Download Constructive Complete Distributivity II

Constructive Complete Distributivity II
Robert Rosebrugh
Department of Mathematics and Computer Science
Mount Allison University
Sackville, N. B. Canada
R.J. Wood
Department of Mathematics, Statistics
and Computing Science
Dalhousie University
Halifax, N.S. Canada
Abstract
A complete lattice, L, is constructively completely distributive, (CCD)(L), if
the sup map dened on down closed subobjects has a left adjoint. It was known
that in boolean toposes (CCD)(L) is equivalent to (CCD)(Lop). We show here
that the latter property for all L (suciently, for ) characterizes boolean toposes.
Research partially supported by grants from NSERC Canada. Diagrams typeset using
catmac.
1
Introduction
The notion of constructive complete distributivity for a complete ordered set
)( ) was dened in 1] to mean the existence of a left adjoint to
W, :(D ;!
where D is the set of down-closed subsets of ordered by
inclusion. Terminology springs from (i) the fact that (
)( ) is equivalent
to
o _ n^
o
^ n_
(8S D )
j 2S =
f ( ) j 2 Sg j 2 S
L
CCD
L
L
L
L
L
CCD
L
S
S
T S
S
L
T
(which becomes the condition for complete distributivity of , (
D is replaced by P the power set functor) and (ii) the following:
L
Theorem 1 ( ) () ((
choice.
ac
CD
),(
CCD
CD
)( ) if
L
)) where (ac) denotes the axiom of
Somewhat less cryptically, the (
) condition is the restriction of the
more familiar ( ) to down-closed subsets it implies the stronger condition
in the presence of the axiom of choice which in turn is implied by identifying
the two notions of distributivity.
Thus, in classsical lattice theory, with the axiom of choice freely assumed,
it would appear that (
) is simply an easier condition to verify than
( ) However, a considerable body of literature has explained the value
of doing Mathematics in a topos and in this more general context the very
denition above and Theorem 1 suggest that (
) is the relevant notion.
Indeed, in any topos, P is (
) for all while the statement \P is
( ) for all " is equivalent to choice.
The power objects, P , in a topos are not, in general, boolean algebras
and it transpires that booleaness of the topos of \sets" does make a substantial dierence to the theory of constructive complete distributivity. For
a topos E we write ( )(E) to indicate that P is boolean for all in E
op
We write (
)( ) for (
)( op) With this notation, Theorem 18 of
1] becomes:
op
( ) =) ((
),(
))
Somewhat contrary to intuition, the hypothesis is necessary. In this note we
prove:
CCD
CD
CCD
CD :
CCD
X
CD
CCD
X
X
X
X
boo
CCD
L
X
CCD
boo
L
X
:
CCD
2
CCD
:
:
Theorem 11 ( ) () ((
boo
CCD
),(
CCD
op
))
:
Even in a general topos, ( )op is an involution, so
((
))(
)op) =) ((
)op ) (
))
To prove Theorem 11 we will show that (
))(
)op reduces to
(
)op() where = P1 = D1 is the subobject classier of the topos.
This reduction involves showing that
(
)( ) =) (8
ord(E))(( )(ord(E)( )))
where ord(E) is the 2-category of ordered objects in E (As in 1], an order
is assumed to be reexive and transitive, but not necessarily antisymmetric.)
This follows from the observation that powers of (
) objects are (
)
which in turn follows from the fact that arbitrary products of (
) objects
are (
).
CCD
CCD
CCD
CCD
CCD
:
CCD
CCD
CCD
L
X in
CCD
X L
:
CCD
CCD
CCD
CCD
1
The Main Result
For any topos, E the subobject classier, E is a locale (complete heyting
algebra) in E On the other hand, given any locale, , in set the subobject
classier for ; : sh( ) ;! set enjoys ;(sh ) = However, the logic of
E when viewed from E itself is more classical. The fact that E is always
:
L
L
L:
(L)
(
) shows this. In fact, more is true:
Theorem 2 In any topos we have
CCD
?
