Re: Uniform locales in Shv(X)

[Jeff Egger <jeffegger@yahoo.ca>]

Dear Simon,

I was always told that one of the advantages of the "Tukey-style" 
definition (in terms of uniform covers) was that it would be easier to 
generalise toposes than the definition in terms of entourages. And yet, my 
(admittedly cursory) search for this promised generalisation hasn't turned 
up anything yet.

On the other hand, looking at the work of Jorge Picado et al., it seems to 
me---and here, I think, you and I are in agreement---that the only axiom 
which poses any problem is the "admissibility axiom". Specifically, for 
any locale L (internal to any topos E), one can define an object of Weil 
entourages for L: it comprises those elements of the frame underlying LxL 
such that (the nucleus corresponding to) the corresponding open sublocale 
contains (the nucleus corresponding to) the diagonal sublocale. It is 
moreover clear that this object is a meet-semilattice, so it makes sense 
to speak of filters in it---i.e., upward-closed sub-meet-semilattices. A 
Weil uniformity is a filter in this sense satisfying three further axioms: 
square-refinement (which asserts that a certain endomorphism of the filter 
is epi), symmetry (which says the filter is also closed under a further 
unary operation), and the one problematic axiom, "admissibility"; but I 
wonder how important this axiom really is? For locales with enough points, 
it exists to ensure that the topology derived from the uniformity is not 
coarser than the original. Perhaps, if we define a Weil pseudo-uniformity 
on L to be exactly as above minus the admissibility axiom, it is 
possible---at least in the case of a localic topos E=Shv(X)---to describe 
a second locale underlying the pseudouniformity? If so, then we don't 
really need formulate the admissibility axiom in the internal language of 
the topos.

Cheers,
Jeff.



--------------------------------------------
On Thu, 8/21/14, henry@phare.normalesup.org <henry@phare.normalesup.org> wrote:

  Subject: Re: categories: Uniform locales in Shv(X)
  To: "Jeff Egger" <jeffegger@yahoo.ca>
  Cc: categories@mta.ca
  Received: Thursday, August 21, 2014, 7:51 AM

  Dear Jeff,

  I think the problem is that
  (as far as I know) no one has developed a nice
  constructive theory of uniform locale. If
  I'm mistaken about that I would
  be
  really happy to know more about it.

  I have done in my thesis (in chapitre 3 section
  3,
  https://www.imj-prg.fr/~simon.henry/Thesis.pdf)
  the case of constructive
  metric locale,
  under the assumption that the map $Y \rightarrow X$ is
  open. Removing this openness hypothesis seems
  difficult but I stil have
  hope that this is
  possible, and I have a few idea about it...


  For the
  general case it seems to me that the only definition that
  have a
  chance to work properly without the
  openess condition is the definition by
  entourage and this one should be easy to
  externalise: any local section of
  the sheaf
  of entourage will corresponds to an open sublocale of Y
  \times_X
  Y, (containing the restriction of
  the diagonal to the open sublocale of
  $X$ on
  which it is defined ), and it should not be to hard to give
  the
  axioms that these have to satisfy... is
  this the kind of things your are
  looking for
  ?


  Best
  wishes,
  Simon Henry



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