From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/5016 Path: news.gmane.org!not-for-mail From: "Townsend, Christopher" Newsgroups: gmane.science.mathematics.categories Subject: RE: Triquotient assignments for geometric morphisms Date: Tue, 23 Jun 2009 08:27:41 +0100 Message-ID: Reply-To: "Townsend, Christopher" NNTP-Posting-Host: lo.gmane.org Mime-Version: 1.0 Content-Type: text/plain; charset="iso-8859-1" Content-Transfer-Encoding: quoted-printable X-Trace: ger.gmane.org 1245763983 26469 80.91.229.12 (23 Jun 2009 13:33:03 GMT) X-Complaints-To: usenet@ger.gmane.org NNTP-Posting-Date: Tue, 23 Jun 2009 13:33:03 +0000 (UTC) To: "Barney Hilken" , Original-X-From: categories@mta.ca Tue Jun 23 15:33:01 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 1MJ67M-0004b9-Tf for gsmc-categories@m.gmane.org; Tue, 23 Jun 2009 15:32:57 +0200 Original-Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1MJ5PO-0005gL-7n for categories-list@mta.ca; Tue, 23 Jun 2009 09:47:30 -0300 Content-class: urn:content-classes:message Original-Sender: categories@mta.ca Precedence: bulk Xref: news.gmane.org gmane.science.mathematics.categories:5016 Archived-At: Barney As far as I am aware, no generalisation of localic triquotient assignments = to geometric morphisms has been developed. That's not for want of trying on= my part! A na=EFve approach would be to define a weak triquotient assignment on a ge= ometric morphism f:F->E to be a filtered colimit preserving functor that is= required to interact with the inverse image of f in a manner that mimics t= he localic case. For this approach to work in a way that is similar to what= happens for locales we would need to have a similar way of characterising = such filtered colimit preserving functors which, as far as I am aware, is n= ot available (essentially due to the technical difficulty that sheafificati= on is 'two step' for toposes, but only 'one step' for locales). The technic= al problems here are, in my mind, the same as the more well known problems = associated with constructing an upper power topos.=20 My current view on how to solve this problem is to use localic representati= ons of geometric morphisms. This requires us to re-state the theory of geom= etric morphisms as adjunctions between categories of locales and to develop= 'topos theory' relative to these adjunctions. For example 'Grothendick top= os' becomes 'category of localic diagrams of a localic groupoid' in this pa= radigm. The lower power topos construction should guide us to see how its a= ction effects (the category of actions of) localic groupoids; then by upper= /lower symmetry we know what the 'upper' case should be (and hence to weak = triquotient assignments on geometric morphisms). Unfortunately getting the = symmetry to work even at the much simpler level of bounded geometric morphi= sms is proving a headache.=20 If you would like any further detail, please feel free to get in touch.=20 Regards, Christopher=20 -----Original Message----- From: categories@mta.ca [mailto:categories@mta.ca] On Behalf Of Barney Hilk= en Sent: 22 June 2009 16:49 To: categories Subject: categories: Triquotient assignments for geometric morphisms 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? 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