From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/8302 Path: news.gmane.org!not-for-mail From: henry@phare.normalesup.org Newsgroups: gmane.science.mathematics.categories Subject: Re: Uniform locales in Shv(X) Date: Mon, 25 Aug 2014 22:24:54 +0200 Message-ID: References: Reply-To: henry@phare.normalesup.org NNTP-Posting-Host: plane.gmane.org Mime-Version: 1.0 Content-Type: text/plain;charset=iso-8859-1 Content-Transfer-Encoding: 8bit X-Trace: ger.gmane.org 1409064956 30342 80.91.229.3 (26 Aug 2014 14:55:56 GMT) X-Complaints-To: usenet@ger.gmane.org NNTP-Posting-Date: Tue, 26 Aug 2014 14:55:56 +0000 (UTC) Cc: categories@mta.ca To: "Jeff Egger" Original-X-From: majordomo@mlist.mta.ca Tue Aug 26 16:55:51 2014 Return-path: Envelope-to: gsmc-categories@m.gmane.org Original-Received: from smtp3.mta.ca ([138.73.1.127]) by plane.gmane.org with esmtp (Exim 4.69) (envelope-from ) id 1XMIA2-0005L3-Pq for gsmc-categories@m.gmane.org; Tue, 26 Aug 2014 16:55:50 +0200 Original-Received: from mlist.mta.ca ([138.73.1.63]:57719) by smtp3.mta.ca with esmtp (Exim 4.80) (envelope-from ) id 1XMI9U-0005ah-Uj; Tue, 26 Aug 2014 11:55:16 -0300 Original-Received: from majordomo by mlist.mta.ca with local (Exim 4.71) (envelope-from ) id 1XMI9T-0002Ti-W8 for categories-list@mlist.mta.ca; Tue, 26 Aug 2014 11:55:16 -0300 In-Reply-To: Precedence: bulk Xref: news.gmane.org gmane.science.mathematics.categories:8302 Archived-At: 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. [For admin and other information see: http://www.mta.ca/~cat-dist/ ]