From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/5015 Path: news.gmane.org!not-for-mail From: Steve Vickers Newsgroups: gmane.science.mathematics.categories Subject: Re: Triquotient assignments for geometric morphisms Date: Tue, 23 Jun 2009 11:40:49 +0100 Message-ID: Reply-To: Steve Vickers NNTP-Posting-Host: lo.gmane.org Mime-Version: 1.0 Content-Type: text/plain; charset=ISO-8859-1 Content-Transfer-Encoding: 7bit X-Trace: ger.gmane.org 1245763943 26351 80.91.229.12 (23 Jun 2009 13:32:23 GMT) X-Complaints-To: usenet@ger.gmane.org NNTP-Posting-Date: Tue, 23 Jun 2009 13:32:23 +0000 (UTC) To: Barney Hilken , Original-X-From: categories@mta.ca Tue Jun 23 15:32:19 2009 Return-path: Envelope-to: gsmc-categories@m.gmane.org Original-Received: from mailserv.mta.ca ([138.73.1.1]) by lo.gmane.org with esmtp (Exim 4.50) id 1MJ66i-0004JM-KB for gsmc-categories@m.gmane.org; Tue, 23 Jun 2009 15:32:16 +0200 Original-Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1MJ5Sf-00061o-Rp for categories-list@mta.ca; Tue, 23 Jun 2009 09:50:53 -0300 Original-Sender: categories@mta.ca Precedence: bulk Xref: news.gmane.org gmane.science.mathematics.categories:5015 Archived-At: Dear Barney, The weak triquotient assignments go along with the double powerlocale monad PP, since a weak triquotient assignment for f: X -> Y is a map g: Y -> PP(X) satisfying certain conditions that relate to the strength of PP. I believe Townsend has published some work on this. This is similar to how open maps go along with the lower powerlocale P_L (see my "Locales are not pointless"), though with P_L it is made tighter using an adjunction that is not available in the PP case, so the open analogue of triquotient assignment, the map from Y to P_L(X), is characterized uniquely. For open maps we know of a trivial generalization to toposes: a geometric morphism is open if its localic part is an open locale map. You could probably play the same trick with triquotient assignments, and then I think your stability property follows from stability of the hyperconnected-localic factorization. However, there is also a more interesting generalization in the case of open maps, got by generalizing P_L to the symmetric topos construction M. (This is described in the Elephant, but also, in much more detail, in the Bunge-Funk book "Singular coverings of toposes". See also my paper "Cosheaves and connectedness in formal topology".) In fact, Bunge and Funk have proved that for a locale X, P_L(X) is the localic reflection of M(X). The relationship between P_L and open maps transfers to one between M and locally connected geometric morphisms. Since PP is the composite of (commuting) monads P_U and P_L, where P_U is the upper powerlocale, one natural approach to a topos generalization would be try also to generalize P_U. This generalization seems to be missing in our current state of knowledge, though I've had some thoughts about it and firmly believe that it exists. Regards, Steve. Barney Hilken wrote: > Has anyone generalised the theory of (weak) triquotient assignments > from locale maps to geometric morphisms? In particular, does the > pullback (assuming boundedness) of a geometric morphism with a > triquotient assignment have a unique triquotient assignment satisfying > the Beck-Chevalley condition? > > Also, if f:X->Y is a continuous function between topological spaces, > are there any reasonable conditions (other than openness) under which > the interior of the direct image along f is a weak triquotient > assignment for the inverse image map? > > Thanks, > > Barney. > [For admin and other information see: http://www.mta.ca/~cat-dist/ ]