From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/9526 Path: news.gmane.org!.POSTED!not-for-mail From: Steve Vickers Newsgroups: gmane.science.mathematics.categories Subject: Topos theory for spaces of connected components Date: Sun, 4 Feb 2018 10:52:14 +0000 Message-ID: Reply-To: Steve Vickers NNTP-Posting-Host: blaine.gmane.org Mime-Version: 1.0 (1.0) Content-Type: text/plain; charset=us-ascii Content-Transfer-Encoding: quoted-printable X-Trace: blaine.gmane.org 1517756866 23867 195.159.176.226 (4 Feb 2018 15:07:46 GMT) X-Complaints-To: usenet@blaine.gmane.org NNTP-Posting-Date: Sun, 4 Feb 2018 15:07:46 +0000 (UTC) To: categories@mta.ca Original-X-From: majordomo@mlist.mta.ca Sun Feb 04 16:07:42 2018 Return-path: Envelope-to: gsmc-categories@m.gmane.org Original-Received: from smtp2.mta.ca ([198.164.44.40]) by blaine.gmane.org with esmtp (Exim 4.84_2) (envelope-from ) id 1eiLtF-0005Lv-N7 for gsmc-categories@m.gmane.org; Sun, 04 Feb 2018 16:07:33 +0100 Original-Received: from mlist.mta.ca ([138.73.1.63]:37413) by smtp2.mta.ca with esmtp (Exim 4.80) (envelope-from ) id 1eiLuY-0006iI-6T; Sun, 04 Feb 2018 11:08:54 -0400 Original-Received: from majordomo by mlist.mta.ca with local (Exim 4.71) (envelope-from ) id 1eiLtf-0003GL-AP for categories-list@mlist.mta.ca; Sun, 04 Feb 2018 11:07:59 -0400 Precedence: bulk Xref: news.gmane.org gmane.science.mathematics.categories:9526 Archived-At: Topos theory gives a solid account of local connectedness, where each open -= indeed, each sheaf - has a set (discrete space) of connected components. Th= e definition of locally connected geometric morphism covers not only individ= ual spaces but also bundles, considered fibrewise. It also covers generalize= d spaces as well as ungeneralized. Is there an analogous theory for where the space of connected components is S= tone? ("Connected" is now defined by orthogonality with respect to Stone spa= ces instead of discrete spaces.) The obvious example is any Stone space X, for instance, Cantor space, where X= is its own space of connected components. We get Stone spaces of connected c= omponents more generally for any compact regular space - take the Stone spac= e corresponding to the Boolean algebra of clopens. People tend not to notice= the Stone space aspects in the usual examples based on real analysis, since= they are also locally connected. Being a Stone space then just makes the se= t of connected components finite with decidable equality. For any compact re= gular space, we find that each closed subspace has a Stone space of connecte= d components. (By the way, if you wonder what brought me to this, it was from pondering th= e symmetric monad M on Grothendieck toposes. Bunge and Funk proved that for u= ngeneralized spaces its localic reflection is the lower powerlocale, which r= aises the question of whether there is a missing topos construction whose lo= calic reflection is the upper powerlocale. On the other hand, the symmetric m= onad is related to local connectedness. Points of MX are cosheaves on X, and= X is locally connected if there is a terminal cosheaf in a strong sense, wi= th that cosheaf providing the sets of connected components. Perhaps understa= nding the Stone space view of connected components would cast light on this m= issing construction.) All the best, Steve.= [For admin and other information see: http://www.mta.ca/~cat-dist/ ]