From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/9542 Path: news.gmane.org!.POSTED!not-for-mail From: Marta Bunge Newsgroups: gmane.science.mathematics.categories Subject: Re: Topos theory for spaces of connected components Date: Thu, 8 Feb 2018 20:04:40 -0500 (EST) Message-ID: References: Reply-To: Marta Bunge NNTP-Posting-Host: blaine.gmane.org Mime-Version: 1.0 Content-Type: text/plain; charset=utf-8 Content-Transfer-Encoding: quoted-printable X-Trace: blaine.gmane.org 1518186242 11147 195.159.176.226 (9 Feb 2018 14:24:02 GMT) X-Complaints-To: usenet@blaine.gmane.org NNTP-Posting-Date: Fri, 9 Feb 2018 14:24:02 +0000 (UTC) Cc: categories@mta.ca, marta.bunge@gmail.com To: Steve Vickers Original-X-From: majordomo@mlist.mta.ca Fri Feb 09 15:23:57 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 1ek9al-0002Q7-Ja for gsmc-categories@m.gmane.org; Fri, 09 Feb 2018 15:23:55 +0100 Original-Received: from mlist.mta.ca ([138.73.1.63]:40171) by smtp2.mta.ca with esmtp (Exim 4.80) (envelope-from ) id 1ek9bw-0004NB-Dj; Fri, 09 Feb 2018 10:25:08 -0400 Original-Received: from majordomo by mlist.mta.ca with local (Exim 4.71) (envelope-from ) id 1ek9az-0003pq-Fq for categories-list@mlist.mta.ca; Fri, 09 Feb 2018 10:24:09 -0400 In-Reply-To: Precedence: bulk Xref: news.gmane.org gmane.science.mathematics.categories:9542 Archived-At: Dear Steve, You wrote: > Topos theory gives a solid account of local connectedness, where each > open - indeed, each sheaf - has a set (discrete space) of connected > components. [...] > > Is there an analogous theory for where the space of connected > components is Stone? ("Connected" is now defined by orthogonality > with respect to Stone spaces instead of discrete spaces.) I have only partial answers to your question.=20 Consider F a bdd S-topos, not necessarily locally connected. There are two = instances of non-discrete localic generalizations of the discrete \Pi_0(F) = of connected components that may be relevant. They were both reported in my= lecture "On two non-discrete localic generalizations of \pi_0=E2=80=9D at = the Colloque Internationale =E2=80=98Charles Ehresmann : 100 ans=E2=80=9D, = Amiens, 2005. An abstract is included in Cahiers de Top.et Geo.Diff.Cat 46-= 3 (2005). A fuller account of my lecture can be found in my Research Gate p= age. It consists of two unrelated parts.=20 The first part (otherwise unpublished) reports my construction of the total= ly (paths) disconnected topos P_0(F) of path components of F by collapsing = paths to a point. It was also the subject matter of a lecture that I gave a= t UNIGE Seminar in 2003, and of another that I gave at the Workshop on the = Ramifications of category Theory, Firenze, 2003.=20 =20 The second part (in collaboration with J. Funk, published as =E2=80=9CQuasi= components in topos theory : the hyperpure-complete spread factorization=E2= =80=9D, Math.. Proc. Camb. Phil. Soc 142. 2007 ) contains a construction of= the zero-dimensional topos \P_0(F) of quasicomponents of F.=20 Both reduce to the usual (discrete) in the case of a locally connected topo= s F.=20 With best regards, Marta ----- Original Message ----- From: "Steve Vickers" To: categories@mta.ca Sent: Sunday, February 4, 2018 5:52:14 AM Subject: categories: Topos theory for spaces of connected components Topos theory gives a solid account of local connectedness, where each open = - indeed, each sheaf - has a set (discrete space) of connected components.= The definition of locally connected geometric morphism covers not only ind= ividual spaces but also bundles, considered fibrewise. It also covers gener= alized spaces as well as ungeneralized. Is there an analogous theory for where the space of connected components is= Stone? ("Connected" is now defined by orthogonality with respect to Stone = spaces 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 connec= ted components more generally for any compact regular space - take the Ston= e space corresponding to the Boolean algebra of clopens. People tend not to= notice the Stone space aspects in the usual examples based on real analys= is, since they are also locally connected. Being a Stone space then just m= akes the set of connected components finite with decidable equality. For an= y compact regular space, we find that each closed subspace has a Stone spac= e of connected components. (By the way, if you wonder what brought me to this, it was from pondering t= he symmetric monad M on Grothendieck toposes. Bunge and Funk proved that fo= r ungeneralized spaces its localic reflection is the lower powerlocale, whi= ch raises the question of whether there is a missing topos construction who= se localic reflection is the upper powerlocale. On the other hand, the symm= etric monad is related to local connectedness. Points of MX are cosheaves o= n X, and X is locally connected if there is a terminal cosheaf in a strong= sense, with that cosheaf providing the sets of connected components. Perha= ps understanding the Stone space view of connected components would cast li= ght on this missing construction.) All the best, Steve. [For admin and other information see: http://www.mta.ca/~cat-dist/ ]