From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/9527 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: Sun, 4 Feb 2018 16:48:44 +0000 Message-ID: References: Reply-To: Marta Bunge NNTP-Posting-Host: blaine.gmane.org Mime-Version: 1.0 Content-Type: text/plain; charset="Windows-1252" Content-Transfer-Encoding: quoted-printable X-Trace: blaine.gmane.org 1517843090 21515 195.159.176.226 (5 Feb 2018 15:04:50 GMT) X-Complaints-To: usenet@blaine.gmane.org NNTP-Posting-Date: Mon, 5 Feb 2018 15:04:50 +0000 (UTC) Cc: "s.j.vickers@cs.bham.ac.uk" To: "categories@mta.ca" Original-X-From: majordomo@mlist.mta.ca Mon Feb 05 16:04:46 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 1eiiJx-0004k0-43 for gsmc-categories@m.gmane.org; Mon, 05 Feb 2018 16:04:37 +0100 Original-Received: from mlist.mta.ca ([138.73.1.63]:37610) by smtp2.mta.ca with esmtp (Exim 4.80) (envelope-from ) id 1eiiKh-00026o-QV; Mon, 05 Feb 2018 11:05:23 -0400 Original-Received: from majordomo by mlist.mta.ca with local (Exim 4.71) (envelope-from ) id 1eiiJo-0008Bx-2f for categories-list@mlist.mta.ca; Mon, 05 Feb 2018 11:04:28 -0400 Thread-Topic: categories: Topos theory for spaces of connected components Thread-Index: AQHTncoZFhYXdrMm3EqU8NoMLCww3aOUcddV In-Reply-To: Accept-Language: en-CA, en-US Content-Language: en-CA Precedence: bulk Xref: news.gmane.org gmane.science.mathematics.categories:9527 Archived-At: Dear Steve, I have nothing to say about your Stone spaces question in general, except f= or your remarks in the second part of your message about the symmetric mona= d M, where you suggest that the Stone locale view of connected components w= ould perhaps cast light on the missing construction of a topos version N of= the upper power locale P_U, just as the symmetric monad M is a topos versi= on of the lower power locale P_L. In my paper =93Pitts monads and a lax descent theorem=94 (2015), (Remark 7.= 6), I leave it as an open question (more or less) the construction of such = an N. [ The name =93Pitts monad=94 I gave on account on a condition which f= irst appears in a theorem of A.M. Pitts whereby, in a lax pullback with bot= tom map an S-essential geometric morphisms, the top map is locally connecte= d. The S-essential geometric morphisms are precisely the M-maps, and for th= e lower power locale monad P_L, the P_L-maps are the open maps. ] However, toposes are more complicated than locales and a perfect analogue m= ay not be what one should seek Indeed, one can view the symmetric monad M (= classifier of distributions on toposes X, or equivalently of complete spre= ads over X with a locally connected domain) as a topos version of the lower= power locale P_L. There is however another such candidate, which is the b= agdomain monad B_L (classifier of bags of points, or equivalently of branch= ed coverings over X, namely of those complete spreads that are purely loca= lly equivalent to a locally constant cover). See M. Bunge and J. Funk, Sing= ular Coverings of Toposes (2006), (Def. 9.32). In the same source SCT ( 8.3= ) there is a diagram which shows that there are two factorizations of the u= nit X=97> M(X), namely one through the unit X=97> B_L(X) and the other thro= ugh the unit X=97> T(X) where T (classifier of probability distributions, t= hat is of distributions on X which preserve the terminal object, equivalent= ly of complete spreads over X whose domains are locally connected and have = totally connected components, the latter meaning that the connected compone= nts functor preserves pullbacks). In particular, M(X) is equivalent to B_L(= T(X)). It is therefore of interest (to me at least) to find, not just the N that I= mentioned above, but also a monad B_U, as both would presumably be topos v= ersions of the upper power locale monad P_U. In addition, it is of interest= (to me at least) to find versions of a "single universe=94, by which I mea= n an analogue to the double power locale monad P, which as you and C. Towns= end have shown, is such that P(X) for X a locale, can be viewed either as a= composite in either direction of P_L and P_U applied to X, or as equival= ent to the double exponentiation O^O^X (even if X not necessarily exponenti= able) where O is the Sierpinski locael. For O =3D the objects classifier in Top_S, the double exponential is in fac= t relevant already in my first (Algebra Universalis 1995) paper where I con= struct the symmetric topos by forcing methods, in that distributions on X c= an be seen as carved out of O^O^X (suitably interpreted via points). Simila= rly, an =93upper=94 version N of M can be constructed as the classifier N = of local homomorphisms over toposes. The question then in my view is now ho= w to deal with the =93upper=94 version B_U of B_L. The analogues semiopen-o= pen versus perfect-proper (or tidy-relatively tidy) are of course relevant = to this and constitutes work in progress. Best regards, Marta ________________________________ From: Steve Vickers Sent: February 4, 2018 5:52 AM To: categories@mta.ca 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/ ]