ L(~)/An. If ~ = In(R) then we say R is inter­

preted as ~, or ~ is the interpretation of R.

- 10 -

Such an interpretation of .t_ in C{. determines an inter­

pretation in Q. of each well-formed formula If of ~, defined

inductively as follows. Let (xi ' xi1,•••,xin-1) be the free 0

variables in If, with 0 ~ io < i1 < •• fl < i n-1 • If If is atomic,

then it is of the form R(xia(o)'xia(1)'•••,xia(k-1)) for some

unique k-ary relation symbol R and some unique map a of the

ordinal k = {O,•••,k-1} onto the ordinal n = {O,•••,n-1}.

Then If is interpreted as 'lf~1(Ik(R)), where 11' : An + Ak a

is the morphism induced by i.e. k n for j a, 1Tj 0 'If = 'If a (j) cr

Next, if If is ('f'c'f"), with c € {A,v,+}, then there

are unique n', n" < n and unique strictly order-preserrtng

maps a' : n' + n and c" : n" + n such that

€ k.

(Xic'(o)'•••,xia'(n'-1)) and (xia"(o)'•••,xic"(n"-1)) are the

free variables in If' and If" respectively. If If' and If"

' " have been interpreted as <P' € L(~/An and q> 11 € L(Q)/An

respectively, then

-1( ') -1( ") 'Ira' q> c 'Ira" q> '

If = 'f'c'f"

which.is in

will be interpreted as

L(a) /An.

Furthermore, if If 1·s I If' , and If' has been interpreted

as <P' € L(QD/An, then If will be interpreted as

tq>' = <P' +a n € .L(GD/An. A

Finally, if 'l' is (3xm)'f' (or (Vxm)'f'), then we con-

sider the two cases 1) X m is not free in If' , and 2) xm is

f'ree in If ' • In case 1 ) ' If and If' have the same free

variables, and 'l' is given the same interpretation as If'. In

case 2), If' has already been given an interpretation

<P' € L(~/An+ 1 • Moreover, there is a unique j € {0,1,•••,n}

- 11 -

such that ij-1 < m < ij (where II i_1 < m < io " is read as

" m < io " and " in-1 < m < in II is read as II i < m ") .

' n-1 The interpretation of '1' will then be 3l(tP') (or, respec-

tively, Vj(tP')), which is in L((i)/An, where 3 . An+1 -+ An . jth n A n+1 is II projection along the coordinate " i.e. 1Ti 0 j = 1Ti '

for i € {O,•••,j-1} and n 0 J n+1 for i E {j,•••,n-1}. 1Ti = 1Ti +1

The above clearly determines a ur:,ique interpretation in a_

of each well-formed formula of ~ , induced by the given inter­

pretation of ~ in a. An ~-theory is simply a set (or class) T of well-formed

formulas of ft , called the axioms of the theory T. Given an 6t .ft -theory T, a model of T in Cl. is an interpretation of

in Cl for which the induced interpretation of each axiom of T

is valid, i.e. if '1' € T is interpreted as q> € L(Q)/An then

tP is equivalent to id n· A

It is desirable that models of theories have the property

that any !f. -formula which is intuitionistically derivable from

the axioms also receives valid interpretations. This can be

ensured by an assumption on ~' namely that pull-backs of epi-

morphisms are again epimorphisms. Under this assumption the

pre-logos Cl is, by definition, a logos in the sense of Freyd

(see [Fr2 ]).

1.4 Interpreting higher-order languages. In the category

Set there is an operation P on the objects, the power-set ope­

ration, which enables us to interpret second-order (and even

higher-order) languages in this category. The interpretation

- 12 -

must be designed so that variable symbols, which in the

language are required to "range over" n-ary predicates, under

the interpretation are forced to "range over" the object P(An),

where A is the chosen basic object of the interpretation.

Moreover, if R is a relation symbol of the language, admitting

two variables, the first of which is restricted to range over

elements (i.e. a first-order variable) and the second restricted

to range over unary predicates, then the symbol R must be

interpreted as a monomorphism into A x P(A). In Set the opera­

tion P has the property that, for given objects A and B, the

morphisms from A to P(B) correspond precisely to the subjets

of A xB. In particular, morphisms from A to P(1) correspond

to subsets of Ax 1 , and hence to subsets of of A. For this

reason we say that P(1) -which of course is a two-element set -

is a subobject classifier in Set.

We can ask for an operation P in any category with finite

products. We can also, separately, ask for subobject classifiers

in such categories. Carefully stated - and yet very simple -

defining properties of a subobject classifier and an operation P

on objects imply not only that they both are essentially unique,

but, much more importantly, their presence in a category with

finite products forces that category to be a logos.

Thus, it turns out, the categories admitting interpretations

of higher-order theories can be characterized by much simpler

axioms than can those admitting interpretations of first-order

theories.

Not &very logos possesses a subobject classifier or an

operation P with the desired properties. For example, any

- 13 -

Heyting algebra, considered as a partially ordered category, is

a logos, as Freyd remarked in [Fr2 ].

We will now give the above-mentioned definitions of subobject

classifiers and power-objects - as the values of the operation P

will be called.

1.5 The fundamentals of topoi.

1.5.1 Definition: In a category with a terminal object 1,

a subobject classifier ~s an object n for which there exists

a map t : 1 ~ g which, for each object X, gives rise as follows

to a one-to-one correspondence between maps X ~ g and subobjects

of X : For any monic X'~ X there is a unique map Xx,: X~ n,

called the characteristic map for (the subobject represented by)

X' + X, making the following diagram a pull-back diagram :

X' > X

1 t ---:> n

and, moreover, the map t can be pulled back along any map to n,

thus yielding a subobject of the domain of such a map.

1.5.2 Definition : A category with a terminal object,

binary products and a subobject classifier g has power-objects if

for each object X t"here is an object P(X) and a map £x:P(X)xx~ n

such that for any map q,: Y x X ~ n there is a unique map

fq>: Y~ P(X) satisfying £Xo(fq> xX) = q,.

1.5.3 Definition : A topos is a category with a terminal

object, binary products, a subobject classifier and power-objects.

- 14 -

1.5.4 Remarks Any topos has equalizers, for if f,g are

two maps X+ Y then these yield a map <f, g> : X + Y x Y, which

we compose with the characteristic map for the (monic) diagonal

map Y + Y x Y. The result is a map X + n, which classifies a

subobject of X. Any monic X' + X representing this subobject,

is an equalizer for f and g.

Hence any topos has finite limits (in particular pull-backs).

It is known ([Mi]) that any topos also has finite colimits (in

particular, finite coproducts and an initial object 0) and is

cartesian closed in the sense that any endofunctor of the form

- x X has a right adjoint, usually written (-)X ([Ko}),It is also

known (see [KWJ or · [Fr 1 ]) that topoi have unique epic-monic

factorizations,

1.6 Examples of topoi

1.6.1 The category Set_ of all sets and all functions,

where o can be any set with two elements, and power-objects

are power-sets.

1.6.2 The category Set f of all finite sets and all their

functions, with o and power-objects as in Set •

1.6.3 For any small category ~'the functor category

Set #., Here we can construct o, as a functor t& + Set ,

as follows : For a ~-object A, n(A)- n.t.(~(A,-),0)­

{F: ~+Set }F is a subfunctor of ~(A,-)}, by the Yoneda

lemma and the ·defining property of o. By "subfunctor" we mean

that for each B E 1~1, F(B) c: ~(A,B), because subfunctors in

that sense are canonical representatives of the subobjects of

~(A,-). We write F c: ~(A,-) to express this relation. Hence

we define n by O(A) = {F : tfp + Set .I F c: ~(A,-)}, and its

- 15 -

action on a morphism f € ~(A,B) is defined by

n(f)(F)(C) = {g € tf, (B,C)jg of € F(C)}. The map t : 1 + n

is defined by tA (*) = ~(A,-), where the asterisk is the

universal name for the element of "the" one-element set. Then

we construct the power-object of a functor X : Ci; + Set by

the definition P(X)(A) = {F ~+Set IF c: ~(A,-) x X} for

A € ~~~, and for f € 'ff(A,B) P(X) (f) (F) (C) =

{(g,x) € -~(B,C) x X(C) I (g of,x) € F(C)}. Finally, ex:P(X) x X+ n

is defined by (EX)A(F,x)(B) = {f € ~(A,B)!(f,X(f)(x)) € F(B)}

and (eX)A(F,x)(g)(f) =go f for g € ~(B,C). It is easy to

verify that these definitions are meaningful and have the desired

properties.

1.6.4 For a topological space T, the category Sh(T) of

Sets-valued sheaves on T. Here we define n and power~objects

as follows : For an open subset U of T, n(U) is the set of

all open subsets of u, and for an inclusion map u iu' : u' + u,

and open v c: u' u n (iu, )(V) = v n u', The map t : 1 + 0 is

defined by t (*) u = u. For a sheaf X on T, P(X)(U) is the

set of all natural transformations from xlu to olu, and

P(X)(ig,)()..) = )..'U' ={AviV c: U'}. Finally eX: P(X)xX + 0

is defined by (eX)U(A,x) = Au(x) , for A € P(X)(U) and

x € X(U). Topoi of the form Sh(T) are called space-topoi.

1.7 The internal language of a topos.

When interpreting a language (first-order or higher-order)

on an object X in a topos E, we can - and will - replace

L(E)/X by the class E(X,n), the elements of the latter corre­

sponding to the equivalence classes of the former •

- 16 -

We give now a brief description of how some of the logical

structure in E is 'constructed. The logical connectives

can all be obtained from internal operations on o itself,

assuming that a morphism t : 1 ~ o is given, by virtue of

which n is a subobject classifier. First of all let

o : 1 ~ n be the characteristic map of the unique morphism

0 ~ 1 which is a monomorphism because in a topes any morphism

to the initial object 0 is an isomorphism (a well known and

easily established fact, see [Fr1 ]. Then let A : n x n ~ n -.

be the characteristic map of the monomorphism <t, t> : 1 ~ n x b.

Thirdly, let ~ : n x n ~ n be the characteristic map of the

equalizer of A and the first projection n x n ~ n. Finally,

let v o x o ~ n be the characteristic map of the monic part

of the morphism {id0 t 0 } . o + o ~ n )( n where we have . ' t 0 id0

employed the notational convention that tx . x ~ n is the . composed morphism t X ~ 1 -> 0.

1.7.1 Remarks : We write ~~ n x n for the equalizer

of A and the first projection from n x n to n, and call it

the internal orderins on n. It yields a partial ordering on

any E(X,n), as follows : (j) !! t; if and only if <t~>, w > : x ~ n x o

factors through @' i.e. if and only if (j) A 1/J = (!), where I.P A

stands for the composed map X <q>~!J!>> n x 0 ~> n • Similarly,

we write q> v ljl for v o <q>,ljl> , and q> ~ ljl for ~o <q>,ljl> •

We also write for the composed map 0 X ~ 1 -> o. Then,

writing ., : o ~ o for the characteristic map of the unique

(mono-)morphism 0 ~ o, it is easy to prove that iq> = q> +ox

for any· q> E E(X,n).

ljl

- 17 -

It is well known (see [KW] or [Fr]) that n, with the

internal operations t, 0 ,A,v and +, is a Heyting algebra

object in E, i.e. each E(X,n) is a Heyting algebra with the

induced operations described above.

Since any topos has unique epic-monic factorizations, it

follows that it also has existential quantification along any

morphism. It is well known, and shown in [KW[ and [Fr 1 ], that

it also has universal quantification along any morphism. The

construction of the latter is rather involved, and can be found

in [KW]. For a given morphism f : X+ Y in a topos E, 3f

and vf will be considered as maps from E(X,n) to E(Y,n),

consistently with earlier conventions.

The following proposition is easily proved

1.7.2 Proposition : Let x : X' +X and y : Y' + Y

be monomorphisms in a given topos, let ~ be the characteristic

map of x, and let f X + Y be a morphism. Then the diagram

Y'

l 1

commutes if and only if f ox : X' + Y is "onto Y" in the

sense that y factors through the monic of f oX.

In the next proposition the first part is easily proved, and

the second - expressing that the so-called Beck conditions, in

the terminology of [La 3 ], hold in any topos- is well known (see

[ KW] and (St]) •

- 18 -

1.7.3 Fro~os1t1on . In a topes, let . f'

>

(*) g '1 1 g

f >

be a commutative diagram, and let ~= dom(f) ~ o. Then

3f' (q> b g') ~ 3f(q>) og and Vf(q>) o g ~ Yf' (q> o g'). If (*)

is a pull-back then the above two ~-relations reduce to

equalities (the Beck conditions for a top~s).

The next proposition is also well known (see e.g. [KW] or

[St]). It states that the internal language in a topos satis­

fies what in [La 3 ] was called Frobenius reciprocity.

1. 7.4 Proposition : 3f(q>) A 111 = 3r(~ " 111 of) for any

f >

diagram ~ ~ in a topos. 0

The following rule will also be needed later.

1. 7.5 Proposition 3f(q>) .. 111 = Vf(tP. • 111 of) for any

diagram ~f~ in a topos.

n

Proof

hence that 3f(q>) of .. 111 of~ q> • 11J of, so we have that

3f(q>) <> 111 ~ Yf(q> <> 111 of) •

Furthermore, from v f ( q> ""' 111 o f) ~ Y f ( <P -> 111 o f) we get

that Vf(<P,.. wo f) of< tP • 111 of, i.e. that

- 19 -

v f ("' ~ 1/1 o f) o f " <P ~ t.IJ o f , i. e. that

3f(vf{tp-> 1/1 0 f) 0 f 1\ <.p) < Ill , which by 1.7 .4 is equivalent to

vf("' ~ 111 o f) " 3f('P) ~ 1/1 , i.e. to vf("' ~ 1/1 o f) ~ 3f(<P) • 111.

This completes the proof.

The following rule states that one can always change the

order of two consecutive quantifications, if they are of the same

kind :

1.7.6 Proposition For any commutative diagram

in a topos we have that and

Proof . From vf,(\fg(tp)) og' of= vf,(\fg(<P)) 0 f' og . ~ Vg(<.p) og ~<€> we get that vf,<vg(<P)) o g' ~ vf(q:>) , and

hence that vf,(vg(<.p)) < v ,(vf(<P)). - g By symmetry we also have

that vg,(Vf(tp)) ~ vf,(Vg(q:>)). The case of existential quanti­

fiers is proved similarly.

The following proposition will also be helpful

1.7.7 Proposition :Iff: X+ Y is an epimorphism in a

topos, and q:> : Y + n is given, then 3f(q:> of) = <P = vf(q> of).

