Download Exponential laws for topological categories, groupoids

C AHIERS DE
TOPOLOGIE ET GÉOMÉTRIE DIFFÉRENTIELLE
CATÉGORIQUES
RONALD B ROWN
P ETER N ICKOLAS
Exponential laws for topological categories, groupoids
and groups, and mapping spaces of colimits
Cahiers de topologie et géométrie différentielle catégoriques, tome
20, no 2 (1979), p. 179-198
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Vol. XX -2 (1979)
CAHIERS DE TOPOLOGIE
ET GEOMETRIE DIFFERENTIELLE
EXPONENTIAL LAWS FOR TOPOLOGICAL CATEGORIES, GROUPOIDS
AND GROUPS, AND MAPPING SPACES OF COLIMITS
by
Ronald BROWN and Peter NICKOLAS
INTRODUCTION.
One aim of this paper is
give a topological version of the well
known exponential law for categories, which states that for categories A
and B there is a functor category ( A, B) such that for any C, D and E ,
there is a natural isomorphism of categories
to
topological version we use as underlying category of spaces the
category K of topological spaces X and k-continuous maps, by which we
For
our
mean
tinuous maps
the
f:
functions
-&#x3E;
F such that
f a:
C - Y is continuous for all
con-
C - X of compact Hausdorff spaces C into We give
Y) of k-maps X - Y a topology with a subbase of sets
a :
set K (X,
W(a, U )
X
o
f such that f a (C) C U,
Y . 7hen K has an exponential
of those functions
above and U open in
with
a :
C - X
as
law
generalizing to
non-Hausdorff spaces the exponential law of [4] , (The corresponding category of k-spaces and continuous maps is well known - see for example
[10] - but gives less precise results than ours.)
Our
objects
must
in K,
prove
version of
topological
and this
(D, E)
prove that the
a
brings
k-category
structure
it is convenient
to
(1)
is for
us to our
maps of
k-categories,
second
if D and E
(D, E)
are
have these maps defined
that
expository point are.
This
k-continuous,
as
is, category
that
we
means we must
and
to
do this
induced maps. We there-
Ehresmann [2] involving the double k-category 0 E of commuting squares of E , with its horizontal and vertical
compositions 00 and B . M. and MI"’e Ehresmann have pointed out to us
that our Theorem 2 is very close to a special case of Proposition 4.10
fore
use
methods of A. and C.
179
L2j,
of
in
the
(or groupoid ) objects
cartesian closed category is itself cartesian closed. However
a
that
which proves that the category of category
of this
exposition
simplicity and utility
our
special
case
and of its
applications
we
hope
will show
of these results and methods.
exponential isomorphism ( 1 ) is also valid for k-groupoids C,
D, E and hence also for k-groups (though (D, E) is of course a groupoid
and not a group). Our main application of the exponential law is to the
space M (D, E) of morphisms of k-groups with the compact-open topology.
We prove that if D is a colimit I£m Dx of k-groups, then the natural bijection
The
is
if D is
k-homeomorphism. Also,
k-group and E is any k-group,
is
a
a
k-homeomorphism.
ducts and free
k-groups.
In Section 3
the
map 4J
As
of (3) is
we
a
a
consequence
we
are
prove that if D is
homeomorphism.
previous sections,
U D given in [5].
For further references
and
k-groupoid, U D
groupoids
categories
bibliography of [6].
we
is its universal
then the natural map
These results
than that of
of
a
the
on
free
k-pro-
in Section 2.
a
Hausdorff
kw -groupoid,
then
The
and makes
to
obtain results
theory
approach here is more direct
use of the explicit construction
and
refer the reader
applications
to
of
topological
the 80 papers listed in
the
1. AN EXPONENTIAL LAW FOR k-CATEGORlES.
object in this section is to set up an exponential law for topological categories. To do this we need to start with a cartesian closed
category of topological spaces. A number of such categories are available,
Our
but for
purposes it is convenient to use a modification of the k-continuous maps of [4] to allow for non-Hausdorff spaces.
