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 -
BIBLIOGRAPHY
[Bo] Bourbaki, N., General Topology, Part I, Hermann, Paris,
and Addison-Wesley, 1966.
[CK] Chang, c.c. and Keisler, H.J., Model Theory, North-Holland,
Amsterdam-London, 1973.
[De] Demazure, M., Topologie et fais~eaux, Expose IV i Schemas en Groupes I, Lecture Notes in Mathematics 151, Springer-Verlag, 1970.
[EH] Eckmann, B. and Hilton, P., Group-like structures in gene-
[En] Engenes, H.,
ral categories, Math.Ann. 145, 227-255, 1962.
Equality in Topoi,presented at the 80th Annual Meeting of the AMS, San Francisco 1974. Abstract appeared in the Notices of the AMS, Vol. 21, no. 1, p. A-80.
[FN] Fenstad,J.E.and Nyberg, A., Standard and Non-standard Methods in Uniform Topology, Logic Colloquium '69, North-Holland, 1970.
[Fi] Fitting, M.C., Intuitionistic Logic, Model Theory and Forcing,
F~eyd, P.,
-"-
North-Holland, Amsterdam-London, 1969.
Aspects of Topoi, Bull-Austral. Math. Soc.
1, 1-76, 1972.
On Functorializing Model Theory, an unpublished pamphlet, 1974.
[GJ] Gilman, L. and Jerison, M., Rings of Continuous Functions, van Nostrand, 1960.
[Go] Godement, R., Topologie Algebrique et Theorie des Faisceaux, Hermann, Paris, 1958.
[He]
[Is 1 2
- 97 -
Heyting, A., Intuitionism, North-Holland, Amsterdam, 1956.
Isbell, J.R., Uniform Spaces, Mathematical Surveys, no.l2, of the A.M.S., Providence, 1964.
-"- Top and its Adjoint Relatives, Proc. of the Kanpur Topology Conference, Academia, Prague,
1971.
[Jo] Johnstone, P.T., Internal Category Theory, Dissertation,
[Ko]
[KLM]
[KW]
[La 1 1
Kock, A.,
Univ. of Cambridge, 1974.
Stron5 Functors and Monoidal Monads, Various Publications Series no. 11, Aarhus Univer- · sity's Mathematics Institute, Aarhus, 1970.
Kock, A., Lecouturier, P. and Mikkelsen, C.J., Some Topestheoretic Concepts of Finiteness, Preprint
Serier 1973/74, no. 29, Aarhus University's Mathematics Institute, Aarhus, 1974.
Kock, A. and Wraith, G.C., Elementary Toposes, Lecture Notes Series no. 30,Aarhus Universjty's , Mathematics Institute, Aarhus, 1971.
Lawvere, F.W., Quantifiers and Sheaves, Proc. of the Intern.
-"-
-"-
Cong. of Math., Nice, 1970.
Introduction to Toposes 1 Algebraic Geometry and Logic, Lecture Notes in Mathematics
no. 274, Springer-Verlag, 1972.
Equality in Hyperdoctrines and the Comprehension Schema as an Adjoint Functor, in Applications of Categorical Algebra, Proe.
Symp. Pure Math. no. 17, A.M.S., 1970.
[Lu] Luxemburg, W.A.J., A General Theory of Monads, in Applica-tions of Modal Theory to Algebra, Analysis and Probability, Holt, Rinehart and Winston,
1969.
- 98 -
[MH] Machover, M. and Hirschfeld, J., Lectures on Non-standard
[MeL] MacLane, s.,
Analysis, Lecture Notes in Mathematics no.94, Springer-Verlag, 1969.
Categories for the Working Math~matician, Graduate Texts in Mathematics no. 5, Springer-Verlag, 1971.
[Mil Mikkelsen, C.J., Doctorial dissertation, to app2ar at
[Mu] Mulvey, c.,
Aarhus University, Aarhus, Denmark.
Intuitionistic Algebra and Representations of Rings, in Memoirs of the A.M.S., no.l48,
3-57, 1974.
[RS] Rasiowa, H. and Sikorski, R., The Mathematics of Metamathe-
[Sc] Scott, D.,
matics, Monografie Matematyczne, Warsaw,l963.
Extending the Topological Interpretation to Intuitionistic Analysis, in Logic and Foundations of Mathematics, Wolters-Noordhoff Publishing, Groningen, 1968.
[Sh] Shoenfield, J.R., Mathematical Logic, Addison-Wesley, 1967.
[St]
[Til
[Tu]
[Vol]
Stout, L.N.,
Tierney, M.,
Tukey, J.W.,
Volger, H.,
General Topology in an Elementary Topo~, Dissertation, Univ. of Illinois at UrbanaChampaign, 1974.
Axiomatic Sheaf Theory, Lectures delivered at the C.I.M.E. Conference on Category Theory and Commutative Algebra, Varenna, 1971.
Convergence and Uniformity in Topology, Princeton, 1940.
Ultrafilters, Ultrapowers and Finiteness in a Topos, Preprint, April 1974.
~
[Vop 1 ] Vopenka, P.,
-"-
- 99 -
The Independence of the Continuum Hypothesis,
Comment. Math. Univ. Carolinae 2 {suppl.),
Praha, 1964.
The Limits of Sheaves and Applications on Constructions of Models, Bul1.Acad.Polon.Sci.
13, 189-192, 1965.