?
?
?
#
-
D
More generally, for any ordered object
joints.
X
#DD has such a string of adX
Proof. It suces to observe that = D1 = DD0 and argue as in 1,
Theorem 3].
3
(The adjoint string exhibiting set as a total category in 5] arises in a
similar way, starting with the empty category the common thread is what
is called a yoneda structure in 4].)
The situation for op is quite dierent unless E is boolean. Indeed, the
proof of Theorem 10 towards the end of this section shows that if op is
merely a locale then (and hence E) is boolean. That this is somewhat
contrary to intuition, as claimed in the Introduction, is seen by considering
0
1
f
g
B A A CC
=B
B@ A A CA
U
A
U
A
f
< <
< g
in set2. The distributive law for binary meets over binary joins intuitionistically implies its dual. Since both the domain and codomain of the above
example are nite and the lattice structure is \pointwise" it appears at rst
glance that all distributivities are consequences of the binary
ones. However,
W
even
simple example the innitary operations : D ;! and
V op : inDthis
op
;! op cannot be obtained by considering all iterates of their
binary counterparts. Conceptually: is not cardinal nite. Technically: the
op
functions witnessing f
g and f
g as (
) objects of set do
2
not give an arrow of set .
Another illustration of this sort of phenomenon is encountered when one
considers the innite demorgan law. We digress slightly.
For a (not necessarily complete) heyting algebra, (
)( ), we write
op
:
;! for its negation and observe that we always have op a .
In spite of the fact that is a right adjoint, we always have 1 = 0 which is
usefully construed as a fragment of left exactness of op. Indeed, demorgan's
law, (
)( ), ( ^ ) _ , holds if and only if op is left exact,
op
)( ) and completeness of .
in which case is \geometric" modulo (
We say that satises the innite demorgan law, (
)( ) if and only if
op
preserves all infs, which occurs precisely if has a right adjoint. It is
easy to see that is boolean, (
)( ), if and only if a op. Clearly,
the following implications hold for heyting algebras in any topos:
< < < CCD
H
H
DM L
H EY
H
H
H
a
b
a
b
H EY
H
H
H
I DM
H
B OO
(
B OO
(
H
) =) (
B OO
I DM
) =) (
) =) (
H EY
4
DM
op
)
)
H
In general, they are all strict. To see that (
) =) (
) is strict it
suces to consider a nite heyting algebra in set, , for which we have
( )( ) but not (
)( ). From niteness we have ( )( ) =)
(
)( ). Clearly, 3 = f
g will do. However, in set2 illustrates
that (
) =) ( ) is strict. Once again, ( )() follows immediately
from pointwise considerations, but (
)() fails via the same obstructions
op
as (
)().
But is not a mere locale in its own topos. In showing the equivalence
op
of (
)() and (
)() it is convenient also to establish their equivop
alence with (
)() and (
)(). The next two lemmas are crucial.
(In neither case is assumed to be order-preserving.)
B OO
I DM
H
DM
H
I DM
B OO
H
H
DM
H
< < I DM
DM
DM
I DM
CCD
CCD
B OO
I DM
H EY
Lemma 3 (Higgs) ( : ) =) ( = 1 ).
-
2
Proof. Consult Johnstone 2], exercise 3, p.44.
Lemma 4 (Benabou) ( 1 : ;! ) =) ( ( ) = ^ ( )).
Proof. Let :
: -
-
U-
-
so that
be (a representative of) the subobject classied by
!U
U
?
-
1
?
is a pullback and
-
!U
R
@
U
?
1
;
=;
;
the latter since is classied by 1 and 1 . We must show that
= ( (1 ! -) ^- )
@
R
@
:
5
The subobject classied by ^ is (
and only if
) : 1-
!U
U
-
-
so the result follows if
1
(
?
?