our
180
say
Let X and Y be topological spaces and f : X- Y a function. "We
is k-continuous if for all compact Hausdorff spaces C and conti-
f
nuous
maps
composite f a : C - Y
C - X the
a :
spaces and functions form
K(X, Y )
is the
maps X -
of k-continuous
set
we
denote
by K,
so
that
Y . The usual product and
give the product and sum in K . Although the analogous
k-spaces and continuous maps has been considered by a number
of spaces
sum
theory of
of writers,
ces
category which
a
is continuous. These
to
the reader is warned that the literature contains many referen-
and «k-continuous functions* defined in
k-spaces »
senses
dif-
ferent from those used here.
There is
k ( X)
k: K - K where, for X a topological space,
topology with respect to all continuous maps a : C - X
Hausdorff. Then k (X) is an identification space of a
functor
a
has the final
for C compact and
locally compact space
ing for each non-open
aA :
f:
CA , X
set
such that
the
A of k(X)
al (A)
is
now
define
sub-base of open
a
sets
continuous map from
a
sets
CA
obtained
compact Hausdorff
a
open in
not
compact-open
the
of spaces
sum
X - Y is k-continuous if and only if
We
a
namely,
-
CA
CA
by choosand
a
map
( cf. [10] ). A function
f: k ( X ) - Y
is continuous.
topology on K (X , Y) by taking
W(a, U)
for U open in Y and
compact Hausdorff space C
to X ;
here
a:
as
C - X
W(a, U)
maps f: X - Y such that f a( C ) C U . (This
[10].) The exponential law stated in [4] for
consists of the k-continuous
topology
is considered in
Hausdorff spaces and extended
tion
5.6,
can now
be stated for all spaces
(1.1 ) Exponential
is
well-defined,
is
a
The function
are
k-continuous,
non-Hausdorff spaces in
to some
law
in K .
bijection
The
and is
a
space K(X, Y)
as
follows:
exponential correspondence
homeomorphism.
is functorial in the
sense
that if
f *:K( X, Y ) -K (W, Y) is
g* : K ( X , Y ) -K ( X , Z ) is k-con-
then the induced function
continuous and the induced function
[3], Sec-
484
tinuous.
is continuous
continuous in the
are
g*
Further, if g
also is g* . (The
so
circumstances
given
are
proofs
easy; that g* is al-
ways k-continuous follows from the exponential law (1.1) as in
is also easy to prove that if g: Y - Z is a homeomorphism into,
is g*:
f* ,
that
[4]. ) It
so
also
K(X, Y &#x3E; - K(X, Z).
proved in [4] is that if f : W- X is a k-identificaf : k( W ) - X is an identification map), then
Another result
tion map (that
is
a
It
then the
and U
c
on
that,
sets
W(a, U)
for
a
form
is
injective
and its inverse is k-
if
a:
11
is
a
sub-base for the open
of
Y,
C - X continuous, C compact Hausdorff
sub-base for the open
a
sets
sets
of
K (X, Y) .
From this
we
standard way :
natural map
homeomorphism.
Suppose
further that Y X Z is
and Y X Z has its
( 1.3 ) The
a
is, f *
its domain).
be shown
( 1 ,2 ) The
is
into (that
can
’U ,
deduce in
a
if
k-homeomorphism
continuous
is
is,
topology
as a
a
pull-back
subspace
as
in the
of Y X Z . Then
diagram
we
have:
natural map
homeomorphism
where the latter space is the
pull-back
of g* and h* .
objective now might be to extend the exponential law (1.1)
to the cases of topological categories and groupoids. Since the morphisms
are only to be k-continuous rather than continuous, however it seems reasonable to deal instead with k-categories and k-groupoids, in which the
Our
182
structure
To fix the
of
a set
C of
k-continuous.
are
maps
notation,
arrows
and
the
of
composition
pairs ( p , q)
C
k-category
and a’, a, u
spaces
if in addition it is
A
a
respectively),
C -
the usual axioms for
is such
a
set
C of
groupoid
category. It is usual
a
arrows.
