Subject: Perfect double negation

From: categories <[email-masked]>
Date: Thu, 22 Jan 1998 16:40:56 -0400 (AST)

Date: Thu, 22 Jan 1998 17:22:17 GMT
From: Martin Escardo <[email-masked]>

Dear toposophers,

Every sober space has a smallest dense perfect subspace.  In many
cases this is precisely the subspace of maximal points in the
specialization order. In general it is larger than that.

This is approached via locales. Every locale A has a smallest dense
perfect sublocale, which is spatial if A is. It is obtained by the
perfect coreflection of the double negation nucleus. 

The construction seems to generalize from locales to toposes. But what
does it produce (instead of subspaces of maximal points)? And what is
its logical content? Could toposes of sheaves help one to understand
this?

Thanks in advance for any clue. Details follow.

Martin Escardo
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Every finite T_0 space has a smallest dense subspace, namely its
subspace of maximal points in the specialization order. And every
locale has a smallest dense sublocale, induced by the double negation
nucleus, as it is well known. But it is also well known that this is
hardly ever spatial, even if the given locale is. I have considered a
modification of this construction.

Let's say that a nucleus on a locale is perfect if it preserves
directed joins. If we say that a perfect map is a continuous map
f:A->B such that the right adjoint f_*:O(A)->O(B) of the frame map
f^*:O(B)->O(A) preserves directed joins, then a nucleus is perfect iff
it is induced by a perfect map. Let's also say that a sublocale is
perfect if the inclusion is perfect.

It turns out that every locale has a smallest dense perfect sublocale.
Classically, one can show that spatial locales are closed under
perfect sublocales.  Hence every sober space has a smallest dense
perfect sober subspace. This goes as follows.

One knows that the set NA of nuclei on a locale A, with the pointwise
ordering, is a frame. Denote by FA the set of perfect nuclei. Then one
can show that FA is a subframe of NA. (The join of a set of perfect
nuclei is computed by taking the pointwise (directed) join of the
finite compositions of the given nuclei.)  [[Digression: F is
functorial on the category of compact and stably locally compact
locales with perfect maps. For such a locale A, the locale FA is
compact regular, and if A is compact regular then FA=A. For example,
if A is the topology of lower semicontinuity (=Scott topology) on the
unit interval, then FA is the Euclidean topology on the unit
interval. But this is another story.]]

Thus, there is a coreflective adjunction between FA and NA, which in
one way includes FA into NA and in the other assigns to each nucleus
in NA the join of the perfect nuclei below it. (If A is compact and
stably locally compact then there is a simple formula for the perfect
coreflection of nuclei on A.)

Thus the smallest dense perfect sublocale is induced by the perfect
coreflection of the double negation nucleus. But what is it?  Let's
call it the support of A and denote it by Supp A. The support always
contains the subspace of maximal points (oh, I should say Max Pt A is
included in Pt Supp A---but I'll refer to a locale as space for
terminological simplicity).

In many cases it consists exactly of the maximal points. For instance,
if A is compact and stably locally compact then this is the case iff
the subspace of maximal points is compact, and in this case Supp A is
a compact regular locale. (I observe at this point that compact stably
locally compact locales are closed under perfect sublocales). Some
examples and counterexamples can be useful: (i) Let A be the Scott
topology induced by the prefix order on finite and infinite sequences
over {0,1}. Then Supp A is the topology of Cantor space. (ii) Same but
sequences over natural numbers. Then Supp A = A, and not Baire space
as one could expect from the previous example, because Baire space
fails badly to be locally compact. (iii) Let A be the Scott topology
on compact real intervals ordered by reverse inclusion. Then Supp A is
the topology of the Euclidean real line. (iv) The previous example
doesn't fit in the above characterization as it is not compact. If we
add an artificial bottom interval, then we get a compact stably
locally compact locale. But Supp A is then the topology of the
Euclidean real line with a bottom element in the specialization
order---bottom doesn't get removed in this case. (v) Let UA be the
upper power locale of a compact regular locale A. Then Supp UA = A.
This again fits in the above characterization. 

(The above claims [[except the ones in double brackets]] are proved in
the paper "Properly injective spaces and functions spaces", which
should have been called "Perfectly injective spaces and function
spaces". It is going to appear soon. Meanwhile it is available at
http://www.dcs.ed.ac.uk/home/mhe/pub/papers/injective.ps.gz)

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Some questions to finish:
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Perfect double negation on the topology of a space removes the partial
points of the space in many cases. What does it do to toposes?  Any
clue is very much appreciated.

In another direction, double negation takes us from intuitionistic
logic to classical logic. Is that behind any correlation
intuitionistic logic<->partiality, classical logic<->totality? Can we
make this precise by taking toposes of sheaves?

The inclusion s:A->Supp A, being a continuous map, induces a geometric
morphism S=Shv(s):Shv(A)->Shv(Supp A). Since s_* preserves directed
joins, one could guess that S_* preserves directed colimits. Would
that be the case? Have these "perfect" geometric morphisms been
studied?