C AHIERS DE
TOPOLOGIE ET GÉOMÉTRIE DIFFÉRENTIELLE
CATÉGORIQUES
I GOR K ŘÍŽ
A direct description of uniform completion in locales
and a characterization of LT -groups
Cahiers de topologie et géométrie différentielle catégoriques, tome
27, no 1 (1986), p. 19-34
<http://www.numdam.org/item?id=CTGDC_1986__27_1_19_0>
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Vol. XXVII-1 (1986)
CAHIERS DE TOPOLOGIE
ET
GÉOMÉTRÍE DIFFÉRENTIELLE
CATÉGORIQUES
A DIRECT DESCRIPTION OF UNIFORM COMPLETION IN LOCALES
AND A CHARACTERIZATION OF LT-GROUPS
by Igor K0159Í017E
RESUME.
(au
sens
ne
une
Isbell a construit la completion uniforme d’un locale
de fermeture absolue) a 1’aide d’hyperespaces. Ici on donautre
construction, plus directe, par g6n6rateurs et
relations.
Comme
application, on obtient "1’enveloppe locale" d’un groupe
topologique, qui permet de caract6riser les groupes topologiques
qui sont des locales. Dans le cas commutatif, une telle caract6risation
été
a
obtenue
dans
un
article
par
Isbell,
Kriz,
Pultr
et
Rosický.
The question of uniform completeness in locales (in the sense of
absolute closedness) was discussed first in Isbell’s paper [3]. Among
other results, Isbell gave a description of the uniform completion. Since
the classical Cauchy-filter approach does not yield the absolute
closedness property in locales, it was replaced by another construction
the idea of which was to employ hyperspaces. This is also one of the
reasons why Isbell called completion in locales "hypercompletion".
The main purpose of this paper is to introduce another description
of uniform completion in locales, which does not use hyperspaces and
in fact is even more straightforward than the original topological construction. It -simply consists of writing down generators and defining
relations. In connection with this we will also prefer the natural terminology without the prefix "hyper".
construction of a "localic
characterisation of those
which
It
are
localic.
should
be noted that for the
topological groups
commutative case a characterisation of this type was obtained in [4].
As
hull" of
an
a
application, we give a general
topological group, which yields
a
1. PRELIMINARIES.
1.1. Frames and locales. Basic facts concerning locales
in [5] (as well as in other papers, e.g. [1, 3J).
A frame is
a
complete
can
be found
lattice A in which the distributive law
19
holds for any a E A, S C A. We will write I(A) or 1 for VA and O(A)
or 0 for VO. A frame morphism is a mapping f : A -&#x3E; B of frames A, B
preserving joins and finite meets. The category of frames will be
denoted by Frm. The lattice Q(X) of all open sets of a topological space
X ordered by inclusion is a frame and a continuous mapping f : X -&#x3E; Y
induces a frame morphism
Thus, Q becomes a contravariant
topological spaces to Frm.
To make Q
Loc = Frmop of
has a right adjoint
functor
from
the
category Top of
covariant functor, one introduces the category
locales, and writes Q : Top -&#x3E; Loc. The functor Q
a
where T(A) is the set of all frame
with the natural topology (see e.g. [5]).
morphisms
(.) together
A -&#x3E; 2 = Q
There is a natural parallelism between the concepts in Top and
(see [3, 5J). It is, however, often not of much use in technical
parts of the proofs presented. For that reason it is of an advantage
to write morphisms as in frames, although we think in locales. Let us
give some examples, which will be useful later on :
of B
A frame morphism f : A -&#x3E; B is called an embedding
into A iff it is surjective. An embedding is called closed if we have
Loc
A
It is
morphism
easily
f is dense
checked that
Now let A be
a
The frame A is called
I.I.I.
Pcoposition.
a
iff
dense closed
frame. For a, b
regular,
The
dense
E
emnbedding
is
an
isomorphism.