category C in which C and
and m are k-continuous.
morphism f:
with functions
objects, together
map)
C , where C X C is the subset of C X C,
a p = a’ q , and where m ( p , q ) is written p q ;
X
confuse the category with the
A
(small) category consists
and
satisfy
must
of
a
final and initial maps,
unit
such that
these functions
to
m :
O b (C)
a set
a’, a : C- Ob(C) (the
u:Ob (C)-&#x3E;C (the
first recall that
we
Further, C
p-1
and the inverse map p-
C - D
of k-categories
is
Ob (C)
a
are
k-groupoid
is k-continuous.
consists of
a
pair
of k-con-
tinuous functions
which
commute
is written
with the category
M ( C, D).
This
set can
structure.
The
set
be identified with
a
of these
subset
morphisms
of K ( C, D),,
the space of k-continuous functions between the spaces of arrows, and
give M(C, D)
We
object
now
the compact-open
wish
of
k-category ( C, D) having M ( C, D)
space and with k-continuous natural transformations
commuting
of
topology.
to construct a
this purpose it is convenient
arrows
squares in D
to
to
[2] defining
subspace of D4
follow
be the
of D such that p q,
rs
are
defined and
arrow
space of
two
183
as arrows.
as
For
first the space 0 D
of
quadruples
equal. This
k-categories with object space D .
the horizontal category In D , has initial and final maps
the
we
space is
One of
these,
respectively,
and
unit map
composition
The vertical category
D has initial and final maps
B
unit map
and vertical
composition
let X be the domain of the composition Eel , with its topology
subspace of DDxDD. Then X is the arrow space of a k-category
Now
as a
with
object
space D X D and with
composition
composition E3: X - D D is then a morphism of k-categories.
effect, we are expressing the statement that o D is a double k-category.
The vertical
In
PROPOSITION
1. For
with
o
ture
maps induced
bject
k-categories C,
space M (C,
by
D) and
arrow
D there is
space M (
ture
13 D
The
on
initial,
m
k-category (C,D)
D), and with
struc-
the vertical category structure on 0 D . I f D is a
C, D) with the induced inverse map. Finally, if D
groupoid, so also is (
is a topological category or topological groupoid,
P ROOF.
C,
a
so
also is
( C, D).
final and unit maps of the vertical category
struc-
D D are morphisms between the horizontal category 03D and
184
D
itself. They therefore induce k-continuous maps between
M ( C, D) , and these we take as
( C , D) . The composition in ( C , D) is
and
the
M ( C, m D)
initial, final and unit
maps of
the k-continuous map
homeomorphism given by (1.3). The axioms for a k-category are easily verified. Similarly, if D is a k-groupoid the inverse map
on B D is a morphism for (11 D , and induces a k-continuous map
where
is the
a
making ( C , D)
If the
( C, D) .
a
k-groupoid.
functions of D
structure
This proves
are
continuous
so
also
are
those of
Proposition 1.
T H EO R EM 2
( T he exponential law for k-categories). I f C , D and E,
k-categories,
there is
a
natural
isomorphism of k-categories
which is continuous and has k-continuous inverse.
logical
category,
then O -1
are
Further, if E
is
a
topo-
is continuous.
complete proof of the theorem is quite lengthy, and we therefore omit a number of straightforward details.
For k-categories A, B an application of (1.2) shows that we may
regard the space (A , B) as a subspace of K ( A , B)4. In particular, we
can regard ( C, (D, E)) as a subspace of K(C, K(D, E)4)4 and hence
P ROOF. A
of K(C, K(D, E))16 .