Proof Since f is epic it suffices to prove that

- 20 -

3f(c.p of) of = c.p of = vr<"" of) of • The inequalities

3f (c.p o f) o f ~ c.p of ~ v f (<Po :f) o f follow from the adj ointnesses

3f -I - o f -1 vf , and the converse inequalities follow from

v f (c.p o f) ~ "" ~ 3f (c.p o f) , which again follow from the mentioned

adjointnesses.

The following fact will be needed in our later discussions

of various notions of non-emptiness.It says,in effect, that the

external property of having global support is equivalent to

internal non-emptiness.

1.7.8 Proposition Let m : X' ~ X be a monomorphism

in a topos, and let r , m be the corresponding map 1 ~ P(X).

Then X' ~ 1 is an epimorphism (i.e. X' has global support) r -,

if and only if' 3P(eX) o m = t, where p is the projection

P(X) X X ~ P(X).

Proof : Let @ ~ P(X) x X be a monomorphism classified

by eX, and look at the following diagram, where the three

square cells clearly are pullbacks

1 X X' ---1-x--..m--....... > 1 x X ___ __.._ ___ > 1

v 0------·>P(X) X X

1 ____ t ____ > n#

- 21 -

So by the Beck condition for 3 (1.7.3), 3p(Ex) Qrm•=

1 x m is a monomorphism classified 3P,(eX o(rm,x X)). But

by ex o (rm1 x X), so

part of p' o ( 1 x m) •

3P , ( ex o ( r m-. x X) ) classifies the monic

Hence 3P 1 (exo (rm, x X)), and hence

is t if and only if the monic part of p' o (1- x m)

is an isomorphism, i.e. if and only if p' ·o (1 x m) is an

epimorphism, i.e. if and only if 1 x X' has global support.

But clearly 1 x X' has global support if and only if X' does,

so the proof is complete.

1.7.9 Remark We have seen, in 1.6.4, that, in a

space-topos Sh(T), the morphisms from 1 to n correspond to

the open subsets of T, and it is easy to see that this corre­

spondence preserves the natural ordering. So Sh(T)(1,n) can

be identified, as a Heyting algebra, with the lattice of open

subsets of T. The latter is rarely a Boolean algebra, in fact

it is so if and only if-complements of open sets are also open

in T. Hence the Heyting algebra object n is generally not a

Boolean algebra object, i.e. one for which the Heyting algebras

of morphisms into n are all Boolean. Sometimes it is, however,

as in Set, and when that is the case in a topos E we say E is

a Boolean topos.

The following equivalences are well known

1.7.10 Proposition

equivalent

(1) E is Boolean,

For a topos E the following are

(2) E satisfies the principle of the excluded middle,

in the sense that <P v t <P -= tx for all <;:> : X + n,

- 22 -

(3) ~ 0, = id 0 ,... + 1""1 n 0 ., ., ,

(4) {~} : 1 + 1 + o is an isomorphism.

1.8 Set-theoretic concepts. In sets, eX : P(X) x X+ n

is the membership relation, restricted to the elements and the

subsets of X. For if ; 1 + P(X) X X is given, correspn-

ding to (A ,x) € P(X) X X, then e:x o; = t if and only if X € A.

We will think of the E IS X as local membership relations in any

topes. This interpretation enables us to define, in the internal

logic, most of the concepts definable in classical set theory.

However, different but classically equivalent definitions

(meaning that in Sets they yield the same concept) will often

yield different concepts in other topoi. For examples of this,

see the discussions of finiteness in [KLM] and [VOL), and the

discussion in [Mu] of definitions of a field object and of the

real-number object.

We now define some concepts that we will make

extensive use of later, using the internal logic as described

above.

1. 8.1 Definition : Let X be an object. The singleton

map {•}: X+ P(X), written {•}x when necessary, is the map

corresponding (by the defining property of the power-object) to

the equality predicate X x X + n • The union~

u : P(P(X)) + P(X), written 'UX when necessary, is the map corre­

sponding to 3P 2 (eP(X) o p 3 A e:Xo p 1 ) : P(P(X)) x X+ n, where

p 1 , p 2 and p 3 are the obvious projections from

- 23 -

P(P(X)) x P(X) x X to the subproducts P(X) x X, P(P(X)) x X

and P(P(X)) x P(X) respectively. Finally, given a map

f : X + Y, P(f) is the map P(X) + P(Y) corresponding to

3P(X) x f(ex) : P(X) x Y + o.

The following proposition is well known. It was proved

by Kock in [Ko], and (with more detail) by Stout in [St].

1.8.2 Proposition : The operations X+ P(X), f + P(f)

constitute a covariant f .. mctor P. The transformations {a~}

and u are natural transformations to P from the identity

functor and P o P respectively, and (P, { •}, U) is a triple

(monad) on the topos, meaning that UX o {•}P(X) = idP(X) =

UX o P({•}x) and UX o UP(X) = UX o P(UX) for any object Xr

1.8.3 Remark : P is called the covariant power-object

functor because there is also a contravariant Eower-object functor

* p , which acts as p on objects, and whose action on maps is

* P(Y) -+- P(X) such that, for f : X + Y, p (f) . is the map . corresponding to the composed formula

P(Y) x X P(X) x f> P(Y) x Y _e...;;;y_> o

* We prove now that P is, in fact, a contravariant functor.

* Since it is clear that P (idx) = idx, we prove only that

* * * f "' P (gof) = P (f)oP (g) for X-> Y __g_> Z. So we must show

* * that P (f) o P (g) is the map corresponding to

p ( Z ) X X p ( Z ) X f > p ( Z ) X Y p ( Z) X g > p ( Z) X Z i> 0 •

- 24 -

But this means showing that

* * Ex 0 (P (f) oP (g) X X) = Ez 0 (P(Z) X g) 0 (P(Z) X f) •.

Now the left-hand side in this equation is

* * e:x o (P (f) x X) o (P <g) x x) ,

which, by the defining property of e:X, is

* Ey 0 (P(Y) X f) 0 (P (g) X X) '

which, of course, is

* e:yo(P (g)xY) o(l'(Z)xf),

which, by the defining property of e:y, is

e:zo(P(Z)xg)o(P(Z)xf),

as desired.

1.9 Interpretation in some topoi.

1 • 9.1 In Set , if we discuss formulas on X in terms

of the subsets of X which they define, then tX is X itself,

oX is the empty set, negation is complementation in X, meet

is intersection, join is union, and the arrow operation is

described by X' • X" = (X' n X") U (X' X'). For a function

f : X+ Y, and a subset X' of X, 3f(X') is

{y € Yl3x EX' f ( x ) = y } , and V f (X ' ) is { y E Y I f ( x ) = y • x e: X ' } •

Clearly, Set is a Boolean topes. In Setr , the topes of finite

sets, the above discussion can be repeated verbatum.

- 25 -

1. 9. 2 In Set ~, recall how n was defined (1.6. 3).

The logical operations on n must then be described as follows

t 1 ~ n has already been defined,

o 1 ~ n oA(*) is the empty subfunctor of ~(A,-),

-.

v

n ~ n 'JF)(B)= {fE.ct'(A,B)IVCE I~I,VgE~(B,C):gof¢F(C)l:

n x n ~ n : AA(F,G)(B) = {f E~(A,B) lvc E 1~1 ,

Vg E~(B,C) go f E F(C) n G(C)},

vA(F,G)(B) = {fE~(A,B).I either n(f)(F) ,

or n(f)(G), or both, is ~ (B,-)},

.. n x n ~ n ... A(F,G)(B) = {fEqg(A,B)Ivc E 11:1,

Vg Ef1(B,C) g of E F(C),.. go f E G(C)},

and the natural ordering on n is simply the subfunctor relation

~ is the subfunctor of n x n defined by

~(A) = {(F,G) E n(A) x n(A)IF c G}.

Then we describe the quantifications along a given map

f E Set ~(X, Y). We need the fact that epic-monic factorizations

of Set ·~-maps can be done componentwise. To see this, use the r.d f' A given f, and for each A E 1~1 let X(A) >> Y' (A)"-'?Y(A) ..

be the epic-monic factorization of fA. Then observe that this

defines a subfunctor Y' of Y, with f' : X ~ Y' a natural

transformation, which is clearly epic and makes f' X --> Y' C-> Y

equal to f.

Now let ~ : X~ n be given, defining a subfunctor X' of

X. Let Y' c Y be the image in Y (componentwise, by the above)

- 26 -

of the composed map X''+ XL> Y. Then 3r(tO) Y + n is the

characteristic map o.f Y' • Explicitly

3f(q:~)A(y)(B) = {h € c:g(A,B) l3x € X' (B) Y(h)(y) = fB(x)}

for A,B € 1~1 and y € Y(A). And vf(<P) can be described

explicitly by

Vf(<P) A (y)(B) = {h € ~ (A,B) !Vx € X(B) : Y(h)(y) = fB(x.) • x € X' (B)}

A well known fact (see [Fr]), which can be verified using

the above description of the logical operations, is that Set ~

is Boolean if and only if ~ is a groupoid, meaning that all

~-maps are isomorphisms.

Then we describe the set-theoretic constructions defined in

1.8. For X € !Set ~~, P(X) and £X have been described

before (1.6.3 ), so we describe here the singleton map

{ •} : X + P(X) For A € 1~1 and x € X(A), {•}A(x) is the

subfunctor of ~(A,-) x X defined by

{•}(x)(B) = {<h,x') € ~(A,B) x X(B),X(h)(x) = x'} •

Then we describe the union map U : P(P(X)) + P(X), as

follows (where G c ~(A,-) x P(X)) : UA(G)(B) = {(h,x') E~A,B) xX(B),3Fc~(B,-) xX: (h,F) € G(B) &

(vc E I~ IHvg: B +C)(g,X(g)(x')) E F(c))}.

In a space topos Sh(T), n has been defined so

that n(U) is the set of all open subsets of U, for each open

U in T. The logical operations are then defined by

t(U)(*) = U, ou(*) = ¢ , -,U(U') = int(U'-U') (notice here that

- 27 -

it makes no difference whether we use "interior in T" or

"interior in U"), "u(U',U 11 ) = U' nU", v0 (U 1 ,U 11 ) = U' UU",

and •u(U',U") = (U' nU") Uint(U,U').

From this it is immediately clear that Sh(T) is a

Boolean topos if and only if each open subset of T is also

closed. This, of course, is equivalent to discreteness for

spaces whose singletons are closed.

The quantifications along a given rna? f € Sh(T)(X,Y)

can be described as follows : Given X' G+ X, 3f(X') is the

sub-sheaf of Y described by 3f(X')(U) = {y€Y(u)J3x€X'(U) :

f 0 (x) = y} and Vf(X') is the one described by Vf(X')(U) =

{y€Y(U),'fx€X(U): fu(x) = y • x€X'(U)}.

Having already described P(X) and for sheaves X

on T, in 1.6.4, we describe now the singleton map

{~}: X+P(X), as follows: {•}u{x)v(x') = u{v' cviX(i~,)(x)=x'} for open U c T, x € X(U), open V c U and x' € X(V).

Finally, we describe u : P(P(X)) + P(X), as follows

uu<n>v<x') = u{v' c vlv~ E P(X)(V) : ~v(x') = v .. nv<~> = v},

for U open in T, n € n.t.(P(X)Iu' n0 ), V open in U and

x' € X(V).

~ 28 -

2 NOTATION CONVENTIONS AND PRELIMINARIES.

2.1 Notation conventions and definitions. In this

chapte~ the discussion is confined to one topes, E, whose

objects are denoted X, Y, Z, X', etc.

When we write an object explicitly as a product, as

for example X x Z x X x Y, the order in which the factors

appear indicates which canonical product is referred to. Thus

the example just mentioned is (the object part of) the given

limit of the functor D defined on the discrete category

{1,2,3,4} by D(1) = D(3) =X, D(2) = Z and D(4) = Y.

If n is a positive integer, and xn appears as a n factor in a product : ••• x X x • • •

' then that product

should be read as ••• x X x ••• x X x •4•, where X is repeated n

times as a factor. Thus X2 x y3 ·is to be considered as a

product of five (not two}. factors.

So when an object is exhibited as a product of n factors

(n > 0), the notation determines which object is to be considered

as the ith factor (0 < i ~ n). The given projection from such a

product to its ith factor is denoted by a letter (usually

p,p',~ etc.) equipped with the subscript 1,2,•••i-1,i+1,•••,n •

Thus the product X x Y comes equipped with a projection, p 2 ,

to X, and another p 1 , toY. That is, the subscript lists the

place-numbers of those factors along which we project the product.

This notation will prove to be very economical in the

following chapters, because of the following generalization of

it : If 1 ~ i 1 < 1 2 < ••• < im ~ n (m < n), then the projection

- 29 -

map carrying the subscript i 1, i 2 , ••• , im and defined on a

given product of n factors is always to be interpreted as

the natural projection to the subproduct formed by the factors

whose place-numbers are not among the ik's. Thus, if a map

denoted is said to r.ave as its domain a given product

of at least five factors, say x x y2 x x x z2 '

then our nota-

tion determines the function completely - in the example just

mentioned p2 •5 would be the natural projection to the subpro­

duct formed by the 1st, 3rd, 4th and 6th factors, i.e. p2 •5 is

the unique map X x y2 x X x z2 ~ X x Y x X x Z determined by

the equations p' o p - p 2.3.4 2.5 - 2.3.4.5.6' p p' 0 p = p 1.2.4.5.6' 1.2.4 2.5 1.2.3.5,6 and Pi. 2. 3 o P2. 5 =

Pl.2.3.4 .. 5 •

Sometimes (in fact quite frequently) we will omit the name

of a projection map, when it is part of a formula where it is

composed on the left with another map, in which case it uill

leave its subscripts to that other map before disappearing.

Thus, if f : X x Z ~ Z' is a given map, we will write f 2 • 3

for the composed map X x Y x X x Z P2 .3 >X x z ~> Z'.

Of course, this convention leads to ambiguities, for we would

write f2.3 also ror the composed map

X x Y' x X' x Z P2 .3> X x Z ~> Z' , but we will avoid con­

fusion by always making it clear \'lhat the domains are.

In the same spirit we will ·write 3 i1.i2.•••.im

(\I ) 1 1 .i 2 • • • •. im for 3

pil.i .•••.i 2 m

( 'V ) • Pil.iz.•••.im

It will

- 30 -

always be clear from the context along which projection maps

we are quantifying. Here we can, finally, point to naturality,

rather than merely to convenience and economy, as a justification

for our habit of indicating which factors are left out by a

projection to a subproduct, rather than indicating which of them

remain. For in Set, if A is a subset of X x Y then 31 (A)

is the subset {y € Yl3x € X : (x,y) € A} of Y and v1 (A) is

{y € Ylvx € X : (x,y) E A}. That is, the subscript of the quanti­

fication symbols indicates precisely Which variables are bound by

that quantification.

