Download On the category of topological topologies

C AHIERS DE
TOPOLOGIE ET GÉOMÉTRIE DIFFÉRENTIELLE
CATÉGORIQUES
M ARIA C RISTINA P EDICCHIO
On the category of topological topologies
Cahiers de topologie et géométrie différentielle catégoriques, tome
25, no 1 (1984), p. 3-13
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Vol. XXV - 1 (1984)
CAHIERS DE TOPOLOGIE
ET
GÉOMÉTR1E D1FFÉRENT1ELLE
CATÉGOR1QUES
ON THE CATEGORY OF TOPOLOGICAL TOPOLOGIES
by
Maria Cristina PEDICCHIO
SUMMARY.
study the main properties of the category 9 of all pairs ( Y, Y* )
with Y a topological space and Y4’ a topological topology on the lattice
of open sets of Y . By q, it is possible to classify monoidal closed and
monoidal biclosed structures on Top .
We
INTRO DUCTION.
Isbell in
tors
is
[8]
gave the
of the category
completely
with 2
Top
of
determined
by
a Sierpinski
It is natural then
that
and whose
f-1 : Y* ~
a
pair ( Y, Y*)
to
define
morphisms
X" is continuous
Our aim is
to
of
topological
spaces where
underlying Y * is, up to isoof open sets of Y , and the topology of Y * is a
is, it makes finite intersection and arbitrary un-
space ; furthermore the
morphism, just the lattice
topological topology, that
ion continuous operations.
( Y , Y*) ,
following classification for adjoint endofunctopological spaces. A pair F -1 G : Top ~ TQp
study
a
set
category 9
whose
all continuous
are
objects
maps f :
are
all
pairs
X -
Y,
such
too.
the main
properties
of such
a
category 9 .
completely devoted to recall definitions and properties
about topological functors and initially complete categories drawing mainly
on Herrlich [6, 7 1.
In Section 2, first we prove that ! is a small-fibred, initially complete category on Top , in order to apply the previous results; then we
construct a monoidal closed (also biclosed) structure on 9 .
Section 1 is
3
problem of classification of monoidal
closed structures on the category of topological spaces [2, 3]). As a generalization of our results about structures induced by adjoining systems
of filters [10], we give a complete description both of all monoidal closed
structures and of all monoidal biclosed structures on Top, relating them
to opportune functors from -1 to Top . The author is greatly indebted to
G. M. Kelly for many helpful conversations and advices.
Section 3 is devoted
1. For
A , 11
and
concrete
a
the
over
categor y
where U: A - X is
a
base category X,
forgetful functor,
a
that
is,
we
mean a
faithful,
a
pair
amnestic
functor.
transportable
A X-morphism
or
Top
c : X ~ Y is said
Z in X and for
th at, for each
Set
to
this definition is
D EF INITION 1.1. A
gical category
iff the
( i ) U : A - X is
( ii)
A is
( iii)
phism
be
a constant
all u , v E X ( Z , .Y ) ,
just
concrete
to
the classical
category
=
c . v .
If X is
one.
( A , 11 )
following hold :
a topological functor
morphism provided
c . u
in the
over
X is called
sense
of
a
topolo-
[6;
small-fibred;
every
constant
morphism U ( A ) --&#x3E; U ( B)
underlies
some
A-mor -
A--&#x3E; B .
category X is Set or Top then condition ( iii ) is equivalent to the following
( iv ) For any X with cardinality one, the fibre (J’ ( X ) is a singleton.
If the base
We shall say that
completes
a
concrete
category iff it verifies (i) and
category is
(11)
a
small- fibred, initially
[1, 11] ).
of Definition 1.1 (see
( A , U ) and (B,V) are concrete categories over X , for a funcwith
over X , F ( A, U ) --7 ( B , V ) , we mean any functor F i A - B
If
tor
F
V.F = 11 ,
D EF INITION
an
1.2. A
extension (of (
DEFINITION
functor F
A , U )) if
over
X, F: (A, U) --&#x3E; (B, V), is called
F is full and faithful.
1.3. An extension
E: ( A, U ) --&#x3E; (B, V )
4
is called
a
small-
fibred, initial completion (o f ( A , U ) )
tially complete.
if
(B, V) is small-fibred and ini -
following list exhibits some properties of small-fibred, initially
complete categories over the category Top of topological spaces ; the same
results are well known for topological categories over Set [7].