To define
C X D -
mE,
so
9,
,
note
that
that for each
f of ( C X D, E ) is
(c,d):(x,y)-(w,z)
morphism :
an arrow
a
arrow
of C X D
we
may write
Note that q,
of
f
with
morphisms C X D - E since they are the composites
d’D: m E , E respectively. We require that 0 f be a mor-
r are
dD
185
phism C - D( D, E) ,
so
for any
given
commuting square in (D , E ) where
fined, for d:y-z an arrow of D , by
a
Certainly
each
ki(c)(d)
c : x - w
the
of C define
ki ( c )
is in DE since q,
in
r are
M ( D , D E)
morphisms
de-
and
are
f
maps
into 0 E . To show that
(say) k2 ( c)E M (D, D E) we note that the four
components of k2 ( c) are k-continuous ( since q is k-continuous) and so
k2 (c) is k-continuous. Also, if d d, d2 in D , then we have
=
and
so
As for the verifications that
a s
required.
to
lVl (D, ill E ),
for
that for
k3 (c)
k1 (c), k4 ( c), though
Again,
the
proofs
that
a
is similar
little
k1 (c), k3 ( c), k4(c) belong
to that for k2(c) , while those
different,
are
Of E(C, (D, E)) ;
equally straightforward.
and that
(C X D, E) , are routine though rather lengthy, and we
To check continuity (rather than just k-continuity) of 0 ,
in
omit details.
note
that
by
(1.2 ) the map
of
(1.4) embeds M(CXD, coE)
as
a
186
subspace
of K(CXD, E)4 ;
then
E) ,
ma pinto K ( c, K ( D, E )) 16 , can be written
as a
where the four components of each
on
C
are
the
( u d’)* , ( u d)*
D . Since each
or
of the exponen-
composites
of (1.1) and the maps on K(C, K(D, E))
tial map 0
ua
Bi
O1 x 02 x 03 x 04
as
is
by u a’ or
continuity
induced
continuous, the
of 0 follows.
To construct 4J =
S -1 ,
first define
TT, o , y : D E - E
to
be the maps
which take
respectively.
Thus
and
7T
is k-continuous. Now
c : x - w
in
( D, E),
It is
C,
of
write
given
g (c)
and define for
straightforward
Q
to
are
the initial and final maps of m E and y
an
arrow g
as a
commuting
is
a
( C, (D, E))
and
an arrow
in C
check that (D is the inverse of 0 . The
topological
7r, Q,
an arrow
square
( c, d ) : (x, y) - ( w, z)
of 4Y follows from the continuity of
and if E
of
O-1
X
D :
k-continuity
k-continuity of y ;
continuous, and hence so
and the
category, then y is
is 4).
COROLL ARY 1.
tion
For any
k-categories C, D,
a
natural
bijec-
M(C, (D, E))- M(D, (C, E)).
From Theorem 2 and the facts about
we
E there is
groupoids
in
Proposition 1,
deduce:
COROLLARY 2
k-groupoids,
(The exponential law for k-groupoids). 1 f C, D , E
there is
a
natural
are
isomorphism of k-groupoids
which is continuous and has k-continuous inverse. Further,
187
if
E is
a
to-
pological groupoid,
then 9 is
a
homeomorphism.
particular, we obtain an exponential law for the case when C ,
D, E are topological groups, although of course (D, E) remains a topological groupoid and not a group.
In
isomorphism 0 of Theorem 2 and its kcontinuity could be proved a little more easily by first constructing a natural bijection Jl (C X D, E)- M ( C, ( D, E)) and applying a standard argument using associativity of the product. Such an argument does not easily give continuity of 0 or (for the case when E is a topological groupREMARK.
oid) of
The existence of the
0-1 .
2. APPLICATIONS TO COLIMITS.
application is to determine up to k-homeornorphism
the space M (D, E) , where E is a k-group and D is a colimit of k-groups.
The use of k-groupoids rather than just k-groups, however, is not orrly
required by our method of proof, but has the advantage of easily giving
results on free k-groups. By generalising still further to k-categories i+e
obtain results on k-monoids as well as k-groups.