For n > 2 we will write Pn(X) for the nth iterated

power-object of X, i.e. Pn(X) = P(Pn-1(X)).

Given a map f in E, we will sometimes write dom(f)

for its domain, and ran(f) for its range (co-domain).

To any monomorphism i: dom(i) ~X corresponds a map

1 -+ P(X), which we will denote r il.

Given such i, and an object

when necessary, for the composed map

Y, we write i, or i(Y)

'i' Y->1 >P(X).

Finally, we write eX for the characteristic map of the

diagonal monomorphism t.X : X ~ X2• See 2.2 for a further dis­

cussion of this.

2 .1 . 1 Definitions :

(i) rx : P(X)2 -+ P(X2) is the map corresponding to

£2.4 A £1.3 P(X) 2 X X2 -+ Q '

(ii) nx P(X) 2 ~ P(X) is the map corresponding to

£2 A £1 : P(X)2 x X ~ n '

- 31 -

(iii) P(X)2 + n is

the diagram p

P(X)2 X X 3 > P(X)2

~:/<•<•z••,l

(iv) <x: p2(X) 2 + n is V 3 (e 2~~ 3~(e 1 • 3 A c1 .~)),

obtained from the diagram

p2(X)2 X P(X)2 --> p (X) p4

P(X)

. (v) Ax: P2(X) 2 + P2(X) is the composed map

p 2 (X) 2 r P (X) > p ( p (X) 2 ) p ( 0X) p 2 (X) ,

(vi) crx : p2(X) x X --> p2(X) is the map corresponding

to e A e p2(X) x P(X) x X --> n (or, more accurately, 3 1

to the latter followed by the natural isomorphism

p2(X) X X X P(X) --> p2(X) X P(X) X X) ,

(vii) stX : P2(X) x X-'-> P(X) is the composed map cr

p2 (X) x x _x_> pz (X) _u_> P (X) ,

(viii) stx : p2(X) --> p2(X) is the map corresponding to

the vertical arrow in the diagram

- 32 -

p3 p2(X) X P(X) X X --------> p2(X) X P(X)

<p'····~ 3 (e•<p ,st >) 3 1·3 2 P(X) 2

(ix) p2(x)2 -> n is the composed map

p2(X)2 < -------> n •

2.1.2 Remarks : When the situation permits it - and it

often will - we omit the subscript 11 X11 from the symbols for the

maps defined above.· This will make more room for other subscripts,

in particular for numbersequences inherited from projection maps.

In fact, we made use of this convention above, in parts (iv), (viii)

and (ix) of 2.1.1.

We will sometimes refer to some of the operations and rela-

tions defined in 2.1.1 by the names usually attached to them when

interpreted in Set. Thus rx is the rectangle map, because in

Set it associates to each pair (A,B) of subsets of X the subset

A x B of xz.

The maps nx and ~ are the intersection operation and

the containment relation, respectively. The map ~X is the

refinement relation for families of subsets of X, and

the meet operation for such.. In Set, Q., ~X!}; iff each

is contained in some B € ~ , and a Ax~ = {A n B I A € CL

Ax is

A € Q,

& B € ~}.

Finally, ~X is the star-refinement relation for families of

- 33 -

subsets of X - in Set a <: ~ iff for each x E X there is

BE ffiJ such that 1for all A E Q._,x E A • A c B.

2.2 Preliminaries : In 1.2 we introduced the

equality predicate on an object Y, meaning the diagonal mono­

morphism 6y : Y + Y2 • However, since the events are now taking

place in a topos, we will refer to the ·~haracteristic function of

written ey : y2 + n as the equality predicate on Y.

We will make use of the following properties of the equality pre­

dicate, of which part (i) is clear, and part (ii) was proved in

[En] :

2.2.1 Proposition : For any object Y,

(i) : ey is a symmetric, reflexive and transitive binary

relation on Y, and for any map ~: Y + n ,

( ii) (the substitutivity property).

In formal proofs in the internal language of a topos we

will make frequent use of the following lemma, whose proof is so

simple that we omit it •

2.2.2 Lemma : In any category with finite products

let X ,••~,x be given objects, let 1 n

J,K,L c {1,•••,n} be

such that L c K n J, and let the maps in the following diagram

be the obvious projections to subproducts

- 34 -

n n xi -> n x

i=1 i€J i

l ! n x~...:....:--. >n xl..

i€K i i€L

Then this diagram commutes, and if, moreover; L = K n J and

K U J = {1,•••,n} , then it is a pullback diagram.

The following propositions will also be helpful.

2.2.3 Proposition : For any ~ X + n in a topos we

have that 31 ( ~ 2 A ex) = ~ = v 2 (ex • ~ 1 ) •

Proof

diagram

X'

t

1 X

For the first equality consider the following

X'~

·1

X 'x t ---~---:> X' X X

lpX P1 --------> X

where ~ is a monomorphism classified by ~. The rectangular

part is clearly a pull-back, so the monic part of

which is equal to ~' hence is monic

already, and, by assumption, is classified by <P - is classified by

- 35 -

3 (~ A eX). The first equality follows. 1 2

For the second equality notice first that the inequality

" < " is equivalent to ~2 A ex < ~ - 1 , hence follows from 2.2.1

(ii). For the inequality " > " notice that, if lfJ < v <ex -><.p ), ' - 2 1

then lfl2 A e < <P , i.e. 3 (lf! " e ) ~ <P, S,:J lfJ ~ <P by the X - 1 1 2 X

above. Hence v2<ex .. lP ) ~ lP, and this completes the proof. 1

2 .. 2.4 Corollary : Given morphis~s X _[> Y ~> n in a

topos, we have that V (eyo<f ,p > • lP ) = <.pof. 2 2 1 1

Proof: Consider the following diagram

<f2aE1> > y2

f

p{jp; y

where the rectangular part is clearly a pull-back. Moreover,

<P 1 = ~ ,o<f ,p >, so v (eyo<f ,p > • <P) = v ((ey ~ <P ,)o<f ,p >)= 1 21 2 21 1 2 l 21

= V , ( e1· • c.p , ) o f, where we have used the Beck condition for V 2 1

(1.7.3). The corollary now follows by 2.2.3.

The following lemma says that <x is a reflexive and

transitive relation on P2 (X).

2.2.5 Lemma ep2 (X) _< <x, and ,~ A "' < '""" • "'\3 1 1 - ~2

Proof The first statement is equivalent to

e < £ ~ 3 (E A C ) , 3 - 2 4 1.3 1·2

- 36 -

i.e. to

0 A £ < 3 (E A C ) , 3 2- ~ 1.3 1.2

Since e < c , it suffices to prove that

0 A E < 3 (E A 0 ) 3 2 4 1.3 1.2

Now e A E < E by the reflexivity of e, so it suffices to 3 2 1

prove that

£ < 3 (E A 0 ) 1 - 4 1.3 1,2 '

which follows from 2.2.1 (iii).

Then we prove the second part of the lemma, i.e.

transitivity of {. We will follow the pattern of the

following proof in Set Given three families, [, y

and ~' of subsets of X, assume that U < V and V < W,

i.e. that

(VU € U)( 3V € y)(U c V) A fVV € lH 3W € W)(V c W). Then

(VU € [)(3V € V)(3W E ~)(U c VA V c W) , so

(VU € U)(3V € V)(3W € ~)(U c W) by transitivity of c.

That is , (3U € ~)(3W € W)(U c W) , or U < W • = =

So consider the following diagram, where we have tried

to indicate how the "method of elements" makes it easier to

parallel - in the general case - the above proof in Set. In

the diagram we have named all the projections simply by their

indices, since we will not need any more specific names for

them. However, some are equipped with certain marks, for the

P2 (X)3xP(Xl:__ 4.5.6

'4

xP(X)

~

p2(X)~2 ~ .

. '. -~

-4" L

- 37 -

purpose of instant recognition. Notice that all projections

in the diagram are split epimorphisms, and that all parts of

the diagram involving only projections commute. In the formal

deduction below, "a " and " tp" are abbreviations for the

maps 3 (E A C ) and E • 3 (E A C ) : £ • a , 4 1.3 1.2 2 4 1.3 1.2 2

respectively, from p2(X)2 x P(X) to n.

We will need the following observation in the proof

below. Some explanatory comments are given in square brackets.

: 3 (( E - A C ) ) A V (tp ) 5 1.3 1•2 3 5 1·4

[by 1.7.3, because the

following are pullbacks

by 2.2.2 !

p2(x)2 x P(X) <.2. p2(X)3 x P(X) ~> p2(x) 2 )

: 3 5{£ A C· ) A V (~ ) 1.3.4 1.2.3 5 1.4

< 3 (t A C A tp ) 5 1.3.4 1.2.3 1.4

·[by 1.7.4 , because

v (tP ) < <P ] 5 1.4 5- 1.4

< 3 (£ A C A a ) 5 1.3.4 1.2.3 1.4

[because a A ( a•b) ~ b always] •

- 38 -

We are now ready for the formal deduction

= v (~,p) 1\ ~ 3 3 1

= v '(tP ) "'< 4 3 1

[by 1.7.3, because

p2(X)2 xP(X) ~ P2 (X)3 xP(X)

13 14' p2(x)2 <---3 __ _ p2(X) 3

is a pullback, by 2.2.2 ]

= V ((e: • a ) A V (~) ) 4 1 2.3 3 3 1.4

< V (e: ~ (a 1\ V (~} )) - 4 1 2.3 3 3 1.4

[because ( a,.b) "c ~a => (bAc)

always ]

< V (e: • 3 (e: /\C /\a )) - 4 1 2.3 5 1.3.4 1.2.3 1.4

[by the above observation]

: V ( E ... 3 (e: 1\ C 1\ 3 (e: /\C ) )) 4 1 2.3 5 1.3.4 1.2.3 4 1·3 1·2 1·4

: V ( € •3 ( E 1\ C 1\ 3 ( € 1\ C ) ) ) 4 1 2.3 5 1 3 4 1.2.3 6 1.2.4.5 1.2.3.4 . .

[by 1.7.3 and 2.2.2 applied to the cell

p2(X)3 X P(X)3 1. 4 --> p2(X)2 X P(X)2

6 I 4 I v v

p2(X)3 xP(X) 2 1 • 4 P2 (X)2x P(X)] ->

- 39 -

= V ,(e •3 (e A3 (c: A£ AC: ))) [by 1.7.4] 4 2~3 5 1o3o4 6 lo2o3o6 1o2o4o5 1o2•3o4

< V (e •3 (e A3 (& Ac: ))) (by transitivity of c:] - 4 1 2.3 5 1.3.4 6 1.2.4.5 1.2.3.5

< v (e ~ 3 (e A c: )) Lfby 1.7.3 and 3 5 (~ 5 ) ~ •] - 4 1 2.3 5 1 1.2.4 1.2.3

= V ,(e • 3 ((e A c: ) )) 4 2.3 5 1 1.3 1.2 2

= V (e ~ 3 (e A c: ) ) 4 1 2.3 4 1.3 1.2 2

= V 1 ( .( £ ~ 3 ( £ A C: ) ) ) , 4 2 4 1.3 1.2 2

: V (e ~ 3 (E A C: )) 3 2 4 1 • 3 1 .2 2

= ~ 2 , as desired •

[by 1.7.3 and 2.2.2,applied to

the cell

p2 (X) 3xp (X) 22>P2 {X) 2 x P(X) 2

! 5' ! 4

p2(X)3xP(X) ~>P2(X)~x P(X)]

[by 1.7.3 and 2.2.2, applied to

the cell

Next we will prove the internal version of the following

fact in Set : 'Q_ ~X, A y < *}:{, • U ~ *w • We need to observe

that c:o<st ,st > = 2 1

V (e• (st )(X)<> e• (st x X)), by definition 4 2 1

of " c:" (see 2 .1 • 1 (iii) ) , and so that

- 40 -

e:o(st xX) = e:o((U•cr)xX)::: 3 ·((e: Ae: )•(crxP(X)xX))::: 3 3 1

=3((e:•(crxP{X))Ae: ):::3((e: AE) AE )= 3 4 1.2 3 2 1 4 1·2

::: 3 (e: A E A € ) , 3 2.4 1.4 1.2

where £is the composed map X x P (X) twist > P(X) x X _e:_> n •

The following lemma says, in Set, that

U <Y ~ (vx- E X)(st(x,~ ) c st(x,v )).

2.2.6 Lemma : ~ < V (c o <st , st >) • \- 3 2 1

Proof: See the following diagram for the reading of this

proof (page 41), where the maps 6twist and 5twist are,

respectively,

and

The statement of the lemma is equivalent to

< < Co<st ,st > ,i.e.to < < V (e:c (st xX) • e:o(st xX)) by 3- 2 1 3- 4 2 1

the above remarks. And, since st x X = (st x X) and 2 2

st x X = (st x X) , it is furthermore equivalent to 1 1

( ~ ) < 3 9 ( € A E A € ) .. 3 1 ( E A E A e: ) , i.e • to 3 4 - 3 2.4 1.4 1.2 z 3 2·4 1•4 1•2 1

( *) : ( <\ ) A 3 ( E A € A E ) < 3 ( e: A E A € ) • 3 4 . 3 9 2.4 1.4 1.2 2- 3 9 2.4 1.4 1.2 1

- 41 -

~· ··~~ p2(X)2xP{X)2

4

p2(X) xP(X)

i p2 (X) 2xx-3~-->P2(f) 2

oxP{X)xX

uxX

p ) x X . £ .

· .. /.