The
PROPOSITION 1.4.
over
(2) A
a
small- fihred, initially complete category
following hold :
is finally complete ;
is complete and cocomplete
and U : A 4
Top preserves limits
colimits ;
( 3 ) An A-morphism is
tive
is
then the
Top,
(1) A
and
I f ( A, U )
a
monomorphism (epimorpbism) iff
(surjective),’
( 4) A is wellpowered and cowellpowered;
( 5) U: A Top has a full and faithful left adjoint
--+
D
it is
injec-
( = discrete
structure) ;
( 6 ) U : A 4 Top has
structure)
a
full
and
faith ful right adjoint
K
( = indiscrete
,
A-morphism f, f is arz embedding i f f f is an extremal
monomorphism i f f f is a regular monomorphism f is a quotient map iff
f is an extremal epimorphism i f f f is a regular epimorphism;
( 8 ) For any topological space X , the fibre U -1 ( X ) of X is a small
complete lattice ;
( 9 ) Any A-object A , with U A 1:- ø and A D U A, is a separator;
( 10 ) An A-object A is a coseparator iff there exists an embedding
of an indiscrete object with two points into A ;
( 11 ) An A-object A is projective i f f A = D U A ,
K UA.
( 12 ) An A-object A is injective iff U A A O and A
( 7)
For any
,
=
.
=
P ROO F.
( 1 ), ( 2 ) and ( 3 ) follow from [6 ], Sections 5 and 6.
( 4 ) is
true
for
Top
is well- and
cowellpowered.
( 5 ) and ( 6 ) follow by Th eorem 7.1 of [6 ] .
( 7 ) follows easily by the definition of regular and extremal morphism.
( R) For any pair Ax , Ax of objects in U-1 ( X ) , let us put
5
l1x
A x
iff there is
with this order
any
family
a
A-morphism g: A x
U-1 (X)
relation,
( Ai x )iE in
the
forms
fibre,
,7C’
small
a
inf ( A ix )
with
U(g) = IX ;
lattice
complete
is the 11-initial
and, for
lifting
of the
source
( 9 ), ( 10 ), ( 11 ) and 12 ) follow by properties of discrete and indiscrete
0
structures.
2. Let
recall the
definition from
following
topology 7’ on
is a topological topology
Y
space
us
[8].
the lattice 0 Y of the open
A
of
sets
a
topological
iff it makes finite intersection and ar-
bitrary union continuous maps (we use the symbol Y" for the set O Y topologized by T ). Isbell [8, 12] has proved that any pair of adjoint endofunctors F j G of Top is determined (up to functorial isomorphism) by a
pair ( Y, Y*,), where Y = F ( { * }) and Y*=G(2),with 2 aSierpinski
furthermore G(2) has,
space ;
tion) and Y* is
a
We define
underlying
as
topological topology
category 9
on
set,
just
O Y
(up
to
bijec-
this lattice.
t = Isbell
category) as follows :
Objects of 9 are all pairs (X, X*), with X c Top and X * a topological
topology on OX. An g-morphism f : (X, X * )--+ ( Y , Y *) is a continuous
map f : X--+ Y such that f-1 : 0 Y OX is continuous from Y * to X * (we
shall write f * for f-1 ). We call V the forgetful functor V : 9--+ Topo defined by
a
--&#x3E;
and
call U the
we
Then,
if
adjoint
Adj
forgetful
U:9--+ Top defined by
functor
objects are the pairs F U CT of
morphisms are the pairs of nat-
denotes the category whose
endofunctors of
Top , and
whose
ural transformations
(a,B ):
Isbell’s result
( F, G)--&#x3E; ( F’ G ’ ) ,
implies
THEOREM 2.1. T’he
the
with
following
a:
F % F’ and
B:
theorem :
categories Adj and 9
6
are
equivalent.