A write a A b iff
iff
frame
20
morphisms of regular
frames
are
exactly the monomorphisms
by the regular frames.
Proof. See
in the full
subcategory
of Frm
generated
[4].
Let Sp denote the subcategory of
Q(X) where X are topological spaces.
Sp is characterized by
Frm generated by the objects
It can be shown (see [5] ) that
1.2. Uniform locales. (For the details, see [9, 101) A cover of a frame
A is a set S C A with the property that VS
1. The system of all
covers of A will be denoted by C (A). Let us call a U E C (A) a refinement of a V E C(A) (and write U - V) iff
=
If
U,
V
E
C(A)
then
we
have
a cover
Write for
We write
Obviously
iff there exists
The
a
U
E
U such that U x
is
pair
y . Put
called
a
uniform
frame iff
we
The
a
have
system U is called a uniformity on A. A uniform basis
system UC C (A) satisfying (iii) and (iv). Similarly as in Top,
have
the
least
uniformity generated by
a
given
uniform
basis
is
we
(see
[8]).
Let
(A, U), (B, V )
be uniform frames. A frame
21
morphism
f : A -&#x3E; B
is called
uniformly
continuous with respect to
where "#" indicates the
We call f
a
image
uniform
U , V iff
function. In that
iff
embedding
it
is
have
write
surjective and,
We will often speak about uniform locales,
in the same way as locales relate to frames.
1.2.1. Lemma. Each uniform locale is
case we
we
moreover,
related to uniform frames
regular.
"completely regular",
Proof. See [8]. We could really claim
be never used in the sequel.
but it would
0
general, the products in Top and Loc need not coincide. Thus,
question arises as to what is the relation between groups in Top and
Loc. It was shown in [ 4 ] and [7] that none of the concepts contained
the other. We will be particularly interested in the fact that a topologic1.3.
In
a
al group need not be
Denote
by
"@
"
a
group in Loc.
and "+" the
sum
in
Frm,
resp. in
Sp. (Realize
that
In the sequel, an L-group means a cogroup object in Frm, while a Tgroup designates a cogroup object in Sp. (Note that if X is a topological
group, then S2 (X) is a T-group and if A is a T-group or an L-group,
then T(A) is a topological group.)
Thus,
an
L-group
inverse L : A + A and
identitites
an
A
a
consists of
unit point E
where V : A
A
T-group
a
::
multiplication 03BC : A -&#x3E; A a A,
satisfy the usual
A -&#x3E; 2 which
® A -&#x3E; A
is the codiagonal and
consists of similar mappings
22
6
is the initial
morphism.
identities. We define homomorphisms of L-groups
the
obvious
way [4]. Note again that a homomorphism
(T-groups)
of T-groups is always a pre-image function of a homomorphism of
topological groups. An embedding of L-groups will be an L-group homomorphism which is an embedding.
satisfying analogous
in
Proposition. L-groups are regular frames.
Proof. Indeed, they are unifomizable (see [4, 7] ).
1.3.1.
1.3.2.
Proposition. Any embedding
Proof. See [4].
Let
is
(A, 03BC)
us
an
of
L-groups
0
is closed.
0
specify what we mean by saying that a T-group
L-group. First of all we should claim the existence of a
now
lifting
where T is the standard restriction mapping. Since T is evidently a
dense embedding (see [4]), it follows directly from 1.1.1, 1.3.1 that
whenever 03BCl exists, it is uniquely determined and makes (A, 03BCl) an Lgroup. T-groups which require this property will be called LT-groups.
2. SUBLOCALES AND FACTOR FRAMES.
In this section we give a general description of factorization in
frames. The main idea of our approach belongs to Johnstone [ 6] ,
who really proved the full strength of the theorem, but restricted the
statement by unnecessary assumptions. In this paper we present the
theorem in the full generality, although the case we really need
is much closer to that of Johnstone [6].
2.1.