- 42 -

Now we have, by repeated applications of 1.7.3 and 2.2.2,

that

{ < ) : {V { E ,. 3 { E 1\ C. )) ) : V {( E ,. 3 ( E 1\ C. )) ) : 3 4 3 2 4 1.3 1.2 3 4 ..a. 2 4 1.3 1.2 4 4

1\C. ))) )=V (& •3(e 1\C:) )= 1.2 ~ 5twist 4' 2:3:5 4 1.3 1.2 3.5

.. : 'Y 1 ( E ,. 3 ( ( E 1\ C. ) ) ) : V ( £ .. 3 ( E /\C. ) ) ,

' 4 2 • 3 • 5 5 1 • 3 1 • 2 3. 6 4 ' 2 • 3 • 5 5 1 • 3 • 4 • 6 1.2 .3 p

and also that

3 ( E 1\ E 1\ E ) : 3 ( ( E 1\E . 1\ E ) ) : 3 (£ Ae 1\£ ) -3' 2.4 1.4 1.2 2 4' 2.4 1.-4 1.2 2 4 1 2.3.5 1.2.5 1.2.3

and that 3 (e: 1\E 1\E ) =3 (e 1\E AE ). 3' 2.4 1·4 1•2 1 4t 1·3·5 1·2·5 1·2·3

We are now ready to prove (*) above

(~ ) /\3 {e 1\E 1\£ ) 3 4 3t 2.4 1.4 1.2 2

: V ( E A 3 ( E A. C. ) ) /\3 ( E A E A £ ) 4' 2·3.5 5 1.3.4.6 1.2.3.6 4' 2.3.5 1.2.5 1.2.3

< 3 ((e •3 (E 1\C. ))/\£ A£ 1\E ) 4' 2.3.5 5 1.3.4.6 1•2•3•6 2.3.5 1.2.5 1.2.3

< 3 {3 (e: /\C. )Ae: AE 1\E ) 4 1 5 1.3.4.6 1.2.3.6 2.3.5 1.2.5 1.2.3

<3 (e ·A3(e 1\C. AE AE )) 4' 2.3.5 5 1.3.4.6 1.2.3·6 1·2·5·6 1·2·3·5

< 3 (e: A3 (e: 1\E 1\£ )) 4 1 2.3.5 5 1.3.4.6 1.2.4.6 1.2.3.4

- 43 -

:3 (e A 3 ((e: Ae: Ae: ) ) 4 1 2.3.5 5 1.3.5 1.2.5 1·2·3 411

=3 (e: A3(e: Ae AE ) ) 4' 2.3.5 4 1.3.5 1.2.5 1.2.3 4 1

< 3 (e: A £ A E ) 4 1.3.5 1.2.5 1.2.3

: 3 (e A e: A e: ) 3' 2.4 1.4 1.2 1

, as desired.

Using 2.2.6, we can now prove the following lemma •

2.2.7 Lemma: .; A <* ~ < * • ""a 1 - 2

/

Proof: The statement of the lemma is equivalent to

V (co<st ,st >) AV (3 (e: A c o<st ,p >)) 3 1 2 1 3 3' 3 11 1.4 2.3 1.2.4 1

< V (3 (e: A co <st ,p >)) , - 3 1 311 1.4 2.3 1.2.4 2

which must be read with an eye on the diagram below.

The formal deduction of the above goes as follows

V (co<st,st>) AV 1 (3 11 (e: ACo<st ,p >)) 3 1 2 1 3 3 3 1.4 2.3 1.2.4 1

=V (co<st,st> A3 11 (e ACo<st ,p >)) 4 1 2 1 3 4 1·2·5 2.3 1.2.4 1

< 't/ (3 11 (( co<st ,st >)"A e AC o<st ,p > )) - 4' 4 2 1 3 4 1.2.5 2.3 1.2.4 1

= V ( 3 (co <st , st > A e A c o <st , p > ) ) 4' 4 11 2 1 3.4 1.2.5 2.3 1.2.4 1

- 44 -

< V (3 (e A c o <st ,p > )) - 4 1 4 11 1.2.5 2.3 1.2.4 2

= V (3 11 ((e ) A co <st ,p > )) 4' 4 1.4 2 2.3 1.2.4 2

: 'if ( 3 ( E A C 0 <st , p >)) 3' 3 11 1.4 2.3 1.2.4 2

as desired.

(Q_,Y., li., W, X)

p2(X)3xP(X)xX

2 3.4 2

,

2

3: (u,y,w,x) 3" 1: (Y,W,W,x)P2 (X)2xp(X)xX--=--··-·- __ __..3:::...'----> P 2 (x) 2

2 : ( Y,, W, W, X )

p2(X)xP(X)

In the above we have made use of the following two facts

1) <* = V (3 (e A c:o<st ,p >)) 3 9 3 11 1.4 2.3 1.2.4

, and

2) c:o<st ,st > A c:o <st ,p > < c: o <st ,p > , 2 1 3.4 2·3 1·2·4 1 ' 2.3 1.2.4 2

which we now prove.

.. 45 -

For 1) we need the following diagram

(![,2.,U, V~x) 4 • (U ,y,u ,x) P~(X)2 x P(X)2 x X >P~(X)2xP(X)xX

s·~ 4"'

--~3~ (!I_,U ,x) ~ (TI,y_, V,x) ·(U ,Y.,V) p2(X) x P(XJ xX p2(X)2xP(X)xX-'~'---=>~ p2(X)2xP(X)

3'

~ 4 (u,y,u) 3 cy,y) - >P 2 (X)2xP(X)----~---->P 2 (X) 2

~ ~~t ~ Clt,Y:V c..

p2(X)xP(X)

stxP2(X)xP(X)

st xP (X)

- 46 -

in which we have that

< * = "<o (~x p2(X)) : v (e: .. 3 (e: 3 2 4 1.3

AC: ))o(stxp2(X)) 1.2

= V ((e: • 3 (e: A c: ))o(stxp2(X) x P(X))) 3 2 4 1.3 1.2

= V (e:o (st x P(X)) ~ 3 ((e "c: ) 0 (st x p2(X) x P(X)2))) 3 2 1 .!±. 1.3 1.2

= v ( 3 ( 0 0 <p 's t >) "* 3 ( £ " c: ) ) ! 3 11 1.3 2 9 2' 4 1.3 1.2

= V 3 (eo <p st > ) .. 3 ( e " c: ) ) 3 4 111 1.3' 2' 2"' 4 1.3 1.2

= v (V (eo <p ,st > • 3 (e: "c: ) )) [by 1.7.5] .3.. 4m 1.3 2' 2 1 n 4 1.3 1.2 4'"

: V (V 119 ( 0 o <p , st > ... 3 ( ( e: 1\ C: ) ) ) ) 3 4 1.2.4 2.3 4 1 1.3 1.2.5 1

: V (V ( 0 o <st ,p > • 3 ( e: 1\ c: )) ) 3 4 ,,, 2. 3 1 2 4 4 1 1 • 3. 5 1. 2. 5

- 0 •

= V (V (e o<st ,p > • 3 (e: "c: ))) [by 1.7.6] 3 1 3 191 2.3 1.2.4 4 1 1.3.5 1.2.5

: V (V 3 ( E A c: o <st . p > ) ) ) 3 9 3 1 " 4 1 1.3.5 2.3' 1.2.4 3'

[by 2.2.4]

: V (V ( 3 ( e: A c: o < st p >) ) ) 3 1 3"1 3" 1.4 2.3' 1.2.4 3"'

='V (3 (e: Ac:o<st ,p >)) 3 7 3 11 1.4 2.3 1.2.4

[by 1. 7. 7]

as desired.

- 47 -

For 2) we merely observe that it is an instance of the

transitivity of c : P(X) 2 + n • This completes the proof of

2.2.7.

The following lemma will be needed later, in the

description of the completion. In Set it expresses the fact

that x E st(x,!:l_ ) if x E U!!_ , for any x E X, U E P 2 (X). =

2.2.8 Lemma : The comp~sed map

p2 (X) x X U x X >P (X) x X _e:_> n is equal to the composed map

p2(X) x x <st,p 1 >> P(X) x X ~> n

Proof By definitions, e: o ( U x X) = 3 ( e: A e: ) and 2 3 1

e: o <st,p > = e: o (U x X)o<cr,p > = 3 (e: A e: ) o <cr,p > 1 1 2 3 1 1

= 3 ((e: Ae: )o<cr ,p' ,p' >) = 3 (e: o <a ,p' > A e: ) = 2 3 1 2 1.3 1.2 2 2 1.3 1

3 ( e: A e: "· e: ) = 3 ( e: A e: ) , where we have also used the Beck 2 3 1 1 2 3 1

condition for 3 (1.7.3)~ applied to the pullback diagram

p2(X) X P(X) X X <cr2,P'1.3,P'1.2>

p2(X) X P(X) X X

p2(X) X X ------------------> p2(X) X X

Corollary : The composed map

<U 2,idx> x x p2(X) X X > P(X)2 X X - 3 > P(X)2

< e:o <st,p > : P2(X) x X+ n • 1

> n is

- 48 -

Just observe that e. ( <U ,idx> x X) < 3 2 -

Proof

< e I\ E ) o < < u , i dx > x x ) < e: o ( < u , i dx > x x) = e: o ( u x x) , by 3 1 2 - 2 2

comparing successive pullbacks of subobjects.

The following lemma is the general version of the easy

observation in Set that st (x,u A V) = st (x, !I) n st (x, v )

2.2.10 Lemma The following diagram commutes

P2 (X) 2 xX Axx

> p2(X)xX

1 <st ,st > 2 1 1 st

P(X) 2 n > P(X)

Proof : One proves that the two routes around the diagram

correspond to the same map P2(X) 2 xX 2 + n , namely to

3 (e: /I.e: Ae: /I.e: /I.€ /I.e: ) , 3.4 2.4.5.6 1.2.4.6 1.2.4.5 1.3.5.6 1.2.3.6 1.2.3.5

where the quantification is along the projection

p2(X)2 X P(X)2 X x2 3 • 4 > P2(X)2 xx2 . This is straightforward,

and we omit it.

The following lemma expresses the fact, in Set, that if

JI <, U' and

- 49 -

2 • 2 • 11 Lemma : The map < A "" 3.4 '1•2.

is < the composed map

p2 (X) 4

<.A ,A > 2.4 1·~ > _ __:.. __ > n •

Proof : By the adjointnessess ( ) 1 V and 5 5

-A~ 1 ~ •- , and Beck's condition for v, one sees that the

given statement is equivalent to

< A <\ A E o( < A A > x P(X)) 3.4.5 1.2.5 2. 2.4 1.3

< 3 (E A c: ) • (<A , A > x P(X)) , which is established 4 1o3 lo2 2.o4 1o3

by straightforward application of the standard methods.

3. UNIFORMITIES

3.1 Basic concept's and facts. The idea of a uniform

structure on a set X is due to Andre Weil, and was first exposed

in [We], as a means for discussing uniform continuity and uniform

convergence in connection with spaces not necessarily metric.

Weil's approach, developed further by Bourbaki ([Bo]) and Kelley

([Ke]), was made in terms of entourages, or neighborhoods, of the

diagonal in X x X.

Weil showed that the family of uniformly continuous pseudo-

metrics on a uniform space suffices to determine the uniformity on

the space. This has been used by some for a different approach to

uniformities, defining them as families of pseudometrics satisfying

certain axioms (see, for example, [GJ]).

- 50 -

Finally, Weil showed that the so-called uniform coverings

of a space also suffice to determine the uniformity, and he gave

a list of axioms on families of coverings of a set, characterizing

those such families that arise from uniformities on the set, as

the families of all uniform coverings.

This last approach to uniformities, modified by Tukey ([Tu])

and developed very far by Ginsburg, Isbell, Frolik, Hager and

others {see [Is]), is the one we will use in describing what a

uniform structure is on an object in a topos. So in the rest of

this chapter, except where otherwise stated, X will be an

(arbitrary b~t fixed) object in a (fixed but arbitrary) topos E.

A uniformity on a set X, in the sense of Tukey, is a

family u of collections of subsets of X (i.e. u E P3 (X)),

satisfying the following axioms : 1) Each U E u covers X, =

2) if ~'~ E u then U A~ E u, 3) if U E u and ~ < ~ E P2 (X),

then :JL E u, 4) if U E u then V "( * U for some y_ E u, and

5) u is non-empty.

When translating the above axioms into the language of a

topos, we encounter a problem with axiom 5, because there are so

many different (and generally inequivalent) ways of describing

non-emptiness. We will handle this problem by first defining uni­

formities without any form of axiom 5 (thereby admitting ¢ as a

uniformity on any set a definitely unusual admission), and

then listing several (in fact three) different versions of axiom

5 as possible extra hypotheses on uniformities.

- 51 -

3.1.1. Definition : A monomorphism u into p2(X) is

a uniformity on X it' it satisfies the fol'lowing axioms . .

(u 1):

(u2):

(u3):

(u4)

(u5 ) 1

(u5 ) 2

(u5 ) 3

u p2F The diagram dom(u) -->

l ri~

1 > P(X) commutes '

the composed map dom(u) 2 u2 A > p2 (X) 2 ---> p2(X)

factors through u '

the intersection (pullback) of

dom(u) x P2(X) u x P2 (X) > P2(X) 2 and

0 --> p2(x)2 factors through

P2 (X) x dom(u) pl (X) x u > p2 (X) 2 '

the diagram

dom(u) u nl ----> P2 (X) <---- dom(u) x P2 (X)

1 131 (<* o (uxP2 (X))))

1 t --'""'"-----'> S1 commutes.

Moreover, u is an !-uniformity , 1 € {1,2,3}, on X

if it sauisfies (u5i) below :

dom(u) is not an initial object,

dom(u) has global support, i.e. dom(u) ~ 1 is epi ,

fi.dP(X~ : 1 ~ P2 (X) factors through u •

- 52 -

Obviously, (u5 3 ) • (u5 2 ) • (u5 1 ) in a non-degenerate topos,

i.e. one where 1 is not initial, for in that case there are no

morphisms from 1 to 0 (see [Fr1 ]). Moreover, these are the

only implications that hold in general between the (u5i)'s.

3.1.2 Definition : If u is a uniformity on X, we

will write uX for the pair (X,u), and call it a uniform space

object (uso) in E.

3.1. 3 Definition : If uX and vY are usc's in E,

and f E E(X,Y), we say f is uniformly continuous (~.) from

uX to vY if P(P*(f))•v factors through u.

The following proposition is an immediate consequence cf

the functoriality of * PoP :

3.1 • 4 Proposition The usc's in E, with all the u.c.

morphisms, form a category Unif(E), which has a canonical forget-

ful functor to E acting as the identity on morphisms.

3.2 Bases for uniformities. Classically, i.e. in the

topes Set , a basis for a uniformity u on a set X is a

subfantily b of u with the property that for each U E u = there

is V E b with V < U. This property can be formalized in the ==- = =

language of E, as in the following definition :

3.2.1 Definition Let uX be a uso, and b some

monomorph:'.sm into P2 (X). We say that b is a basis for u if

- 53 -

it factors through u and makes the following diagram commute

dom(u) u > p2 (X) < pl

dom(b) x p2(X)

l l 31( < 0 (b X P2(X)))

1 t

> 0

3.2.2 Proposition : Any uniformity is a basis for itself.

If a basis b for a uniformity u on X factors through a mono­

morphism b' into P2(X), and b' factors through u, then b'

is also a basis for u. If b is a basis for two uniformities, u

and u' '

on X, then u and u' are equivalent (meaning that

they factor through each other).

