aspects of locale theory

aspects of locale theory

[Paul Taylor]

Steve Vickers has bounced me into replying to the questions about
locales that Rafael Borowiecki (aka Hasse Riemann) asked, even
though I had said that I would do so when I was ready with some
other comments on that subject.   The following also includes the
answer to a question that I myself raised last year, which has
further consequences for the appropriateness of locale theory
as an account of general topology, but I am not going to say what
these are until I am actually ready.

In my "autobiography", I had said,

> Another [problem] is how to embed the category of locales in a CCC
> WITHOUT using illegitimate presheaves (Vickers and Townsend) or
> the axiom of collection (Heckmann).  When I wrote the original version
> of this posting a couple of weeks back, I thought I could solve this
> one.  I am still hopeful, but it turns out to be a powerful question,
> cf Ronnie's (2) above.

Rafael asked me,
> Why should presheaves be illegitimate?

I am not saying that presheaves in general are illegitimate in the
plain English sense of the word.   The work that I was referring
to uses the category of all functors from the opposite of the
category of locales to the category of sets.   Since the category
of locales is "large", this functor-category is super-large or
"illegitimate", where the quoted words have a technical meaning.

As Steve Vickers has already explained, he goes to some trouble
to avoid the problems, essentially by exploring only a tiny part
of the presheaf category.  He really only uses the exponentials
Sigma^X of locales, which are the functors  Loc(-xX,Sigma).

The rest of the comments concern the paper
   @article{HeckmannR:carcec,
   title={A Cartesian Closed Extension of the Category of Locales},
   author={Heckmann, Reinhold},
   journal={Mathematical Structures in Computer Science},
   year={2006}, volume={16},  pages={231--253}}
which was inspired by Dana Scott's equilogical space construction.

However, Reinhold's "equivalence relations" are what the presheaf
category provides.  So a relation on a single locale requires data
from every object in the category.  (The details are rather
complicated, and I don't recall them exactly at the moment.)
The collection of morphisms between two equilocales is defined
as the image of a class in a set.

> Then, i suppose the axiom of collection is valid at least in the CCC.

No.  The axiom of collection says roughly that, given a function
from a class to a set, its image is a set.   This is quite a
strong axiom of set theory, and is certainly not valid in something
as logically weak as a CCC.

> But what is so bad about the axiom of collection in this case?

The vast majority of ordinary mathematics can be done in an elementary
topos with a natural number object, maybe together with assumptions
of excluded middle or the axiom of choice.   This is roughly but
not quite equivalent to Zermelo's set theory -- NOT ZFC, which adds
the substantially more powerful axiom-scheme of replacement.

Rather than using sledgehammers (adding more and more powerful axioms),
most categorists would prefer to re-examine the problem to look
for more delicate ways of doing things.

On a different aspect of locale theory, I asked on 22 July 2008,

> Where can I find a published proof that
>     if    X --->> Y   is a (not necessarily regular)  epi  of locales
>     then so is   X x Z --->> Y x Z   for any locale Z?
> NB  (I know that) this is not true for general pullbacks of locales!

Peter Johnstone told me, essentially, that I was assuming excluded
middle, in the form that every locale is open (as he would say) or
overt (using my word).  In fact, the answer is negative even with
excluded middle, as Till Plewe pointed out to me.   (Till no longer
studies categories or locales, but is still doing academic research
in Japan, or at least was last year when I was in touch with him.)

The (basis of the) counterexample that Till pointed out is described
in Peter's book (Stone Spaces) in section II 2.14.   It is the
locale QE of rationals with the Euclidean topology, that is, the
frame of open subsets of the reals, quotiented by their effect on
the rationals, so for example  (3,pi)v(pi,4) = (3,4) in QE since
pi is irrational.   I also write QD for the rationals with the
discrete topology, so QD is homeomorphic to N.

The point is that QE has enough points, indeed  QD-->>QE is epi,
but QExQE doesn't.

This means that   QDxQE --> QExQE  is not epi.

Paul



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