Re: Uniform locales in Shv(X)

[henry@phare.normalesup.org]

Hi,

My point about the entourage approach was that it does not require
overtness at least for the basic definitions:

For example, you can state the axiom "for all entourage a there exists an
entourage c such that c.c <= a$ as" using the following form instead of
the composition of entourage:

pi_12 ^* c (intersection) pi_23^* c <= pi_13^*a

where pi_12, pi_23 and pi_13 denotes the three projection from X^3 to X^2.

the uniformly below relation can also be defined as: a is uniformly below
b if there exist an entourage c such that pi_1^* a (intersection) c <=
pi_2^* b and hence you can state the admissibility axiom without
overtness.

The absence of overtness still yields a lots of problems: for example the
uniformly below relation is no longer interpolative. (and I agree that it
is not clear at all that this is the good way of doing things)

When focusing on overt space everything work properly and one can defines
for example completeness and completion (it is actually a direct
consequence of the results about localic metric space in my thesis) but at
some point it yields other problems related to the fact that subspaces of
uniform spaces are no longer uniform space and this gives examples of
things that should be uniform spaces but which are not overt.


If you are willing to restrict to open maps then using the entourage
approach give a not to awful answer to your initial question: a relative
uniform structure on an open map f:Y ->X, is the given by a collection of
a family of open sublocales of Y \times_X Y which correspond to open
sublocales of the form {y_1,y_2,x | x \in U, (y_1,y_2) \in S(x) } for U an
open subset of X and S a section over U of the sheaf of entourage of Y in
sh(X). You just need to write down all the axioms that these things has to
satisfy but they are going to be relatively natural (and if you are only
interested to a notion of "basis of entourage" they I think they are not
going to be to complicated)

Cheers,
Simon

> Hi Toby,?
>
> Many thanks for your comments. ?
>
> Over the weekend, I noticed a (probable) mistake in my previous posting:
> namely, the assertion that _only_ the admissibility axiom could require
> overtness of the locale  in question. ?In fact, the natural way to define
> composition of entourages (qua relations) is to apply first the inverse
> image of?
> ? 1xDeltax1 : LxLxL ---> LxLxLxL,?
> and then the open image of?
> ? 1x!x1: LxLxL ---> LxL. ?
> Thus the intuitive way of writing down the square-refinement axiom also
> requires L to be overt. ?(Btw, when discussing that axiom  in my previous
> post, I carelessly wrote "epi" where I obviously meant "cofinal".)
> ?Perhaps there is some other way of expressing that axiom, but I no longer
> care: I've reconciled myself to the idea that uniformity requires
> overtness.
>
> I also made some progress on?
>
>>> 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.
>
> Namely, I have an idea of how this might be done, but it requires a detour
> through gauge spaces  (=uniformities defined via families of
> pseudometrics, for those not familiar with the terminology) which I'm not
> sure is constructively valid. ?In any case, I like this idea: in
> particular, if it works, then one could apply it to the case where L is a
> discrete locale and investigate to what extent the idea of a uniform space
> as a _set_ together with a family of entourages fails to capture the
> general idea. ?
>
> Cheers,
> Jeff.



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