The right adjoint of
B -&#x3E; A (recall that
by f* :
we
obtain
a
natural
1-1
a
frame
f*(x)
=
morphism f :
). Putting
f(y) VXy
correspondence
23
A -&#x3E; B will be denoted
for a surjective f:
between
the
embeddings
and
the nuclei (see [1 ]).
A -&#x3E; A satisfying
(Recall
that
nucleus
a
is
A
on
mapping
a
j:
for any two a, b
and obtain
inclusion.
2.2.
A
E
A.) If j
frame
a
a
a
join-basis
nucleus
A -&#x3E;
A, then
on
Aj ,
whose
a
a
if
for any
Denote
an
is the
for
any a, bE A
of A.
precongruence relation
Take
put
right adjoint
precongruence relation if
2.2.1. Observation. Let S C AxA and let A’ be
is closed under finite meets. Then
is
one can
frame A is any subset A’ C A which satisfies
We call a subset R C AxA
with a R b the set
is
a
embedding j :
of
join - basis
is
on
a
join-basis
of
A, which.
A.
R C AxA. An element
a E
A will be called R- coherent
two a, b E A with a R b we have
by =&#x3E;
the
operation
of
implication
in A
given by
2.2.2. Lemma. L et R be a precongruence on A and let b E A be R coherent. Then the element a =&#x3E; b is R -coherent for any a E A.
Proof. Let
a
x
R y. Since R is
a
precongruence
set Q C A such that
Now
24
relation,
we
can
choose
2.2.3. Theorem. L et R be a precongruence on A . Then the set A(R)
of all the R-coherent elements together with the induced ordering
is a frame and there exists a nucleus j : A -&#x3E; A such that A j = A(R) .
Moreover, j : A -&#x3E; A(R) is universal among the join-preserving
mappings f from A to complete lattices B satisfying
(More exactly, for any
f j : A(R) + B such that f
Proof.
Define j :
A -&#x3E; A
such f
=
there exists
a
unique join-preserving
fj o j .)
by
Since a (finite or infinite) meet of R-coherent elements is
R-coherent again, we obtain
Let
us
show
since the
while the
we have
as
that j
right
hand element is R-coherent.
right
required. joins
preserve
preserves finite meets. We
hand elements
Conversely,
R-coherent
by
It remains to prove the universality
and let a R b -&#x3E; f(a) = f(b) . Put for a
Since s(a) is R-coherent
and consequently
On the other
are
obviously
hand,
and
s(a) &#x3E; a ,
since f preserves
25
joins,
we
we
evidently
have
it holds
Lemma 2.2.2. Thus
of j .
E
Let f : A -&#x3E; B
A
conclude that s(a) ? j(a) ,
have
f(s(a)) f(a).
2.2.4. Corollary. Preserve the notation of 2.2.3 and assume that
R
R(S, A’) for a join-basis A’ and a relation S C AxA . Then for
any frame morphism f : A -&#x3E; B which satisfies
=
there exists
unique
a
frame
morphism g : A(R) -
Proof. Since f preserves finite meets,
we
B
satisfying
f = g o
j.
have
2.3. Let A be a meet-semilattice. Denote by 4-A the set of all downward-closed ("decreasing") subsets of A. Then 4- A is a completely
distributive lattice (in particular, a locale) and we have a mapping
Moreover,
one can
prove
2.3.1. Proposition. The mapping ! gives rise to a reflection from the
0
category MSL of meet-semilattices to Frm (see [11J).
Of course, the word "reflection" applies only to the properties
of the mapping § and does not indicate that Frm should be a full
subcategory of MSL. Considering 2.3.1, one sees that Proposition 1.1
from [6] is a special case of 2.2.3 for a certain type of precongruence
relations on !A.
