Date: Wed, 19 Oct 1994 16:03:58 +0500 (GMT+4:00)
From: categories <cat-dist@mta.ca>
Subject: coherent locales

Date: Wed, 19 Oct 1994 18:08:33 +0100 (MET)
From: Paul Johnson <johnson@king.math.muni.cz>

Dear categories,

In ``Stone spaces'', Johnstone notes that the assumption that
every coherent locale is spatial is equivalent to the
prime ideal theorem for distributive lattices.

Is it generally agreed that the assumption that the spatial
part of every coherent locale has coherent topology is
also equivalent to the above two axioms?

Cheer, PBJ.

Date: Fri, 21 Oct 1994 12:35:02 +0500 (GMT+4:00)
From: categories <cat-dist@mta.ca>
Subject: Re: coherent locales

Date: Thu, 20 Oct 1994 16:46:15 +0100 (MET)
From: Paul Johnson <johnson@king.math.muni.cz>

>
> Date: Wed, 19 Oct 1994 18:08:33 +0100 (MET)
> From: Paul Johnson <johnson@king.math.muni.cz>
>
> Dear categories,
>
> In ``Stone spaces'', Johnstone notes that the assumption that
> (A) every coherent locale is spatial
>
> is equivalent to
>
> (B) the prime ideal theorem for distributive lattices.
>
> Is it generally agreed that the assumption that
>
> (C) the spatial part of every coherent locale has coherent topology
>
> is also equivalent to (A) and (B) above?


In a hopeless reply to  my own querie,
it seems (after one sleepless night)
less and less likely that (C) ==> (B).

Ironically, since the trivial locale is coherent (and spatial),
for a coherent locale A to have a non-coherent spatial part,
it must have a point.

Conversely, axiom (C) still appears quite strong,
and I mean for tangible reasons:
For example, (C) holds iff the passage from coherent spaces
to distributive lattices has a left adjoint in coherent maps,
thus identifying what I would call the
``constructively-spatial'' distributive
lattices as reflective in all.

But might (C) hold without assuming any choice???
Apparently, most of this battle can be fought at the
level of Boolean algebras, but the reductions I have
yet to have sorted out.


Cheers, PBJ.