Proof : By 2.2.3, < is a reflexive relation, so

<\ o ( u x P2 {X)) is greater than or equal to the characteristic map

of dom(u) xu : dom(u) 2 -> dom(u) x p2 (X). Now the latter - and

hence also the former - of these two maps is "true on dorn(u)" when

existentially quantified along p 1 : dom(u) x p2(X) --->P2(X)2, I f

because pl u

dorn(u) 2 -->> dom(u) -> P2(X) is the epic-monic

factorization of

dom(u)2 dom(u) xu > dom(u) x P2 (X) .!J_> p2(X)

(notice that, in any category, projection maps of the form

A2 +A are epimorphisms).

The second statement of the proposition is a triviality.

For the third statement, it clearly suffices to show that

if b is a basis for u, and also factors through u', the u

i I l

' :

- 54 -

factors through u'. So we assume that 3 1 ( < o (b x P2 (X))) is

"true on u")) which by 1. 7. 2 means that u factors through the

monic part, k : K ~p2(X), of the composed map

pl @ n(dom(b) x P2 (X)) -> p2(X)2 -> pZ(x) •

So it suffices to show that k factors through u'. And it does,

beca:use @ n (dom(b) X p2 (X)) -> P2 (X) 2 factors through

@ n : ~dom(u') x pz (X)) -> pz (X) 2, by assumption (see the diagram

below), so k factors through the monic part, k' K' + P 2 (X),

of the composedmap ~ n (dom(u') xp2(X)) + p2(X)2 '!:.!.> p2(X) ,

and k' factors through u' because

@) n (dom(u') xP2(X)) + P 2 (X) 2 factors through

P2 (X) xdom(u') P 2 (X) xu'> P 2 (X) , by axiom (u3) •

and p o (P 2 (X) xu') = u' o ,., by naturality of projections. 1

..,

@'~om(u') x p2 (X2.2.: -- ->p2(x) x dom(u') .

; / / .. ~ 1T lp2 (X) x U'

/ . ~ 0 . ~ . ,.p2(X)2

"' /

.....

dom(u) ~ u --------~~----~---=::- v ~ p2(X)'

-. This completes the proof of the proposition.

- 55 -

Corollary : If u is a uniformity on X, and

b is a basis for u, then u is the smallest uniformity on

X through which b factors. Hence, if dom(b) is an initial

object, then so is dom(u).

3.2.4 Proposition ; A monic b into P2(X) is a

basis for some uniformity if and only if the following two

diagrams commute :

dom(b) b > p2(X)

(b1) 1 1 u rid .,

1 X > P(X) , and

dom(b) 2 b2 > p 2 (X ) 2 ____A_> p2 (X) < P:i dom(b) x p2(X)

(b2) 1 1 3 1 (<*o(b xP 2(X)))

1 t

> 0

in which case that uniformity, unique by last proposition, is the

subobject of P2(X) classified by the vertical map in the diagram

dom(b) x p2 (X) _________ P~~---------:> p2(X)

3 (<o(bxP2(X))) 1

(b3 ) 1

(b3 ) 2

(b3 ) 3

- 56 -

Moreover, that uniformity satisfies (u5 ) if and only if 1

dom(b) is not an initial object,

if and only if

it satisfies (u5 ) 2

dom(b) has global support, and, finally, it satisfies

(u5 ) if (and, provided the topes satisfies AC, only if) 3

there exists a mpa from 1 to dom(b).

Proof : First suppose b is a basis for a uniformity u

on X. S,ince b then factors through u, (b1) commutes by axiom

(u1) for\ u, and, since b 2 factors through u2, for (b2) it

suffices to prove that the following diagram commutes

dom(u) 2 u2 > p2(x)2 A > p2(X) <

p1 dom(b) x p2 (X)

1 1 31 ( <* o(b x P2 (X)))

1 t

> n

so by axiom (u21 it suffices to prove that 3 ( < * o (b x P2 (X))) 1

is "true on u". By assumption, 3 ( < o (b x P2 (X))) is "true on u 11 , 1

so the desired result now follows with the help of (u4) and 2.2.7.

Now suppose only that b makes (b1) and (b2) commute, and

let u(b) be a monomorphism into p2 (X) with characteristic map

3 (<;o(bxp2(X))) . p2(X) + n . We must then show that 1 ) u(b) . 1

is a uniformity on X and 2) b is a basis for it.

To prove 2), we show first that b factors through u(b),

or, in other words, that 3 (<\o(bxP2(X))) 1

is "true on b". This1

- 57 -

by 1.7.2 is equivalent to b factoring through the monic part

of A~> dom(b) xP2(X) ~> P2(X), where a represents the

sub object of dom(b) x 1? 2 (X) classified by < o (b x p2 (X)). Notice

that a can be obtained as the pullback of ($) -> P2 (X) 2 along

b x P2 (X). Now we show more than is strictly needed, namely that

b factors through p 1 o a itself. First,

6 dom(b) dom(b) > dom(b) 2 dom(b) x b> dom(b) x P2 (X) is a

factoring of b through p • 1

Second,

(b xP2(X))o(dom(b) xb) obdom(b) = b2 obdom(b) factors through

6p2 (X), and hence through @ -> P2 (X) 2 (by reflexivity of ~;

eee 2.2.5). So (dom(b) x b) o t.dom(b) factors through the pullback

of @-> P2 (X) 2 along b x P2 (X), yielding a factorization of b

through p o a. 1

Then, before completing the proof of 2), we prove 1), and

first of all we observe that u(b) satisfies (u1), by the above

and by axiom (b1) for b.

To show that u(b) satisfies (u2), notice that the sub­

object u(b) 2 of P2 (X)2 can be obtained as the intersection of

the pullbacks of u(b) along the two projections P2 (X)2-->> P2(X).

So the characteristic map for u(b) 2 is

3 1 (~o(b.xP 2 (X))) 1 A 3 1 (-<o (bxP 2 (X))), which is equal to

3 ( ( < A < ) 0 ( b 2 X p 2 (X) 2 ) ) • Now ' by 1.2 3.4 1.2

2.2.11 '

the sub-

object classified by this (i.e. u(b) 2 ) factors through the sub-

object classified by

3 (<\ o<,A A >o(b 2 x P2 (X) 2 )) = 3 (<\ o((A ob 2 ) XA )) 1•2 3.4' 1.2 1.2 3.4 "1.2

< 3 ( < * 0 (b X A ) ) < 3 ( < 0 (b X A ) ) • - 1 1 - 1 1

- 58 -

Hence A 0 u(b) 2 : dom(u(b)) 2 -+ P 2 (X) factors through the subobject

classified by 3 ( < 0 (b X p2 (X)))' 1

i.e. through u(b), as desired.

Now, to show that u(b) satisfies (u3), observe that the

characteristic map of u(b)xP 2 (X) is 3 1 (<\o(bxP 2 (X))) 2 , and

that of p2 (X) x u(b) is 3 1 ( < o (b x P2 (X))) 1 , so we must prove that

3 1 (<o(b xP2 (X))) 2 A< ~ 3 1 (<(o(bxP2 (X))) 1 •

This follows from transitivity of < , as follows

= 3 (< o(b xP2(X)2) A< ) 1 3 1

(by Frobenius reciprocity)

= 3 ((<; A< )o(b xp2(X)2)) 1 3 1

< 3 << o (b xp2(X)2)) - 1 2

(by transitivity of ~ )

=3(<o(bxP2(X))), 1 1

as desired.

The following diagram where i € {1,2} illustrates the aoove proof:

I

dom(b) x p2(X)2 ------=-P....._ ___ ·--3> p2(X)2

dom(b) x p2(X)----.::..P...._ __ _

. • "\: p2 (X)

p2(X)3-~2(X)2

So u(b) satisfies (u3).

- 59 -

For (u4) it suffices to observe that, by (b2),

3 ({o(bxP2(X))) = 3 <<*o(bxP2(X))) < 3 C<*o(u(b)xP2(X))), 1 1 - 1

so 3 1 (<*o(u(b) xP2(X))) ou(b) ~ 3 1 (<o(bxP2(X))) ou(b), whfch

factors through t 1 ---> n by construction of u(b)e

This completes the proof of 1). Nm1 we observe that the

rest of the proof of 2) - that b in fact is a basis for u(b) -

is a triviality, by the way we defined bases in general.

To complete the proof of 3.2.4, we show that u(b) satis­

fies (u51 ) if and, assuming AC when i = 3, only if b satisfies

(b3i)' i € {1,2,3}.

For i = 1, this is a triviality, since 0 ~ P2 (X) is a

uniformity on X always.

For i = 2, "if" is trivial, and for "only if" it suffices

to shol'l that dom(b) x P2 (X) has global support, which it has

because a subobject of it, namely ($) n (dom(b) x p2(X)), maps by

an epimorphism to dom(u(b)), which we assume has global support.

This is illustrated in the diagram

@n (dom(b) x p2(X)) --> d:::::~b::~·-).-.··-----1~->-2-~> 1

v 1 b X p2(X)

@ --------------> p2(X)2

<I v Q

- 60 -

For 1 = 3, assume first that there is a map 1 -+ dom(b).

Then there is one 1 ~> dom{u(b)). Now notice that the

following diagram commutes :

1

' so (u5 3 ) follows from (u3) for u(b). Then assume there is a map

c : 1 -+ dom(u(b)), and th&t the topos satisfies AC. Then the

epimorphism (5} n (dom(b) x P2 (X)) ->> dom(u(b)) splits, and

we get a map 1 --> dom(b) as follows :

1 c > dom(u(b)) ~@ n (dom(b) X P2 (X))

l dom(b) xp2(X) --2--> dom(b) •

This completes the proof of 3.2.4.

- 61 -

4. THE COMPLETION

Filters. Traditionally, a filter on a set X is a

non-empty set of non-empty subsets of X, closed under enlargements

and binary intersections.

These properties can be expressed in the language of any

topos, for any object X and any monomorphism F : dom(F) + P(X).

However, we must then again deal with the am~iguity of non-emptiness,

only this time we have it two-fold. For the filter itself, and for

its members, we can interpret non-emptiness as (a) being.non-1nit1al~

(b) having global support, and (c) having a global section (i.e. a

point or a morphism from 1). For the members of the filter we may

also consider a fourth, possibly natural, non-emptiness-condition,

namely (d) "F is disjoint from {¢}", in the sense that

0 > dom(F)

1 ox

1 1 > P(X) is a pullback.

Clearly, (c) • (b) ~ (a), and it is easy to see that also

(d) • (a).

It is instructive to interpret the above four conditions

on F in a space-topos, Sh(T), so let X be a given sheaf on the

space T, and let e.e.(X) ~> T be the associated espace etale

over T. Then a monomorphism F into P(X) yields a local

homeomorphism f : e.e.(F) +e.e.(P(X)) , which, being one-to-one,

is the embedding of an open subset of e.e.(P(X)). Now, as far as

- 62 -

F itself is concerned, condition (a) above means simply that

e.e.(F) is nonempty, condition (b) means that the composed map

e.e.(F) ~> e.e.(P(X)) P(x) > T is onto, and condition (c)

means that there is a continuous map (a global section)

s : T + e.e. (F) satisfying ?(x) of o s = idT.

For an interpretation of conditions (a)-(d) on the "members"

of F, consider the family F of those open subsets of e.e.(P(X))

that correspond to global sections of P(x) of. Then condition

(a) means that ¢ ~ F, condition (b) means that each U E F is

mapped onto T by P(x), (c) means that for each U E F there is

a global section of P(x) mapping T into u, and (d) means

that no open subset of e.e.(P(X)) has image in T disjoint from

the image of any member of F ~ 111hich is equivalent to the conditton

that each U E F is mapped by P(x) onto a dense subset of T.

The latter condition is what some call ","t' -dense" (see, for

example [ Fr 1 ] ) •

We will follows Stout ([St]) in adopting condition (c) for

members of filters, but the nonemptiness of the filters themselves

will be dealt with as that of uniformities and uniformity-bases, by

allowing consideration of three different possibilities.

Thus we have the follottdng.

4. 1 .1 Definition : Let X be a given object in a topes.

A filter on X is a monomorphism F into P(X) satisfying the

condition that the diagram

- 63 -

commutes for each of the following three choices of ~ :

(f1) :

quantification is along p2(X) xP(X)2 2.3 > p2(X),

(f2) V (E A c • E ) , where remark to (f1) applies , 2, 3 3 1 2

(f3) v (E,.. 3 (E )) , where the quantifications are along 2 3 3 1

p2(X) X P(X) X X - 3-> p2(X) X P(X) - 2-> p2(X) •

Moreover, a filter F on X is an i-filter on X if it

satisfies (f4i) below :

(f4 ) 1

(f4 ) 2

(f4 ) 3

dom(F) is not an initial object,

dom(F) has global support ,

fidxl : 1 + P(X) factors through F.

4.1.2 Remarks : Axiom (f1) means that "F is closed under

binary intersections", (f2) means that "F is closed under

elargements", and (f3) means "there is something in each member

of F". Moreover, (f4 1 ) means that "F is not empty", (f4 2 )

means that "there is something in

"F has an element (a "point") 11 •

Fn '

and (f43) means that

We will need equivalent versions of (f4i), i E. {1,2,3},

which are statements of the same form as (f1) - (f3). This is

provided by the following proposition.

- 64 -

4.1.3 Proposition : A monomorphism F into P(X) satisfies

(f4i) if (provided, when i = 1, that the topes is two-valued) and

only if the following diagram commutes

where \jJ - for i = 1 - is the map 1 o char( roP(X~ ) , for i = 2

\jJ is 3 2(E), where the quantification is along p2(X) x P(X) 2-> p2(X),

and for i = 3 w is the composed map

p2(X) <id,idx>

___ _.._...;;.;;..._ __ ,> p 2 (X) X p (X)

Proof : For i = 1, assume first that dom(F) is an initial

object. Then ~ o rF1 = .., o char(loP(X)l) o rF; = I o chart""op(x)l) o oP(X) /

: = 1 o t • t , io the diagram does not commute.

Conversely, assuming the diagram does not co~~ute, and that

the topos is two-valued,. we have that char(roP(X)I) .o rF-, = t,

since .., o charcr oP(X)l) o rF1 ~ t • So, in fact, F is (equivalent

to) oP(X)' i.e. dom(F) is initial.

For i = 2 the proposition is simply an instance of 1.7.8.

Finally, for i = 3, it suffices to observe that the following

diagram commutes :

P(X) A

1

1 -----~---> 1 X p (X)

1 p2(X)

1rFlx P(l{)

> p 2 (X) X P(X) EP(X)

--------~~~----~> n

- 65 -

4.2 Cauchy filters. Given a uniform space uX in Set,

and a filter F on X, F is called a Cauchy filter on uX if

it contains a member of each uniform cover of uX, i.e. if it

intersects each ~ € u (see [Is]). In attempting to internalize

this definition, to make it meaningful in any topos, we encounter

the same problem as the one we discussed in connection with the

filter axioms, namely the uncertainty of which degree of non-

emptiness is correct, useful or in other ways desirable.