3. COMPLETION OF UNIFORM LOCALES.
give a description of uniform completion in terms
defining relations. Throughout this paper, the completeness means the natural categorical phenomenon and not a
generalization of the topological Cauchy-filter construction. It
coincides with Isbell’s concept of "hyper-completeness" (see [3J).
In this section
of generators and
3.1. Let (A,
of all pairs
with
a E
A,
U)
U
E
we
uniform locale. Denote
be
a
U ,
where k and
c are
Put
26
by
given by
S C ! Ax ! A the
system
(Realize that + A’
is
meet-closed.) Recalling
the notation of
2.2.3,
write
Now both ! A and A are locales. Since A C ! A, we should be careful
when indicating their localic operations. There is no trouble with the
meets, which are preserved by the nucleus and hence coincide.
On the other hand, there is a natural way to distinguish joins : we
"
for A, while in + A we simply use the setreserve the symbol " V
theoretical symbol " U ".
It
Hence,
follows immediately from the definition of S
we have a dense embedding p : A -&#x3E; A given by
that + A’ C A.
Put
We
see
easily
that
Uo
is
a
system of covers, for
In fact we obtain a uniform basis. Denote by U the corresponding
uniformity (see [8]).
The pair (A, U ) together with the mapping p : A -&#x3E; A (which is
easily checked to be a uniform embedding) will be called the completion of
a
uniform locale
3.2. Let f : (A, U ) -&#x3E;
Define g : A -&#x3E; B by
(B, U’ )
putting
(recall
we
(A, U ).
be
that elements of the
obtain a frame morphism f
a
uniformly
A -&#x3E;
are
R-coherent). Using 2.3.1,
B, satisfying
mapping f o o k : A -&#x3E; B preserves also
unique frame morphism f 1: ! A -&#x3E; B satisfying
3.2.1. Lemma. We have
Proof. Use 2.2.4.
First,
morphism.
form ! x
0:!
Since the
a
continuous frame
note that
27
finite meets,
we
obtain
Now, compute
Take a U E U . Choose a W E U to satisfy
conclude the proof by another calculation :
3.2.2. Theorem. For
there exists
the diagram
a
a
unique
uniformly
frame
continuous
morphism
W.W
U
(see 1.2)
and
morphism
f :
(A, U) -&#x3E; (8, U’) completing
Moreover, this morphism is uniformly continuous.
that the frames A, B, A, B are uniform and hence
deduce the uniqueness part of the theorem from I.1.1
and from the density of p . Taking into account that
Proof.
regular,
Realizing
we
obtain the existence part_ from 3.2.1
prove the uniform continuity of f . Let
we
Choose again a W E U to
satisfy W.W
28
and 2.2.4.
U. We
compute
It
remains
to
_We will call a uniform locale
p : A -&#x3E; A is an isomorphism in Frm.
3.3. Theorem. A frame (A, U)
f : (B, U1) -&#x3E; (A, U) is closed.
(A, U) complete
is complete
if the
completion
iff each uniform
embedding
Proof. If we assume that each_ uniform embedding f : B -&#x3E; A is closed,
is in particular the dense p : A -&#x3E; A. Thus, p is iso.
We prove the converse. Assume that A is complete. It suffices to
prove that any dense
so
is an isomorphism (in other cases
of f to the closure of A in B,
dense. Put j B = f * o f .
we
see
3.3.1. Observation. We have
Proof. Take
a
U
E
U
1
with U a
b. Put
We have
hence
and hence
finally jB(a) A
3.3.2. Lemma.
For
c
A
a E
=
0.
we
have
Proof.
3.3.3.