)
(1 ! )- is a pullback this follows easily from the information now assembled.
Corollary 5 ( 1 and ( ) = ) =) ( = 1 )
Theorem 6 (
I DM
)() =) (
B OO
:
)()
Proof. Assume that : ;! has a right adjoint, : ;! . The
counit for the adjunction gives 1 . We have ( ) = since is a right
op
op
adjoint, and thus ( ) = = . It follows from Corollary 5 that = 1
so that is a (split) monomorphism. By Lemma 3 we have op = 1 , hence
= 1 op = op op = op1 = op, and hence is boolean.
Lemma 7 For a (
object of ord(S)
L
CCD
) object of ord(S) and in S,
I
I
L
is a (C C D)
:
Proof. A product of (
) objects is (
) 3], or observe that (
)
is equivalent to saying that, for any family S of down-closed subsets of , we
have
o _
^ n_
j 2 S = (\S )
and this identity is constructively inherited by powers.
CCD
CCD
CCD
L
S
S
Lemma 8 For a ( ) object of ord(S) and an object in ord(S) we
have ord(S)( ) is a (
) object of ord(S)
L
X L
CCD
X
CCD
:
6
Proof. The arrow j j ;! in ord(S) gives rise to ord(S)( ) ;!
ord(S)( j j ) = j j, the inclusion of order preserving arrows in all arrows
D X
D X
L
L
X
X L
X
from to . Since is complete the latter has both left and right adjoints,
given by Kan extensions. Since jX j is (
) the adjoint string exhibits
ord(S)( ) as a ( ) object. See 1, Proposition 10].
X
L
L
L
X L
CCD
CCD
Theorem 9 ((
CCD
),(
CCD
op
)) () (
CCD
op
)()
Proof. We always have (
)(). See 1, Corollary 4]. So the equivalence
op
op
of (
) and (
) trivially implies (
)(). For the converse,
op
op
assume (
)(), that is (
)( ), and (
)( ). We need to show
(
)( op). From
CCD
CCD
CCD
CCD
CCD
CCD
CCD
CCD
L
L
L
-
-
W+ ?
?- D
L
#
application of ( ) : ord (S) ;! ord(S) gives
- #
W ? (D ) = (ord(S)( )) = ord(S)( )
+
?
By Lemma 8, we have (
)(ord(S)( )) and so by 1, Proposition
op
L
op
op
L
11], we have (
CCD
Theorem 10 (
L
co
op
)(
L
op
op
op
CCD
op
L
CCD
op
op
L
L
)
op
L
op
:
:
)() =) (
B OO
)().
Proof.
As noted in 1], a complete object is a locale, (
W
if : D ;! is left exact hence (
) implies (
)( ), if and only
). In particular,
our hypothesis makes op a heyting algebra, the negation for which, a priori
distinct from op, we write as ;( ) : ;! op. Necessarily, ;op ; ( ) 1
and ;op ; ( ) = , giving, by an application of Corollary 5, ;op ; ( ) = 1 .
Thus op, and hence , is boolean.
L
L
L
CCD
LOC
L
LOC
(The implications of Theorems 6 and 10 are trivially reversible.) Combining Theorem 10 with Theorem 9 and Theorem 18 of 1] we have:
7
:
Theorem 11 ( ) () ((
boo
CCD
),(
CCD
op
))
:
References
1] Barry Fawcett and R. J. Wood. Constructive complete distributivity 1.
Math. Proc. Cam. Phil. Soc., 107:81{89, 1990.
2] Peter T. Johnstone. Topos Theory. Academic Press, 1977.
3] Robert Rosebrugh and R. J. Wood. Constructive complete distributivity
3. to appear, 1991.
4] Ross Street and R. F. C. Walters. Yoneda structures on 2-categories.
Journal of Algebra, 50:350{379, 1978.
5] R. J. Wood. Some remarks on total categories. Journal of Algebra,
75:538{545, 1982.
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