For the sake of compatibility, we will use the same kind

of nonemptiness as the one we chose for axiom (f3) in 4.1.1.

Thus we have the following definition.

4.2.1 Definition Let uX be a uso in a topos, F

a filter on X. Then F is a Cauchy filter on uX if the following

diagram commutes

1

where the quantifications are made along the maps

p2(X)2xP(X) - 3-> p2(X)2 - 2-> p2(X), and ii is the composed map

p2(X) -> 1 ru, > p3(X) •

A Cauchy !-filter on uX is a Cauchy filter on uX, which

is also ani-filter on X, i € {1,2~3} •

- 66 -

4.2.2 Notation : In the next section we will need the

notion of a minimal Cauchy filter, by which is meant, in Set, a

Cauchy filter F on a uniform space uX, with the property that

if F' is also a Cauchy filter on uX, and pv c F, then F v = F.

For the purpose of internalizing this definition, let us

write (Cf) for the conjunction of the four defining forn1ulas for

Cauchy filters, and (Cfi) for the conjunction of (Cf) with the

remaining defining formula fori-filters, as given in 4.1.3. Thus

(Cf 1 ) is available only in two-valued topoi, while (Cf2 ) and

(Cf 3 ) are always available.

Explicitly, (Cf) is the morphism

V (e: A e oo> e:o(P 2 (X)xn)) 2.3 3 2

1\ V. (e: A C ~ e: ) 2.3 3 1 2

A V (e: ~ 3 (e: )) 2 3 1

(Cf ) - in twa-valued topoi - is 1

3 (e: A£)) 3 2 1

(Cf) A I o char(oP(X) , (Cf ) is (Cf) A 3 (e:) 2 2

( C f 3 ) is ( C f ) A e: o < i dp 2 ( X) , i dX > •

P 2 (X) ~ n ,

and

With this notation established we are ready to define minimal

Cauchy filters.

4.2.3 Definition A Cauchy filter F on a uso uX in a

topos is a minimal Cauchy filter on uX if the following diagram

- 67 -

commutes

where the quantification is along the projection p2(X)2 --1-> P2(X).

For i € {1,2,3}, a Cauchy i-filter F on uX is a minimal

Cauchy 1-filter on uX if it makes commutati7e the diagram obtained

from the above by replacing II ( Cf) il

2 with "(CF.) " •

1. 2

The object of minimal Cauchy filter~ on uX is the-subobject

yuX auX

-~:> p2(X) classified by (Cf) A v ((Cf) A c ~ e) : p2(X) ~ n 1 2

(notice that, for each uX, we have to choose such a monomorphism

auX from its equivalence class).

The object of minimal Cauchy i-filters on uX is the subobject

a yiuX i,ux > p2(X) classified by (Cfi) A v1 ((Cfi) 2 Ac~ e) : ? 2(X) ~ n.

As shown in [Bo] for the topos Set, every Cauchy filter on a

uniform space contains a unique minimal one. This is true in general,

and to express it we need the following definition

4.2.4 Definition : St p2(X) x P(X) ~ P(X) is the map

corresponding to the formula

3 ( e: A e: A e: A e: ) : p2 (X) x P (X) x x ~ n ' 2.4 3.4.5 1.3.5 1.2.5 1.2.4

where the quantification is along the projection

p2(X) x P(X)') x x2 2"4 > p2(X) x P(X) x X • (In Set, St(U,V) is

- 68 -

the set of all x for which there are x' and U with U E ~

and x' E U and x' E V and x E V, i.e. it is

u{U E ~ I u n v t ¢}.)

4.2.5 Proposition : If F is a Cauchy filter on a uso

uX, let Fm be a subobject of P(X) classified by the formula

3 (c 0 ((St 0 (u X F)) xP(X))) : P(X) + n , 1 • 2

where the quantification is along the projection

dom(u) x dom(F) x P(X) L2 > P(X). Then is also a Cauchy

filter on uX, and it is contained in very Cauchy filter on uX

contained in F. Moreover, if u is an i-uniformity, and F is

ani-filter, then Fm is also ani-filter (i E {1,2,3}).

Proof : (sketch) Axiom (f2) is trivially satisfied by Fm,

and axiom (f3) follows from the formalization of the fact that "each

member of Fm contains a member of F" '

i.e. from the fact that

which itself follows from the fact that

the composed map

p2(X) X P(X) < p ,St>

---'--> p (X) 2 c

---""""'> n

is tp 2(X) xP(X)" Axiom (f1) follows from the fact that the map

p2(X)2 X P(X)2 Ax n > p2(X) X P(X) _S_t_> P(X)

is < the map

p2(X)2 X P(X)2 <St 2 4 ,St >

• 1.3 > P(X)2 ----'-n ___ > P ( x)

The fact that Fm is Cauchy on uX is easy to prove in Set (see

[Bo]), and essentially - except for the time and space needed-

- 69 -

just as easy in general. The same holds for minimality of Fm

see [Bo].

FinallY:~ for the statement about i-filters, just notice

that there exists a morphism from dom(u) x dom(F) to dom(Fm)'

because dom(u) x dom(F) u x F > p2(X) x P(X) 3t ---> P(X) factors

through F . m

4.2.4 Notation We write nux for the composed map

X <U.{o}> > p3(X) x P(X)-r-> P(P 2(X) x X) P(st) > p 2(X) •

4.2.5 Proposition For any uso uX, nux corresponds to

the map

3x t(char(X xu) A e ) :X X P(X) ..... 0 X S 3 2

where the quantification is along the map

X x st X X p2(X) X X -> X X P(X)

Proof : Consider the diagram on page 70, where the top

and right-hand edges form the map corresponding to nux· The two

square cells are clearly pull-backs, and it is also easy to check

that the three triangular cells below them commute - the left-hand

one by definition of -u and { 0}' the middle one trivially, and

the right-hand one by definition of r (2.1.1(i)). The desired

statement now follows by the Beck condition for 3 (1.7.3).

- 70 -

A ~------------~------------------> n

4.2.6 Remark : In Set, for a given point x in a uniform

space uX, nux(x) is the family {st(x,[)l[ E u} of subsets of X • . -

We provenow in Set that nux(x) is a minimal Cauchy filter on uX,

and that it is a minimal Cauchy i-filter if u is an !-uniformity

on 'x ( i E { 1 , 2 , 3 } ) •

Firstly, niX(x) is closed under binary intersections, because

u is closed under binary meets, and st(x,U) n st(x,y) = st(x,U A V)

always (see 2.2.8). Secondly, if A~ st(x,~) E nux(x), then

A= st(x,U u {A}) is in nux(x), because ~ u {A} is in u if ~

is. Thirdly, st(x,U) is non-empty for each ~ E u, because

x E st(x,U) for any non-empty family U of subsets of X, and U

is non-empty since it covers X. Fourthly, assuming u is a

1~-unifor.mity on X, u contains a covering u -o' so n ( x) contains uX st(x,[0 ), i.e. nux(x) is also non-empty, i.e. nux(x) is a

- 11 -

1-filter on X. The cases i = 2 and i = 3 coincide with the

case i = 1 in set. Fifthly, given u € u, we must show =o

that nux(x) n u t ¢. Let y_ E u be a star-refinement of u =o -o'

and let u E U be such that st(x,V) c u . Then, with 0 =o 0

~ = ! u {U } , we have that U E u and U = st(x,y) u U = 0 0 0

st(x,[) e nux(x).

Finally, let F be a Cauchy filter on uX, contained in

nuX(x). If u is empty then so is nux(x), in which case

F = nux(x). If u is non-empty, let uo € u, and let u € F 0

Then X E u o' since u is of the form st(x,V) with y_ € u. 0

n

Hence u c:: st(x,U 0 ), so st(x,1[0 ) € F since F is closed under 0

enlargements. Hence F = nux(x) also in this case, so in each

u =o

case nux(x) is minimal among the Cauchy filters on ux. And then,

of course, when u is a 1-uniformity, in which case we have seen

that nux(x) is a Cauchy 1-filter, it is clearly a minimal such.

This completes the proof, in Set.

The following proposition states that the fact w~ just proved

in Set holds in any topos, i.e. if uX is a uso in a topos then

nux factors through auX' i.e. nux takes its values among the

minimal Cauchy filters on uX.

4.2.7 Proposition If uX is a uso in a topos, then

nux factors through aux· If, moreover, u is ani-uniformity

on X, and - in the case i = 1 - the topos is two-valued,

i € {1,2,3}, then factors through

Proof : We must prove that ~ o nux = tX for each ~ in

the set of defining axioms for minimal Cauchy (i-)filters on uX.

.

- 72 -

As in Set, axiom (f1) is true on nux by 2.2.8. We omit

the detailed verification.

For axiom (f2), we again imitate the proof in Set. By 4.2.5,

v (e A c • e ) o n X = tx if and only if the pullback of 2.3 3 1 2 u

along p2(X) x P(X)2 2 • 3 > p2(X), h d ith w en compose w e A c ~ e , 3 1 2

yields "true". And that pullback, of course, is simply

nux x P(X)2 : X x P(X)2 + p2(X) x P(X)2, so we must show that

(e3 A'\ .. E2) 0 (nux X P(X) 2) = tx xP(X)2" This is equivalent,

by the usual manipulations~ to c o ( (st o (u x X)) x P(X))

~ 3 1 (eP(X) o ((st 2 o (u2 x X) x P(X))) in the lattice of maps

dom(u) x X x P(X) + n , where the existential quantification is

along the projection map dom(u)2 x X x P(X) - 1-> dom(u) x X x P(X).

The last statement above follows from the fact that the subobject

of dom(u) x X x P(X) classified by c o ( (st o (u x X)) x P(X)) is

contained in (i.e. factors through) the equalizer of p 1 • 2

st o <U ,pw > o (u x X x {a}P(X)) : dom(u) x X x P(X) + P(X) • 2 1. 3

This is the general version of the statement in Set that

and

st(x,~ U {A}) = A provided st(x,u) c A, =

for elements x, subsets

A and coverings ~ of the set X. In Set we would prove this by

first observing that x E st(x,~) for any covering U, so that

st(x,U) c A • x E A, and hence st(x,U U {A}) = st(x,U) U A = A.

Moreover, ~ 0 {A} E u by axiom (u3) of 3.1.1.

In an arbitrary topos, the first observation has been made

already, in ?.2.9, and for the second one we need only show the

general version of the fact that U(U U {A}) = U U U A =

for any

- 73 -

subset A and family U of subsets of a set X. By 1.8.2,

U{A} = A is valid generally, so it suffices to prove distributivity

of internal union over binary union : U(~ u Y) = u~ u y. But this

follows immediately from the fact that, for any morphism f : X ~ Y,

3f : E(X,n) ~ E(Y,n) has a right adjoint and hence preserves - v -,

the (binary) coproduct in the partially ordered category E(X,n).

For axiom (f3), we must show that

v Ce: ~ 3 (e) ) on X= tx, i.e. that v ,((e<-3 (e ))o(nuxxP(X)))= tx, 2 3 1 u 2 3 1

because the following diagram is a pullback :

X "ux

--------------~> p2(X)

2 ' 2

X X p (X) "ux x P(X)

------> p2(X) X P(X)

But the latter statement above is equivalent to

E 0 (n xxP(X)) < 3 (e) 0 (n xxP(X))' u - 3 1 u where the left-hand side

is equal to

eo (P(st) x P(X)) o ((r o <u,{•}>) x P(X) =

= 3p(p2 (X) xX)xst ( Ep2 (X) xX) o ( (r o <ii, { • }>) x P(X))

= 3p 3 (X) x p (X) x s t ( e: p 2 (X) xX o ( r x p 2 (X) x X) ) • ( < ii' { • } > x p (X)

= 3p3(X)xP(X)xst(E2.4 A e:1.3) 0 (<u,{•}> xP(X))

= 3x t < < e: A e ) o ( < ii , { • } > x P 2 ( x ) x x ) ) xs 2·4 1·3

- 74 -

so the statement to be proved is equivalent to :

where the right-hand side is equal to

3 (e ) o (p2(X) x st) o Cn X x P2(X) x X) = 3 1 u

= 3 (e o(P 2 (X)xstxX))o(n Xxp2(X)xX) If 1 u

= 3 (e: o (stx X) ) o (n XxP 2 (X) xX) If 1 u

= 3 ( e: o ( u x x) o (ox x X) ) o ( n X x P 2 ( x) x x) If 1 u

= 3 (3 (e: A e: )o (ox:.<:) )o(n xxP 2 (X)xX) '+· 2 3 1 1 u

= 3 (3 ((e: Ae:) o (ox P(X) xX)) ) o (n X xP2(X) xX) If 3 3 1 1 u

= 3 (3 (e: o (ox P(X)) A e: ) ) o (n X x P 2 (X) x X) If 3 If 1.2 1 u

=3(3(e: AE) AE ))o(nx xp2(X) xX) If 3 2 1 If 1.2 1 u .

=3(3(e: AE Ae: ) )o(nxxP 2 (X)xX) If 3 2.4 1.'+ 1.2 1 u

= 3 ( 3 , ( e: A € A e: ) ) o (nux x P 2 (X) x X) 4 If 1.3.5 1.2.5 1.2.3

= 3 ( e: A £ A e: ) o (nux x P 2 (X) x x) 4.5 1.3.5 1.2.5 1.2.3

= 3 ((e: A e: A e: ) o (nuxx p2(X)x X x P(X) x X)) 4o5 1o3o5 1.2.5 1o2o3

: 3 (e: A£ A e: ) , 4.5 1.3.5 1o2o5 1o2o3

so the statement to be proved is equivalent to

£p 2 (X) 0 <U,p > A (eX) ~ 3 £ A E A £ ) 1.3 2 4.5 1.3.5 1.2.5 1.2.3

Now, by 2.2.1, we have that

e:p2(X) o <u,p > A e < e: o (u x p2(X) x X) 1. 3 2 3

and by axiom (u1) for uniformities we have

- 75 -

and the latter is < 3 ( e: A € ) , where the quantification 4 1.3 1.2

is along X x P 2 (X) x X x P (X) - 4 > X x P 2 (X) x X • But clearly

3 ( e: A e: ) < 3 ( e: A ....,.e: A e: ) so we are done 4 1.3 1.2 4.5 1.3.5 1.2.5 1.2.3 '

with axiom (f3).

Next we look at the axiom for Cauchy filters, i.e. 11{e show

that v 2(e:o<u,idP 2(x)> 1 .. 3 3 (e: 2 Ae: 1 )) = tx.