G’ --&#x3E;G
( 9,
P ROP OSITION 2.2.
egory
over
a
( Xi , X*i) is
sink in
Top ,
with respect
suffices
and
to
to
stitall-libred, initially complete
to
we
a
denote
I-objects
of
X* the
by
following
set
and
OX with the initial
topology
a
topological topology, it
diagram
commutative
topology to the continuous composite
the union map).. Since, for any
of the initial
apply properties
(the
family
To show that X* is
( f*i )i E I .
consider the
r i X f*i )
same
applies
to
f*i . h * is continuous for
q-morphism (for any i ). This
with respect to
(fi)i E I
h *: Y* -&#x3E; X* is continuous iff
an
9-morphism
is the
q-final
iff h.
fi
is
an
X
structure on
Any finally complete
is
a source
in
category is
a
initially complete ;
the
following
topological topology
is
9-initial
structure on
structure
X with respect
i , then h is
shows that (X, X*)
any
.
Top and 4) denotes
is
then the
cat-
Top .
P ROO F. If
is
LI ) is a
on
to
X
set:
on
an
with respect
if
0.B, and
4-morphism
to
all the functions 1 :
( gi )i E I
for any
is the
i}
4-final
(X, X*)--&#x3E; X, X*E O.
0
REMARK 2.3. If S is
are
objects
inuous map
an
open
of X * such that r
topological topology X* and r, s,
( Sand s 2 r , then s c s . Consider the cont; since U-1 ( S ) is open in the topological
set
of
product
a
,
7
then there
ex-
neighborhood U c U-1 ( S)
ists
a
and
7"7 15.;:; h
sets
open
,
of ( rn ) with
of X*
containing
n E N’ with tn = r if n = l1 , l2 , ... , lh
is in
Ur’
generally
As
tn = s otherwise
10, 2
construction of
a
consider
and denote
by
k t: X--+ 3 , it follows,
(2,2*) is
by
the
a
constant
0 , 1, 2}
3 =
is open in the final
ogy, so, if
9-initial
structures
by Top-final
struc-
fails.
example,
an
(X,X*)E g,
set
and
The sequence
so 5 is in S.
Note that
tures
r.
topology
2 *f
Remark 2.3, that
g-initial
structure
U ( X , XI), with
map k : 2 --+
the ordered
O 2 . Since the
3 with re spect
on
2 *f
set
is
not a
to
the map
topological topolto k , 2*
2f* .
with respect
Theorem 2.2,
Proposition 1.4 can be applied, hence completeness and cocompleteness of q follow, where limits and colimits are obtained from limits and colimits in Top by q-initial and g-final structures
respectively, with respect to the maps of the limit (or colimit) cone of Top.
Again by Proposition 1.4, U: g --+ Top has a left adjoint D and a right adjoint K with the following properties :
By
for any X E
structure
X*
D
and
( Y, Y * ) E g. If
D ( X ) and by
and X4,
ology
Top
K
determine,
definable
on
(X, X*)
resp., the
OX . For X =
we
denote
by
the indiscrete
coarsest
1* 1,
( X , X*D )
structure
and the finest
the discrete
K ( X ) , then
topological
top-
the fibre
objects), hence I is not
topological. Since Top is topological over Set with the forgetful functor
U : Top 7 Set , then I is small-fibred, initially complete over Set with the
forgetful functor W U . U . We write D and K for the left and right adjoints to FJ and D and K for the composite functors D . D and K . K .
(where
2
denotes
an
indiscrete space with
=
8
two
For
X , Z E Top and
U(X)XU(Y)
set
symbol [ Y, Z]
respect
to
with the
denotes the
( Y, Y* ) E g ,
symbol X ø Y denotes the
topology O ( XO Y ) = Top ( X , Y*), and the
set Top( Y, Z ) with the initial
topology with
the maps
defined
[8, 10]. Furthermore,
note
from
now
on,
we
by
shall
use
letters A , B , C
to
de-
g-objects
will be the
J
the
THEOREM
category [4 ]
2.4. The
(q,
’wherp [Y, Z *]
with respect
g-object D ({ * }) = ({ *} , 2)
to
o ,
1,
category I
{-, -})
is the sett
the maps
{
admits
a
and I
structure
the
g-object
of
monoidal
closed
with
O [ Y, Z I with the final topological topology
PS: Z*--+ 0[ Y, Z !SfOY*’ defined by
topology on Top( X, Y*) with respect to all the
maps {lS} S E O Y* is a topological topology (see Theorem 2.2), then the
tensor product object is in 9. For g-morphisms f A --+ A’ and g : 9 4 B’
P ROO F. Since the initial
we
as
define
the
product f X g .