Our main kind of
The existence of
arbicrary
colimits of
k-categories, k-proupoids
that for the topological
proved
k-groups
case of [6]. However, the results of this section do not require knowledge of the co-completeness of these categories.
may be
or
in
a manner
similar
to
l’e first need:
functor Ob from the category of k-categories, or
category of k-groupoids, to the category K preserves limits and
PROPOSITION 3.
from
the
The
colimits.
PROOF. There
are
functors
P, T , respectively
left and
right adjoints
to
Ob - namely:
point-like functor P : X - X (where the topological groupoid X
object space X and only identity arrows ),
and the tree functor T : X - X X X ( where the topological groupoid
the
has
188
X X X has object
space X ,
arrow
d’,
space X X X and
are
the
projec-
tions).
Hence Ob preserves limits and colimits.
THEOREM 4. Let
D, E be k-categories and
lim DB of a diagram of k-categories DX.
suppose that D is
a
colimit
Then the natural map
x
isomorphism of categories, is continuous and has k-continuous
verse. The analogous ’result also holds for colimits of k-groupoids.
is
an
P ROOF. The k-continuous
continuous
morphisms
To prove that O
is
morphisms DB - D given by
(D, E) - (DB, E)
and hence
of
categories
an
isomorphism
Corollary 1 to Theorem 2,
k-category C natural bijections
verse we use
for any
a
in-
the colimit induce
continuous
morphism
with k-continuous in-
and standard arguments,
to
obtain
The result follows from the Yoneda Lemma.
Theorem 4 and
Proposition 3
COROLL ARY 1. Let the
k-category
now
yield:
D be
a
colimit
lim DB of diagram
a
A
o f k-categories DÀ.
is
a
Then
for any k-category
bijection with k-continuous
for colimits of k-groupoids.
continuous
also holds
For
our next
topological
category
Let D
be
a
E the natural map
inverse. The
corollary, we rephrase the definition
[6] for the case of k-categories.
k-category,
and
or:
189
analogous
result
of the universal
Ob(D)- Y a k-continuous
func-
tion. The universal
category of
k-category Va ( D) is
k-categories as in the diagram
cation map, then
a
pushout
Y is
a
in the
k-identifi-
injection whose inverse is k-continuous on its domain,
the set of morphisms f: D - E for which Ob (f) factors k-con-
tinously through
a.
PROOF. Since o is
a
be the
continuous
which is
is
to
1 f D is a k-category, and a: Ob(D)for any k-category E the induced map
COROLLARY 2.
;
defined
continuous
a
k-identification map,
with k-continuous inverse. Let V be the
injection
pull-
back of o * and
projection V- M (D, E)
is also
injection with kcontinuous inverse. Since Q is the composite of this projection with the
natural map b : M(Uo(D), E)- V , the result follows from Corollary 1.
Then the
Bt1hen the above space Y is
called the universal k-monoid
COROLLARY 3.
induced
by
If
D is
morphism
continuous
singleton,
and is denoted
k-category
a
the universal
of D ,
a
a
and E
D -
a
U(D)
U a (D)
is
a
k-monoid,
by U ( D ).
k-monoid, then the
is
a
continuous
map
bijection
with k-continuous inverse.
This follows
immediately
COROLLARY 4. Let X be
a
from
Corollary 2.
space and i: X -
190
F+(X)
the canonical map
from
X
the
to
free
k-monoid
on
X . Then
for
any k-monoid E the induced
map
is
a
continuous
with k-continuous inverse.
bijection
point-like category on X (cf. the proof of Proposition 3), and 2 denotes the topological category with two objects 0,
1 , one non-identity arrow 0 - 1 , and the discrete topologies on objects
and arrows, then the composite
If X denotes the
PROOF.
injection x - (x, 0 )
k-monoid i : X - F+ (X). So
of the
and the universal
the result follows
morphism defines
from Corollary 3.
the free
k-groupoid version of Corollary 1 to deduce some
on free k-groups and free k-products of k-groups. These results
sharpened for kw-groups in Section 3.
