From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/9541 Path: news.gmane.org!.POSTED!not-for-mail From: Matias M Newsgroups: gmane.science.mathematics.categories Subject: Re: Topos theory for spaces of connected components Date: Wed, 7 Feb 2018 21:34:46 -0300 Message-ID: Reply-To: Matias M 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 1518124122 23746 195.159.176.226 (8 Feb 2018 21:08:42 GMT) X-Complaints-To: usenet@blaine.gmane.org NNTP-Posting-Date: Thu, 8 Feb 2018 21:08:42 +0000 (UTC) To: categories@mta.ca Original-X-From: majordomo@mlist.mta.ca Thu Feb 08 22:08:38 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 1ejtQn-0005UC-9U for gsmc-categories@m.gmane.org; Thu, 08 Feb 2018 22:08:33 +0100 Original-Received: from mlist.mta.ca ([138.73.1.63]:39812) by smtp2.mta.ca with esmtp (Exim 4.80) (envelope-from ) id 1ejtRZ-0008UF-Ji; Thu, 08 Feb 2018 17:09:21 -0400 Original-Received: from majordomo by mlist.mta.ca with local (Exim 4.71) (envelope-from ) id 1ejtQd-0008N3-7k for categories-list@mlist.mta.ca; Thu, 08 Feb 2018 17:08:23 -0400 Precedence: bulk Xref: news.gmane.org gmane.science.mathematics.categories:9541 Archived-At: Dear colleagues, I have some information that may be relevant to the thread started by Steve Vickers. (Details may be found in my article in the recent Freyd-Lawvere issue of the Tbilisi journal.) Steve Vickers escribi=C3=B3: > 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.) Let p:E ---> S be a hyperconnected and local geometric morphism. (The intuition is that E is a topos of spaces and that the inverse image p^* : S ---> E is the full subcategory of discrete spaces.) A construction suggested by Lawvere produces a finite-product preserving and idempotent monad pizero : E ---> E which, I think, is relevant to Steve's question. Indeed, the paper mentioned above gives evidence to support the intuition that: 1) pizero assigns, to each space, its associated space of connected components, and 2) the full subcategory of E given by the pizero-algebras is the subcategory of totally separated spaces. Let me repeat some of that evidence here. If p : E ---> S is, moreover, locally connected then pizero =3D p^* p_! : E ---> E; that is, pizero X is the discrete space of connected components of X. In other words, if p is lc then the pizero construction produces essentially the left adjoint to p^*. A motivating example that is not locally connected is Johnstone's topological topos p: J ---> Sets. For each X in J, pizero X is the totally separated space of `quasi-components' of X. The pizero-algebras are exactly the totally separated sequential spaces. (The construction works in categories that need not be toposes so, for instance, it gives the `correct' result in the case of compactly generated Hausdorff spaces.) Of course, the inclusion of pizero-algebras into E has a finite-product preserving left adjoint. George's mail suggests the question if this reflection is semi-left-exact. It also raises the question if the explicit construction that George gives of the left adjoint to The inclusion functor > 0-Dimensional locales--->Locales is the result of a variant of Bill's construction (using an exponentiating object and a `good' factorization system). I must admit that I don't know how the above connects with the work of Bunge-Carboni-Funk, but Marta mentions the double exponentiation O^O^X (even if X not necessarily exponentiable) > where O is the Sierpinski locael. and that already suggests a connection. Best regards, Mat=C3=ADas. [For admin and other information see: http://www.mta.ca/~cat-dist/ ]