Lemma. For U
E
U
we
have
29
simply
e.g.
consider the restriction
let f be
[3, 4, 6]). Thus,
Proof. Let
and thence
Take
a
Since
p A is iso,
b
E
B. We have
we
a
commutative
diagram
deduce
Put
Considering 3.3.2,
from the density
coherent, since f*
3.3.3 and the inequality f* (0) b (which follows
of f ), we see that v is S-coherent and hence Rpreserves finite meets. On the other hand we
have
by Lemma
implies f(b)
Thus, j B is
3.3.1. Thus, it follows from
E v and in consequence
the
identity
(3.3.1)
that +
f(b)
and hence f is iso.
v
which
0
3.4. Theorem. The completion mapping p gives rise to a reflection
from the category of uniform locales to its full subcategory of
all locales which are absolutely closed under uniform embeddings.
Proof. It follows
directly
from
3.2.2, 3.3.
0
4. A CHARACTERIZATION OF LT-GROUPS.
4.1. Let
(A, il,
L,
£) be a T-group
and let
the corresponding (not necessarily commutative)
For any m E A with e(m) = 1 put
be
30
group
operation.
Then
we
have certain uniform bases
on
A. We denote
standard arguments,
the corresponding
obtain
4.1.1. Lemma. For a, b
Proof.
uniformities
by U R, UL.
By
we
A
E
we
have
0
Straightforward.
Denote the completions of the uniform locales (A, UR ), (A, UL)
resp. AL and put ARL = Arf’A.... (recall that both AR, AL are
subsets of ! A). Clearly ARL is a frame and the diagram of canonical
4.2.
by AR
embeddings
is a pushout.
b y jRL.
The
composition mapping +
A +
ARL
will be
denoted
·
4.2.1. Lemma. The
equalities
hold in ARL for any a,
f : A -&#x3E; B satisfying
n
E A. On the other hand. any frame
31
morphism
for all a,
n
Proof. It is
(uniquely) factorized through j R L.
easily obtained from (4.1.1), 3.1 and 2.2.4.
fA,
4.3.
Let p R:
p RL :·
A RL - A
can
be
pL : AL -&#x3E; A be the completion
be their colimit. The mappings
A R -&#x3E; A,
0
mappings
and let
is taken from 1.3.1 and the asterisk indicates adjunction)
(where ’C
are easily checked to preserve finite meets. Using 2.3.1, we obtain
unique frame morphisms
satisfying
4.3.1.
Theorem. The
morphisms
v, x
can
be
factorized
Proof. By Lemma 4.2.1 it suffices to prove the
relations
are
trivial.
through j RL.
equalities (4.2.2).
The
Computing
and
we
conclude
are
analogous).
the proof for the case of n
In the case of v , we calculate,
32
(the remaining identities
first,
To obtain the last
identity,
assume
and compute
The
symmetry argument concludes
4.3.2. Theorem. VVe
have
an
the
0
proof.
L-group
(A RL, 03BC, l, e)
making diagrams
commutative. Thus, any T-group can be embedded into
a dense subgroup of its spatial reflection.
Proof. We define 03BC, L as factors
T and
pRL are dense and hence
and for that reason (ARL ,ti, L, d is
4.3.3.
is
a
Theorem.
A T-group
A is
an
an
L-group
as
of v, x, through jRL. Recall that
monomorphisms in regular frames,
0
an L-group.
L T-group
iff the
diagram
pushout.
Proof. The sufficiency follows directly from 4.3.2. Thus, let A, 03BCl
A -&#x3E; A e A be an LT-group. Then the following diagram is commutative, since T is dense. This makes (A, 03BCl) a dense subgroup of (A
We deduce that PRL is iso by Theorem 1.3.2.
33
4.3.4. Remark. Note that if the
to
U R
or
UL
Theorem 4.3.3 is
(i.e.,
we
obviously
Acknowledgement.
T-group A is complete with respect
have A R = A or AL = A) the condition of
satisfied and hence A is an LT-group.
I would like to thank A. Pultr and J.
Rosicky
for
valuable discussions.
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2. C.H. DOWKER &#x26; D. STRAUSS, Sums in the category of frames, Houston J. Math. 3
1. J.
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ISBELL, Atomless parts of spaces, Math. Scand. 31 (1972), 5-32.
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34