We
results
will be
by
G is
If
COROLLARY 5.
induced
the
use
now
a
the universal
k-groupoid
morphism
and K is
G -
U ( G)
a
is
k-group,
a
then the map
continuous
bijection
with k-continuous invers e.
This follows from
of
k-groupoids
If
is
a
as
is
B
GÀ
of the
COROLLARY 6.
If
a
family
of
in the category
the universal group
union II
since the colimit
k-categories
k-groupoid (the analogous topological result is in [6]).
{GB }BE A
coproduct G = * GÀ
Corollary 3
U(G) of
the
k-groups, their free k-product is the
of k-groups. This k-group can be taken
k-groupoid ë,
a
continuous
which is the
disjoint
k-groups Gx .
{GB IXCA is family of k-groups,
a
K the natural map
is
as
bijection
with k-continuous inverse.
191
then
for
any
k-group
PROOF. This follows from
is
easily proved
to
Suppose
K
It
can
such that i
since the natural map
homeomorphism.
a
that X is
pointed space. The Graev free k-group
a k-group FGK(X) and a pointed map i : X - FGK(X)
is universal for pointed maps in K from X to k-groups.
now
on X consists of
in
be
Corollary 5
be shown that if
e
a
is the base
point
of
X, i
may be taken
as
the
composite of the injection j: xI (x, e) of X into the tree topological
groupoid X X X with the universal morphism X X X - U(X X X) (cf. [6]).
COROLLARY 7. For any space
X and
k-group
K the map
of pointed maps X,K, with i * induced by i :
continuous bijection with k-continuous inverse.
into the space
is
a
P ROOF. This follows from
is
a
continuous
Y
where
is
a
from
Y ( f):
bijection
once we
have
with k-continuous inverse.
proved
F G K ( X),
that
Now j *
has inverse
morphism (x, y)- f (x) f (y)-1 . So j*
clearly continuous. The k-continuity of Vf comes
composite
XXX- K is the
bijection, and is
regarding it as the
where m *
Corollary 5
X -
is induced
by
the map
abelian
m ’: (a, b)I a b-1
of K X K into K .
pointed space X consists of
an abelian k-group A GK ( X) and pointed m ap j : X - AGK(X) in K such
that j is universal for pointed maps in K from X to abelian k-groups. The
map p: FGK(X)- AGK(X) is a k-identification map.
The Graev
free
k-group
on a
X , Y be paracompact, Hausdorff pointed spaces such
that AGK(X ), AGK(Y) are isomorphic k-groups. Then, for any abelian
group 17 and n &#x3E; 0 , the Cech reduced cohomology groups
COROLLARY 8.
Let
192
are
isomorphic.
K be
PROOF. Let
abelian
an
k-group.
The induced map
k-homeomorphism (since p is a k-identification map). So by Corollary 7 a k-isomorphism AGK( X) - AGK(Y) induces a k-isomorphism
is
a
bijection of pointed homotopy classes [X, K] -[Y, K] .
Let K be the Filenberg-Mac Lane k-group K(TT, n). By [8] (cf. also
[1] chapter 6) the abelian group [ X, K] is isomorphic to Hn(X; 7T ) ,
and hence
a
and the result follows.
E XAMPL E.
and C is
is
Let X be the «Cech circle &#x3E;&#x3E; - that is, X
an arc
isomorphic
to
in
R2B( A UB) joining (0, 0 )
to
AuBuC,
(1, 0 ) .
Z , and it follows that AGK (X) is
AGK([0 ,1]) . Similarly AGK(S1)
=
not
Then
where
H1 (X;
k-isomorphic
Z)
to
k-isomorphic to AGK ([0,1]).
However this method does not distinguish between AGK(X) and AGK(S1)
and we do not know if these are k-isomorphic.
is
not
3. THE UNiVERSAL TOPOLOGICAL GROUP OF A HAUSDORFF
k(ù -G ROU P.