If
r :
X’ --&#x3E; Y’4’ is
an
open
set
of X’ O Y’ , then
f X g: X O Y --+ X’ O Y’ is continuous ; to prove the continuity of
( f X g ) *: [ X’, Y’* ]--&#x3E; [X, Y *] , it suffices to apply properties of the initial
topology. I = ( { *} 2 ) , so
hence
The
associativity
of
o
follows from
Proposition 2.3 of []0],
9
for
topological topology [ Y,Z]*as in
Theorem 2.3 and the 4-morphism {f, g}: {B, C}--+ {B’, C’ j} is hom ( f , g ) ,
for any f : B ’ --+ B and g : c » C’ . If B
( Y , Y*) 6 q , it is known by [8] ,
As for the internal hom
we
get the final
=
that
Let
us
denote
by k: X --+ [ Y, Z]
h * is continuous iff the
are
continuous,
for
the
following composite
any S E
0
for
map o ( h ),
maps
any h:X O Y--+ Z.
ts
for the commutative
Y" , but,
and, by properties of the final topological topology, ts
any S ) iff k * is continuous; hence
and - o B - {B, -} for any R f g,
We end this section with
The
tensor
closed
o
is
not
structures over
(g , - o -, I, { -, - })
diagram
is continuous (for
o
two
further
comments
on
Theorem 2.4.
symmetric, for there exist non symmetric monoidal
Top (for example Greve structures [5 ]), induced by
(see Section 3). A o - preserves
coproducts and coequalizers for any .4 f g, then by Freyd’s Special Adjoint Functor Theorem,
it has a right adjoint, hence the g-structure is biclosed.
3. Consider Top
as a concrete
PROPOSITION 3. 1 (see
category
also [10}). A
10
over
Top itself :
monoidal closed
structure
on
Top,
is
equivalent
PROOF.
to
functor
over
Top,
F :
( Top , 1 ) --+
(g,
U ) such that
By [8)] (or [12 ]) ,
Top . From Proposition 2.3 of [10], ( i )
0
monoidality of ( To p , - D -, {*}).
for X, Y, Z
valen t
to a
Then, for
any monoidal closed
small-fibred,
a
structure
is
(g, -
initial
completion
0 - , I) restricted
to
pair (g,
of
( Top ,
the full
F:
Top --+)
7 ). Further the monoidal
subcategory F ( Top) = Top ,
just (ToP, - D-, {*}) .
EXAMPLES. The canonical
(separate continuity
and
symmetric monoidal closed
pointwise convergence)
ultrafilters of open
the
lotson
where
h ( X) &#x3E; denotes the
PROP OSITION
set
of all open
3.2. A monoidal biclosed
the
sets
to
of I’
structure
of all
any Y E Top
over
Top
PROOF. If
a
monoidal biclosed structure , F and G
11
are
prin-
the top-
containing h ( X ) .
such that
over
Top
of Booth and Til-
is
pair of functors
Top, ( F, G ): Top--+ g,
3.
( i ) and ( ii ) of Proposition 1 and V.F -l V °. G °: Top 4 Top ° .
to a
on
by F ( Y ) =
family
a-structures
Similarly
[2] and Greve 5] are obtained associating
ological topology generated by the family
cipal
sets.
structure
is determined
( Y, Y4) where the topology of Y* is generated by
is
are equi-
structure
the associated functor F is strict monoidal and the
is
and ( ii )
f
defined
by
F
equivalent
satisfies
Sin c e,
for X , Y , Z c
Top,
then if Z= 2,
so V . F -l
V ° . G ° .
and G determines
V.
F -l
V ° . G °
an
Conversely,
F determines
adjunction -DY -l
a
monoidal closed
structure
Y, - &#x3E;&#x3E; with
implies
Since
and (2,2) is
X, hence
a
strong cogenerator of
Top ,
Y 0- -t «Y,"», for. any Y.
Observe that
determined by
a
a
symmetric
pair ( F, G )
then
0
monoidal closed
with G = F.
12
YDX = YD X, for any
structure
over
Top
is
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