Imitating the proof in Set, we use axiom (u4) for uniformi­

ties, and then use the fact, for which a proof was sketched above,

that

st(x,V) c U q st(x,Y u {U}) = U

generalizes to any topos. We have that

= v ( ( e: o ( ii x P 2 ( x) ) ... 3 ( e: A e: ) ) o (nux x P 2 ( x) ) ) 2 3 2 1 •

= v (e: o (u x p2(X)) .. 3 ((e: A e: ) o (n xx p2(X) x P(X)))) 2 3 2 1 u

= v (e: o (u x p2(X)) ... 3 (e: A e:o(n x x P(X)))) 2 3 1 u

= v ( e: o ( ii x P 2 (X) ) .. 3 ( e: " 3xx t (char (X x u) " e ) ) ) . 2 3 1 s 3 2

Now, from axiom (u4) it follows that

e: 0 (u xP 2(X)) < 3 (e: 0 <U,p >A<*) - 2 1.3 1 '

where the quantification is along the projection X x p2(X)2 ~> p2(X)2.

Moreover, from the above mentioned fact we have that

3 (e:o<u,p ,. A~) ~3 (e:P 2 (X)o<u,p > Ae:P(X)o(P2(X)xst)otwist1.3), 2 1.3 1 2 1·3

so it suffices for us to show that the right-hand side above is

- 76 -

< 3 (E A ax t(char(Xxu) A e )) . - 3 1 xs 3 2

This is elementary, and we omit further details.

Xx p2(X) x P(X)

2 --------------------> X

p2(X)2xP(X) twist 1 ·3

'2. 1 p2(X)2xx ... p2(X)2 p2(X)

P2 (X)xst 11

'VI p2(X)

p2(X)xP(X)

l<ii,id>

p3(X)xP ~X)

~ l· • Q

Now we turn to the axiom for minimal Cauchy filters, i.e. we

show that

v 1 ( ( Cf) 2 A c ,.. ep 2 ( x) ) o nux = tx

We start with the following facts, the proofs of which are straight­

forward, and omitted :

- 77 -

v (~:: o <u,idP 2(x)> .. 3 (e A£ ) ) A(~:: • E" ) A 3x st(char(Xxu) A6) 2 1o3•4 3 2 1 3.4 2 1 X 3 2 l

< 3 ( £ 0 < u , i dp 2 (X ) > A 3 ( e: A £ " ~ . ) ) ' - 2 1.3.4 3 2.4.5 1.4"5 1.2.5

and V (£ A C • £ ) A 3 (e: 0 <u,idp2(X)> A3 (£ A£ A€ )) 2 • 3 3 1 2 2. 3 2 1 • 3 • 4 3 2 • '-1.5 1 0 4 .s 1 • 2. 5

< e: 2 ,

both illustrated by the diagram on page 79. Hence

V ( e A c • e: ) A V ( £ o <U, id> .. 3 ( £ A e: ) ) A ( e: • £ ) 2o3 3 1 2 2o3 2 1"3•4 3 2 1 3.4 2 1

A 3x t(char(Xxu) "e ) < e: , xs 3 2 1 - 2

i.e. 'It/ (e: AC ,..e:) AV (e: 0 <U,id> .,.3 (e: A e: )) A (e: cOoE) 2.3 3 1 2 2.3 2 1 3 2 1 2.3 2 1

A 3x t(char(Xxu) "e ) < £ , .xs 3 2 1 - 2

and hence (Cf) A(~:: • £)A 3X t(char(Xxu) A e ) < e: 2.3 2 1 xs 3 2 1 - 2

Since 3x t(char(Xxu) A0 ) < ex, by an easy application of axiom xs 3 2 -

(u1) for u and 2.2.9, it follows that

{Cf) A ( e: • 3X t (char(Xxu) A 0 ) ) A 3X t (char(Xxu) A 0 ) < £ , 2.3 2 xs 3 2 1 xs 3 2 1- 2

and henc9 that

(Cf) AV (e: •3x t(char(X)(u) A e ) ) < 3x t(char(Xxu) A e ) A e , 2.3 3. 2 . xs 3 2 1 3 - xs 3 2 1 2

i.e. that (Cf) AV (e: '¢>£ o {n ·x x P(X)) ) < ~:: o {n X x P(X)) .. £ , 2.3 3 2 u 1 3- u 1 2

by 4.2.5.

- 78 -

p2(X)2 X P(X) X X X P(X) ------3;;....;•::.....4;..._-------> p2(X)2 X P(X)

I ~ ·l I P 2 (X)< 1 • 3 • 4 p2(X)2 xX xP(X) --_-_--3~·-4-----> p2(X)2

; ;U,id> ~ I P 3 (X) xP2 (X) X~ 7xp~

X/ X xP2(X) X xP(X) P 2 (X) x X

\ \

\

\ ~

2 ,p2 (X) } char(Xxu)

3

\~ / E A£ AE 2.4.5 1.4.5 1.2.5

Q '.:::::::.---------

Hence

(Cf) AV (e: •e: o (n XxP(X))) < V (eo (n X xP(X)) • e:) 2 3 2 u 1-3 u 1 2 '

i.e. (Cf) AV ((e: •e: ) o (P 2 (X) x nux x P(X)))<V ((e: •e )o(P 2 (X)xnuxxP(X))) 3 2 3 2 1 - 3 1 2

i.e. (Cf) AV (e .. £ )o(P 2 (X) X n X) < \' (e: .. E )o(P2(X) X n x> , 2 3 2 1 u -31 2 u

Since 0 :'c A j, it follows that

- 79 ...

which is equivalent to

((Cf) AC:)o(P 2(X) x n X) < 0 o (P 2(X) x n X) , 2 u - u

i.e. ,

i.e. v (((Cf) Ac: .. 0)o(P 2 (X) x n X)) = tx , 1 2 u

as desired.

We now turn to the axioms (f4i), i.e. we show that

(f4i) onux = tX provided u is ani-uniformity on X. In fact,

(f4 1 ) onux = tx for any uniformity u on X, i.e. it is always

the case that I 0 char( roP(X)I) 0 nux = tx ' which is equivalent

to char(fOP(X)I) onux = char(oX), because in the following diagram

all four squares are pullbacks :

0 -------------:> 1 ---------:> 1 -----------> 1 -------:> 1

o. P2 (X) xX

X <u,{o}> >P3(X)xP(X) _.;_r_>P(P2 (X) xX) P( st)

In the case i = 2 we must show that 3 (e:} on X= t 2 u X'

assuming that u is a 2-uniformity on X. We have that

3 (e:) on X= 3 (e: o (n X xP(X))) = 3 (3X t(char(Xxu) A 0 )) = 2 u 2 u 2 xs 3 2

= 3 (char(Xxu) A 0 ) , because the triangle at the top of the 2•3 3 2

- 80 -

follownng diagram commutes

2. 3 X X p2(X) X X

' '

~X X P(X)

char(X X u) A 0 3 2

1 "ux x P(X) "ux

\ P2(X) X P(X)

\ j• ~p2(X) '\ -----\~2(€)

Now the subobject of X x p2(X) x X classified by char{X x u) A e 3 2

is <p ,u ,p > : X x dom(u) --> X x p2(X) x X , and 2 1 2

p ' o <p _, u , p > : X x dom ( u) -+- X 2.3 ~ 1 2

is simply p ' : X x dom ( u) -+- X , the 2

monic part of which is the subobject of X classified by

3 (char(Xxu) A0 ). But 2.3 3 2

I p

2 is epic, since dom(u) by

assumption has global support, and since ' p can be written as a 2

composed map X X '

X x dom(u) "dom(u) > X x 1 Q:!

-=--> X • So the monic

part of p~ is idx, so 3 (char(X x u) A e ) = tx, 2·3 3 2

as desired.

Finally, we consider the case of axiom (f4 ), where we are 3

to show that

under the assumption that u is a 3-uniforrnity on X. Consider

the following diagram, where the cells are numbered for easy

t

co 81 -

reference. It will clearly suffice to show that each cell - except

(x) - commutes, and that in cell (x) we. have that

{ o }P(X) o ridx' o !X ~ a o <idP(X) ,idx>· We prove this latter statement

first. It is equivalent to eo((ridxlo:XxP(X))~(E Ae:)o<irL(X)'P ,p >, ... 3 1 --p 1 2

<{P(X)} {•}> 3 x------~~~~~----------:> P (X) xP(X)

41

(i) (ii)

~ p2(X) xX----~------'>P(P2(X) xX)

·x (x)

/ p2 (X)

/ (iv)

{•}p(;Y

f.iyP(X) (v)

i (vi) v r1d1 1 . X

(iii) P(st) st

-----{ _• _} P_(;....X"""'") ____ :> p 2 (X)

(vii)

t (X)2

(ix)

(viii)

by exponential adj ointness and definition of a. We prove·. first

- 82 -

The following squares are easily seen to be pullbacks

1 X X . -------------> 1

P(X) X X

r· d 'l l. X ___ __;;.,;;. ___ > p (X) --------------> 1

e --------------> n

so 1 idJ2xX : 1 xX-+ P(X) x X represents the subobject classified

by eo ( 1ldxlxP(X))o <!P(X)'p 2>, so it suffices to prove that

e:P(X) o (lid~ x X) = t 1xX' which is easy. Then we prove that

eo ((f-idJ(o !X) x P(X)) < e: 3 o <idP(X)'p 1 ,p 2> • By the above we need

only show that

i.e. that - ,[1 ...., e: o < i dP ( x ) , p 1 > o ( x i dx ) = t x x 1 ,

which also is easy.

It remains to comment on the commutativity of the remaining

· · ·cells of the last diagram. Cell (i) is easily checked. For

cell (ii), observe that

ep2(X)xX = e "e = 2.lt 1.3

= € 0 ( {.} X p2 (X)) " € 0 ( { 0} X X) 2 • 4 1 • 3

= e: o(({•}x{o})x(p2(X)xX))A£ o(({•}x{o}')x(P2(X) x X)) 2·4 1.3

= (e: A e: )o(({•}x{•})x(p2(X) x X)) , 2.4 1.3

- 83 .:..

where we have used an elementary property of the equality predicate

(see [En])~ the adjointness relation between and e '

and

naturality of projections.

Cell (iii) commutes by naturality of the transformation

{•} id + P , cell (iv) by definition of st, cell (v) by one of

the triple identities for the triple (P,{•},U) (see 1.8.2),

cell (vi) is trivial, cell (vii) is easy to check, cell (viii)

commutes by adjointness relation between {•} and e, and cell

( ) r- rr. ..., ( )2 ix commutes because < id X ,idx > : 1 + P X factors through

~P(X) : P(X) + P(X) 2 •

This completes the proof of 4.2.7.

4.3 The completion. Classically, a uniform space is said

to be complete if each Cauchy filter on it has a limit, a definition

which makes sense only after a discussion of the topology associated

with a given uniformity, and of limits of filters on topological

spaces. Our approach here will be a different one. We first describe

(what will be called) the completion of any given space, and then use

that to define not only completeness for the original space, but

also separatedness, or the Hausdorff property.

4. 3.1 Notation : For any uso uX , the unique factorization

of nux X+ p2(X) through au X . yuX + p2(X) is denoted .

Yux : X + yuX. The unique (when available) factorization of nux

through 0: i,ux ' i E {1,2,3}, is denoted Yi,uX : X + yiux

4.3.2 Definition A uso uX is complete if Yux :X + yuX

is an epimorphism, and is i-com:elete, 1 E {1 ,2,3}, if Yi uX '

exists and is an epimorphism. :r.1oreover, uX is se:earated if Yux

- 84 -

is a monomorphism (notice that there is no reason to define

"i-separatedness", because yi uX is a monomorphism if and only '

if Yux is).

Remarks : We will define uniformities, u and ui,

on the objects yuX and yiuX respectively, and we will depart from

established conventions by writing simply yuX and yiuX for the ...

resulting uso's, the uniformities u and being understood.

We will then prove that Yux (y1 uX) is u.c. from uX to yuX '

(to yiuX), that yuX (yiuX) is a complete (!-complete) and

separated uso, and that is a reflection of uX into

the full subcategory of UE whose objects are the complete

(!-complete) separated uso's. This reflection, or sometimes,

somewhat inaccurately, the uso yuX (yiuX) itself, is what we will

call the completion (i-completion) of uX •

In order to describe the uniformity ... u in Set, we make two

preliminary definitions . For u € u and F € yuX, Au F is the . = =' subset {F' € yuX IF' n F n u + !l1l of yuX, and ~ is the family

= {Au FIF € yuX} of subsets of yuX .. We prove now that for any given

=' basis b for u, £8ul~ € b} is a basis for a uniformity on yuX.

=

To see that each ~ covers =

yuX, simply observe that F € Au F ='

for all U € u, F € yuX, by definition of Cauchy filter. Then,

given U and = V in = b, let w € b = be such that

observe that 8w<* au A~ follows easily from the following two = = =

lemmas 1) For any filter F on X and coverings U and Y

of X, if .U <V then AY.,,Fc:AV,F" Proof: GivenF!EAy,F' let

u € F' n F n u. Then let v € y_ contain u. Clearly, V € F' nFnV,

sc F' n F nv + !l1 so F' € AV F. ='

2) If

Proof: Let F € yuX, and suppose F E Au F" We claim that 0 0 ='

Au FcAv F . Let F' € AU F" Let u € F n F n u and 0 = =' =' ='

U' E F' n F n u. Then u n U' is in F, and hence is non-empty, = so U u U' is contained in some V E y. Such V is clearly in

F' n F n v, so F' n F n v t ¢, so 0 0

as desired. This

completes the proof of 2) above.

The uniformity on yuX generated by the basis {~ulu E u = =

is denoted u. It is clear that, if b is a basis for u, then .. {f!u lu E b is a basis for u.

= =

We now prove (still in Set) that Yux : uX ~ yuX is u.c.

-1 ) It will suffice to show that {yuX (A!L,F }FE yuX is in u, for

each u € u. So let u € u be arbitrary, and let v w € u be = =o =o'=o such that wo ~· Yo<* u • We claim that w '< { y uX -~ A'Q.

0 , F) } F € y uX =o =o

- in fact, for any w € ~~ and any X € w we claim that 0 =o 0 0

w c Yux-tAu y (x )), i.e. that X E w ... Y uX ( x) n Y uX ( x o) n !I.o + 0 · 0 =o' uX o 0

For, if v € v is such that st(x,W0 ) cV o' 0 =o such that s t ( x 0 , 1,0 ) c U 0 , (and by assumption

then u € Yux(x) n Yux<xo), as we now show. 0

w =o u {U 0}' which is in u since w =o refines

st(x,W0 ) U U0 = U , where we have used that

and u € u is 0 =o

there is such a u ), 0

Let W' be the cover =o it. Then st(x,W') = 0

x E U , which follows 0

from : x0 E W0 c V0 ~ x E V0 c st(x0 ,y0 ) c U0 , and also that

st(x,~0 ) c U0 , which is clear since V0 c U0 • Hence U0 E Yux(x).