In this section
we
consider
only topological groupoids and topolo-
gical groups. Recall also that a kw -space X is one which has the weak
topology with respect to some increasing sequence {Xn} I of compact subspaces with union X . A Hausdorff
k-continuous
map X -
k,,-space X
has the property that any
Y is continuous.
topological groupoid which as a topological space is a k,)-space
is called a kw -groupoid. It is proved in [5] that if G is a Hausdorf f kwgroupoid, then its universal topological group U(G) is a Hausdorff kw group. We shall use the explicit description of the topology of U (G) given in [5] to prove :
A
193
G be
THEOREM 5. Let
Hausdorff k(ù-groupoid
a
and K
topological
a
group. Then the map
induced
by
the universal
i * is
PROOF. It is clear that
i : G - U ( G) is
morphism
a
homeomorphism.
bijection and, as an induced
tinuous, and we need only prove the continuity of ( i *)-1 .
Let f: U(G)- K be a morphism and let
W(A, U)
pact in U ( G)
be
Let
a
a
sub-basic open
neighbourhood
of
f,
so
map, is
that A is
con-
com-
and U is open in K .
Since G is
a
kw -groupoid,
{
Gn}
it has the weak
of compact
topology
subspaces. For
with respect
to
each
pair of integers m , n &#x3E; 0 there is a continuous map p : (Gn)m-U(G) sending
each m-tuple of elements of Gn to its reduced form in U ( G) ([5], page
432). The methods of [5] also show that the compact set A is contained
an
increasing
in
some
sequence
p ( Gn)m,
and the definition of
multiplication
in
U (G)
is such
that
Setting V
f -1 (U ),
=
put
,
so
that N is
Hausdorff,
an
each
open
x
set.
is compact
Since
in C has
a
compact
neighbourhood
contained in N . Since C is closed and hence compact in
finite
is
a
we
have
set
Ux m
g ( Cxj)
there
and
F in C such that
Since U is open and each
...,
(Gn)m
is compact, there
in K such that
194
are
open
sets
Ux1,...
Let W
j = 1,
... , m , and
take
so we
This
F . Then
x E
and
we
prove that
have
completes
the
proof
From Theorem 5
of
gE W ,
for
hc W, and set h *= ( i *)-1 ( h) : we must prove h *( A)C U . If
y E A , then y = p ( c) for some c E C , and then CE C’ for an x E F,
Let
we
sets W ( Cxj, Uxj),
be the intersection of the sub-basic open
kw -groups,
and
on
we can
of
( i*) -1 .
deduce results
topological
on
countable free
groups
applications, namely
products
k,)-spaces [7],
on
Theorem 4. Rather than detail
to
different line of
a
continuity
Graev free
in Corollaries 6 and 7
pursue
of the
these,
abelian
to
we
as
shall
topological
groups.
{GB}JEA
COROLL ARY 1.
If
dorff kw-groups,
and G = E
is
a
countable collection
o f abelian, Hauscategory of topological
GÀ is their sum in the
a Hausdorff
kw-group, and for any
abelian groups, then G is
logical group K the natural map
is
a
topological isomorphism.
PROOF.
Clearly G
group and
k,)-spaces
is
a
abelian topo-
so
is
a
is the
Hausdorff
is compact
quotient
But any
kw-space.
covering,
homeomorphism, by
G’= * GB
of
and
the footnote
so
to
of
a
kw-groups is naturally isomorphic
to
the
sub-
the induced map
countable
195
commutator
map of Hausdorff
quotient
Proposition 3.5
The dual group
COROLLARY 2.
its
by
sum
of
[4].
of abelian Hausdorff
pro duc t of their
duals.
This is the case K
for countable
also
of
sums
=
S’
of
Corollary 1.
locally compact
groups
are
Similar results
due
to
to
this
Kaplan [9] (see
[11] ).
Our
are two
remaining
results
such free groups
group is
are on
([14]
and
free abelian
[7]).
continuous map i : X -
topological
groups. There
The Markov free abelian topo-
AM (X)
into
topological abelian group such that i is universal for maps of X into topological abelian
groups. The Graev free abelian topological group is a pointed continuous
m ap j : X - AGr(X) universal for pointed maps into topological abelian
groups. If X is a Hasudorff kw -space, then so also are AM(X) and
AG(X) [12].
logical
If
a
X, Y are
spaces, then
C (X , Y)
a
will denote the space of
con-
tinuous functions X - Y with the compact open topology. If X, Y are
pointed
spaces, then
C-(X, Y)
is the
subspace
of
C ( X, Y)
of
pointed
maps.
COROLLARY 3.
If
X is
pointed Hausdorff koi-space,
and K is
an
abel-
group, then the maps
ian
topological-
are
homeomorphisms.
P ROOF. The
a
proof
is similar
to
that of
quotients by closed subgroups,
gical group U (X X 2’ ) , where 2’ is
are
Corollary 2,
and
the
FM(X)
tree
since the maps
is the universal
groupoid
on
suggests
a
topolo-
the discrete space
{0,1}.
for i:
EXAMPLE.
Corollary 3
examples
Theorem 11 of
to
X - AM(X)
[15],
which
the spaces X, Y and the
states
class of
counter-
that under mild conditions
topological abelian group K , an algebraic
isomorphism between C(X, K) and C( Y, K) which preserves constant
functions induces a homeomorphism between X and Y . Let EX : AM(X ) - Z
( w ith Z the discrete group of integers) be the homomorphism determined
on
196
by the constant map X - Z with value 1 . Let 4J : AM( X) - AM( Y)
topological isomorphism such that 6y$ = EX. Then the composite
is
be
a
algebraic isomorphism preserving constant functions, and is indeed
homeomorphism, by Corollary 3, if X and Y are k. -spaces. However,
it is easy to give examples of non-homeomorphic X , Y for which such a
(D exists (cf. [7 , Section 5) and X, Y and K satisfy the conditions
of L 151 ( see the erratum to [13] ).
an
a
School of Mathematics and
Computer Science,
University College of North Wales,
BANGOR LL57 2UW, Gwynedd, U.K.
and
Department of Mathematics,
University of Queensland,
ST. LUCIA, Queensland 4067,
AUSTRALIA
197
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285-329.
1955 ),
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structurés, Cahiers Topo. et Géo. Diff. XI ( 1969), 329-384.
3. BROWN, R., Elements
of modem Topology,
McGraw Hill
(Maidenhead), 1968.
4. BROWN, R., Function spaces and product topologies, Quart. J. Math. Oxford
( 2) 15(1964), 238- 250.
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free products of topological groups, J. London Math. Soc. ( 2) 10 (1975), 431440.
6. BROWN, R. and HARDY, J.P.L., Topological groupoids I: Universal morphisms, Math. Nachr. 71 (1976), 273- 286; with DANESH- N ARUIE, G., Topological groupoids II: Covering morphisms and G-spaces, ibid. 74 ( 1976), 143156.
7. GRAEV, M.I., Free topological groups, A. M. S. Transl. 35 (1951); Reprint:
ibid. (1) 8(1962), 305-364.
8. HUBER, P. J., Homotopical cohomology and Cech cohomology, Math. Ann. 144
(1961), 73-76.
9. KAPLAN, S., Extensions of the Pontrjagin duality I: Infinite products, Duke
Math. J. 15 ( 1948), 649-658.
10. L A MARTIN, W. F., On the
State
foundations of k-group theory, Preprint, Louisiana
University.
11. L A MARTIN, W.F.,
Pontrjagin duality for k-groups, Preprint,
Louisiana Ste.
University.
12. MACK, J., MORRIS, S.A. and ORDMANN, E. T., Free topological groups and
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( 1973), 303-308.
13. MACLANE, S.,
Categories for
the
working mathematician, Springer, 1971.
14. MARKOV, A. A., On free topological groups, A.M.S. Transl. 30 ( 1950), 1188; Reprint: ibid. (1)8(1962), 195- 272.
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198