Moreover, st(x0 ,~~) = st(x0 ,~0 ) U U0 = U0 , where we have used that

x E U , which follows from 0 0

which follows from we have proved that

st(x .W ) o '=o

Yux is

x0 E st(x 0 ,y0 ), and that st(x0 ,~) c U0 ,

c st(x ,V ). Hence U E y x<x ), and o=o o u o . u.c.

Next we prove that yuX is a complete and separated uniform

space, i.e. that yyuX : yuX ~ yyuX is an isomorphism, which we show

- 86 -

by producing a u.c. two-sided inverse for it. Appropriately, the inverse

of yyuX will be denoted L, for 11 limit".

So let ~ be a minimal Cauchy filter on yuX. We show that

is a basis for a Cauchy filter on uX.

is nonempty. To see this, observe that

Firstly,

E,, being

Cauchy, has an element of the form AU* F , with F E yuX, for each = !I

:q E u. Since is a basis for u, these elements of F =

clearly form a basis for a Cauchy filter on yuX, so by minimality

of E. they form a basis for F. Hence it suffices to show that

Yux- 1 <Au* F) is non-empty, for each ~ E u and F E yuX. That is, = ,

for such 'Q_ and F we must find x € X such that Yux(x) nFn 'Q.* ~ ¢.

\'Jell, F is Cauchy, so there is some u E F n u*. = Such U is of

the form st(x,U), for some x € X, so for this x, u E y x<x). u~

This proves that Yux- 1 (A) t ¢ for A € E· And 9inee Yux- 1 preserves

containment and (binary) intersections, it is clear that

·{Yux- 1 (A)}A€[ is (a basis for) a filter on X. To show it is a

Cauchy filter on uX, let U € u. Since F is Cauchy on yuX,

E n au t ¢:~

Yux -tAu F)

so let F € yuX be such that Au F € F. We claim that ='

is contained in some member of u**. Now = ='

Let x E U € F n U . 0 0

We claim that For,

-1 ) x € y uX (A!!, F ' given let V E u = be such that st(x,y) E F n 'Q_.

Let x 1 € U0 n st(x,V), where the latter is nonempty because F is

a filter. Then st(x1 ,ll) € ~*, x0 € U0 c st(x 1 ,!!), and

x E st(x,~) c st(x 1 ,[). Hence x E st(x0 ,'Q.*), as desired. And since

rr** is a basic uniform cover of uX, this proves that the filter on

X, constructed above from F, is Cauchy on uX. Let L(F) be the =

unique minimal Cauchy filter contained in the one just constructed.

- 87 ~

Next we show that L(y u.X(F )) = F , y. 0 0

and that y X(L(F )) = F. yu -o -o

always. For the first of these, it suffices to show that each member

of F contains a set of the form y uX- 1 ( Au F) ' with [ E u and 0 ='

F E yuX. Since F is minimal, each of its members is of the form 0

St (B ,Q) for some B E F and U E u (see 4.2.5). We show that 0

Yux- 1 (AU F ) c St(B,[). If -1( x E YuX AU F ) ' there is y_eu such

=' 0

that st(x,y_) E F0 U ~. Since

so x E st(x,V) c St(B,~).

F 0

=' 0

is a filter, B n st(x,Y) + ¢,

To see that it suffices to see that L(F ) =o

is a member of each A E ~0 • And since, for e~ch such A, there

is it suffices to observe that

which is clear since L(F ) =o is Cauchy,

We now use the above discussion in Set to sketch a proof of

the following theorem.

4.3.4 Theorem : Let u be a uniformity [i-uniformity,

i € {1,2,3}] on an object X. Then there is a uniformity

[i-uniforrnity] on the object yuX[yiuX], with respect to which

yuX{yiuX] is a complete [i-complete], separated uso [provided,

when i = 3, that the topos has AC].

Proof (sketch) : Let a : P2 (X) x P2 (yuX) be the map

corresponding to the formula

3(\/{e: .-o3(e: Ae: Ae: ))):P2 (X)xP(yuX)-+-O, 3 4 1.3 5 2.3.4 1.2.4 1.2.3

where the "bi-irnplication", ~,has the obvious meaning, and the

quantifications are along the three projections

' I

- 88 -

p2(X) xP(yuX) x (yuX) 2 x P(X) P2 (X)/ ( yiX)

I -------:> p2 (X) x P ( yuX) x yuX

/. I

J p2(X) x P(yuX) x (yuX)2

Furthermore, let a1 : P2(X) + P2(y1ux) correspond to the formula

P2 (X) x P( y i uX) + Q which is written just as the one given above.

We claim that, if b is a basis for u, then the monic ...

part of a o b dom(b) + P2(yuX) is a basis for a uniformity, u,

on yuX. And if u is an i-uniformity on X, then the monic part

of ai o b dom(b) + P2(yiuX) is a basis for a uniformity, ui' on

yiuX. This is straightforward use of 3.2.4 and the above discussion

in Set. It is easy to see that is an !-uniformity on yiuX,

simply because there exists, by construction, a morphism

dom(u) + dom(ui) compatible with u and ui.

To show that yuX[yiuX] is a complete [i-complete] and sepa-

rated uso, we construct an inverse, L [Li]'

as we did in Set above. First we define L'

map corresponding to the formula

to yyuX [yi uX], just ,yi : yyuX + p2(X) as the

3 2 ( ( E P 3 (X) ) 3 o ( a Y uX x P 3 (X ) x P (X) ) " c: o ( P * ( y uX ) x ( P (X ) ) 1 ) : y y uX x P (X ) + n

[and, quite similarly,

diagram on next page.

Let M • p2(X) + P2 (X) u·

L' : y.y.uX + P2 (X)] , obtained from the i l. l.

be the map corresponding to the formula

3 ( e ", ep ( x} o (( s t o ( u x P ( x) ) ) x P ( x) ) ) : p 2 ( x) x p ( x) + n 2.3 2.4 1

- 89 -

where the quantification is along the projection

p2(X) x dom(u) x P(X)2 2• 3 >P2(X) xP(X). Then let

L : yyuX ~ P2 (X) [Li : yiyiuX ~ P2(X)] be the composed map

a XxP3 (X)xP(X yu

yyuX x p3(X\) x1 P(X)

p3 (X) X P(X)

p4(X) x p3 {X) xP(X) lp*(yuxl x P(X) yyuX x P(X)

3

"{/

p4(X) X p3(X) P(X) 2

3 ( ---- " ----) 2

It is now straightforward (and lengthy) to verify that L [Li]

factors through auX : yuX ~ p2(X)

u is an i-uniformity, and when

[ai uX , i = 3 -

yiuX ~ P2(X), provided

the topos has AC] ,

and that this factorization is a two-sided inverse to

This completes our sketch of the proof of Theorem 4.3.4.

- 90 -

APPENDIX More on structured objects.

In order to avoid the set-theoretic difficulties often

encountered in attempts to construct too large categories, we will

in the following use a category ~ of (some, but possibly not all)

sets, instead of the category Set of all sets. We assume only

that ~(n,m) = Set(n,m) e lSI whenever n,m E lSI, and that

¢ E 1~1· If the set-theoretically inclined reader wishes to assume

that 1~1 is a set, then this will not necessitate any changes in

the following.

c be an =

Now let U : A -+ ~ be a given functor, and let

~-category (i.e. C(A,B) e 1~1 for all A,B e 1~1). Then an A-

object or, more precisely, an (A,U)-object in c,

: Cop = external sense, is a pair (C,G) where C e 1~1 , G

UoG = C(-,C). An A-homomorphism between two such pairs,

=

in the

-+A

(C,G) -+ (C',G'), is a pair (f,n) where f e ~(C,C') and n is

and

a natural transformation G-+ G' satisfying Un = C(-,f), i.e. for

B e l~l and g e C(B,C) we insist that U(nB)(g) = Q(B,f)(g) = = f 0 g B-+ C'. It is easy to see that we get a category of

A-objects in C, and we denote it C ®sA_. It has canonical =

functors (projections) to ~ and making the following a

pull-back diagram (where Y£ is the Yoneda embedding)

Cop Q. CB>s !l --------> a--

j YC 0op

Q ---------------> ~=

Cop U= = 91 U o - 11

- 91 -

There is one obvious criterion for judging whether this,

for given U : ~ + ·~ and for all ~-categories Q, is a good

definition, and that is to ask if the category of &-objects in

S itself is what it should be, namely A itself. =

In fact, we will answer the following precise question

For which (A,U) is there an equivalence of categories

£. + s ®sa making the following diagram commute ? =

(*)

First of all, the functors A + S ®s ~ making ( *) commu-ce Sop = Sop

correspond to functors E : A + ~ satisfying U= o E = Ys o U , =

i.e. satisfying for all ~-objects A and ~-objects n

U(E(A)(n)) = S(n,U(A)). That is, E(A)(n) must be an A-object

with "underlying set" equal to the nth power of the set U(A).

Moreover, with 1 being a one-element ~-object (take, for example,

1 = S(¢,¢) ), the ~-maps i: 1 + n must satisfy :

U(E(A)(i)) = ~(i,U(A)), i.e. for each x E ~(n,U(A)) and this

corresponds to an element (aj) j En E U(A)n U(E(A)) (1) (x) shall

equal S(i,U(A) )(x) = x o i E S( ,U(A)) corresponding to

ai E U(A). In other words, U(E(A)(i)) shall be the projection

of the product set U(A)n to its ith factor.

We give now conditions under which all this can be

accomplished.

We say A has ~-powers if for each A E IAI and n E lsi

- 92 -

there is an object An in A and a map p : n + A(An,A) with

the usual universal property : any q : n + ~(B,A) yields a

unique f E A(B,An) with p(i) of = q(i) for each i E n.

We say U preserves g-powers if, for each n (A ,p) as above,

(U (An), U o p) has the universal property of an nth power of U (A)

in g.

And we say A has = U-canonical ~-powers if, for any A

and n, (An,p) can be chosen so that U(A~) = S(n,U(A)) and

U(p(i)(g) = g(i) for i En and g E S(n,U(A)).

This condition is satisfied if U creates ~-powers, and

only if U preserves ~-powers.

Notice also that, if ! has S-powers, then any choice of

them yields a unique functor E: A Sop

defined on objects + A= = = ' by E(A) (n) n and on rnorphisrns by making of the universal = A use

property of powers in the obvious way.

We can now answer part of our earlier question

A 1. Proposition : Let U : & + S be a functor such that

~ has U-canonical S-powers. Then E Sop

! + A can be chosen

so Sop

that U= o E = Y~ o U. Each such choice of E yields a functor

A + ~ 01?._ b. making ( *) commute.

Proof Straightforward.

To state conditions under which the above produced functor

! + s.. ®sa is an equivalence of categories, we need another defi­

nition :

~ 93 -

We say U : A ~ S reflects = = S-powers if, given any map q

from some n € Is I into some ~(B,A) such that (U(B) ,u o q) th has the universal property of an n power of U(A) in S, (B,q)

must already have had the universal property of an nth power of

A in A.

A 2. Proposition : Suppose that U : A ~ S is a functor = =

reflecting ~-powers, and that ~ has U-canonical powers. Then

there is an equivalence of categories ~~§..@~A making (*)

commute.

Proof The functor A ~ S ®sA described in A 1 takes

an object A in ~ to (U(A),E(A)) € j§_@_§ AI' where E(A) is

a functor Sop ~ A such that E(A) (n) n and a morphism = = A ,

f € ~(A,B) to the morphism

(U(f),E(f)) € (§.@SA)((U(A),E(A)),(U(B),E(B))), =

where E(f) is the natural transformation E(A) ~ E(B) described

by the formula E{f)n = fn : An ~ Bn. If f + g : A ~ B then

so E(f) + E(g) : E(A) ~ E(B). Hence

A ~ S G> SA is a faithful functor. It is also full, for given a =

morphism (h,n) ~ (U(A),E(A)) + {U(B),E(B)), n1 : E(A)(1) ~ E{B)(1)

corresponds to a morphism f : A ~ B, satisfying U(f) = h and

E(f) = n. Hence it remains only to show that each {n,G) € 1~0s~l

is isomorphic to some (U{A),E(A). This can be done with A= G(1).

Then U(A) = U{G(1)) = ~(1,n). For each m € 1~1 let

em : m ~ ~(1,m) be the obvious one-to-one- and onto map. Then let

n : G ~ E(A) be the natural transformation defined as follows

For m € j§_l let nm : G{m) ~ E{A)(m) = Am be the unique map

resulting from the family of a-morphisms

- 94 -

G(em(i)) : G(m) + G(1) =A (i E m).

Now U(G(em(i))) = ~(em(i),n) = - 0 em(i): S(m,n)+§.(1,n) = U(A),

for all i E m. These form a family of projections Ly virtue of

which ~(m,n) is U(A)m. And since U reflects §.-powers this

means that (G(m), (G(em(i)) )i Em) has the universal property of

and hence n - is an isomorphism.

Hence en : (n,G) + (U(G(1)),E(G(1))) is an isomorphism in

~ ® 8 ~, as desired.

This completes the proof.

Next, given a functor U : A + s, and an - - §.-category

let C ®~:A be the full subcategory of C ® 8 A with objects

those (C ,G) E I Q. 0 8 ~) where G preserves S-powers. Again we =

seek conditions on U that will ensure that Q. ®§,A is a good

candidate for a category of ~-objects in Q, and the criterion

is as before.: When is s ? = More specifically,

since E(A) : ~op + A preserves S-powers for all A E 1~1

When is Af"Y"\..,J.(U(A) ,E(A)) A + §. ® SA an equivalence of =

categories ?

As in the proof of Prop. A 2 we need only show that, under

some suitable conditions on u, each (n,G) E Is ®sAl is iso-

morphic to some

A= G(1). Then

power of 1 in

=

(U(A),E(A)). Again, for given (n,G), we set

G(m) = G(1)m =Am, since any set m is the mth

s 0 P, and G preserves mth powers. So in this

case n : G + E(A) is an isomorphism without any assumption that

U reflects §.-powers. Hence we have the following :

... 95 ..

A 3· Proposition : Let U : A ~ S be a functor such = =

that ~ has U-canonical ~-powers. Then there is an equivalence

of categories making the following diagram commute :

A -------------> S I)(\ A

-~~/og~

We note here that related statements were made by John Isbell

in [Is 2 ]. We note also that if U :A~~ creates ~-powers

then the above described equivalence A ~ S ®sA · is an isomorphism =

of categories.

~ 96 -

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