From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/9538 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: Tue, 6 Feb 2018 07:37:04 -0500 (EST) Message-ID: References: <244986425.357598.1517854022932.JavaMail.zimbra@math.mcgill.ca> 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 1517922936 32215 195.159.176.226 (6 Feb 2018 13:15:36 GMT) X-Complaints-To: usenet@blaine.gmane.org NNTP-Posting-Date: Tue, 6 Feb 2018 13:15:36 +0000 (UTC) Cc: Marta Bunge , marta.bunge@gmail.com To: Steve Vickers , categories@mta.ca Original-X-From: majordomo@mlist.mta.ca Tue Feb 06 14:15:32 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 1ej35i-0007Hv-2e for gsmc-categories@m.gmane.org; Tue, 06 Feb 2018 14:15:18 +0100 Original-Received: from mlist.mta.ca ([138.73.1.63]:38271) by smtp2.mta.ca with esmtp (Exim 4.80) (envelope-from ) id 1ej37a-0001Ci-6X; Tue, 06 Feb 2018 09:17:14 -0400 Original-Received: from majordomo by mlist.mta.ca with local (Exim 4.71) (envelope-from ) id 1ej36f-00053X-HD for categories-list@mlist.mta.ca; Tue, 06 Feb 2018 09:16:17 -0400 Thread-Topic: categories: Topos theory for spaces of connected components Thread-Index: akloa9OFFa/dWMHYiMcln0JJI9gdQQ== Precedence: bulk Xref: news.gmane.org gmane.science.mathematics.categories:9538 Archived-At: Dear Steve, Curiously enough, what you describe about how to get a point of MX (a coshe= af, or distribution) is precisely what did in 1990. I communicated it priv= ately to Lawvere at Como 1990, as he was the one who had left the question = as an open one. My method to do that was that of "forcing topologies" (Tier= ney)and not unlike what is done in the Joyal-Tierney paper except for forge= tting the lex part. Unfortunately, when I sent a paper "Cosheaves and distr= ibutions on toposes" containing this result to Peter Freyd for the JPAA, wh= oever refereed it rejected it without giving a reason, but I know that it w= as my fault as the paper was not too clearly written. I subsequently sent i= t to Algebra Universalis where it did appear (same title and alas, still no= t too clearly written) but very much later (1995). That sabbatical year 1995-96 I was spent at the Universita di Genova where = (not coincidentally) Aurelio (Carboni) had just taken a job away from Milan= . Aurelio immediately understood my construction, but thought that it would= be "wiser" to set it in algebraic terms. This resulted in our joint paper = "The symmetric topos", which this time it did appear in JPAA (1995). We did= more than that in that paper, namely to extend it to a KZ-monad and charac= terize its algebras. I therefore abandoned the fibrational point of view wh= ich, as you say, found its way again in my work with Jonathon (Funk) as the= complete spreads with a locally connected domain. However, just this morni= ng (and before reading what you just wrote) I was thinking of using the fib= rational approach again for constructing the topos NX corresponding to the = upper power locale, just as the topos MX corresponds to the lower power loc= ale.=20 It seems to me now that modulo some differences this is what you are trying= to do yourself. Of course it would be "wiser" as Aurelio would have said, = to do it "algebraically" and this is what I did for the coherent monad in m= y "Pitts paper" (2015). So I am pursuing that line as well. I will read the= rest of your message later and maybe respond to your question about it pri= vately. It may take a few days as other things are interfering with my work= at present.=20 Many thanks for your remarks.=20 All the best, Marta =20 ----- Original Message ----- From: "Steve Vickers" To: bunge@math.mcgill.ca Cc: categories@mta.ca Sent: Tuesday, February 6, 2018 4:19:50 AM Subject: Re: categories: Topos theory for spaces of connected components Dear Marta, Here's my thinking on connected components. For M, the paradigm example for how to get a point of MX (a cosheaf, or dis= tribution) is to take locally connected space Y with map p: Y -> X, and the= n to each sheaf U over X assign the set of connected components of p*U. Thi= s gives a covariant functor from SX to Set, and it preserves colimits. If X= is an ungeneralized space, then it suffices to do that for opens U, and th= e extension to sheaves follows. Your theory of complete spreads shows that = that paradigm example is in fact general. The extreme case of p is when X is itself locally connected and we can take= p to be the identity. The corresponding cosheaf is terminal in a strong se= nse: as global point of MX it provides a right adjoint to the map MX -> 1. = The unit of the adjunction provides a unique morphism from the generic cosh= eaf to the terminal one. If X is exponentiable, then (always? In favourable cases?) the cosheaf as d= escribed above can be got by taking points for a map R^X -> R, where R is (= following your notation) the object classifier. This points out Lawvere's a= nalogy with integration, where R would be the real line. Then just as Riesz= picks out the linear functionals as the distributions, we are interested i= n the colimit-preserving ones. In the above account, the role of local connectedness is to ensure that the= connected components genuinely do form a set, a discrete space. What happe= ns if we look for a Stone space instead? Here is my conjecture. 1. For ungeneralized X we should be looking for a Stone space of connected = components of p*U for each _closed_ U. Y will need a suitable condition (st= rongly compact?) as analogue of local compactness. (By Stone duality that c= ould also be expressed by assigning (covariantly) a Boolean algebra to each= open.) 2. Noting that a closed embedding is fibrewise Stone, that assignment will = extend to U an arbitrary fibrewise Stone (entire) bundle over X - that is t= o say, by Stone duality and contravariantly, a sheaf of Boolean algebras. 3. For generalized X that will provide our Stone notion of cosheaf. The ass= ignment from entire bundles to Stone spaces should preserve finite colimits= and cofiltered limits. There's an obvious technical hurdle of how to expre= ss that directly in terms of sheaves instead of entire bundles. 4. If X is exponentiable then this time, by Stone duality, we are looking f= or maps [BA]^X -> [BA] where [BA] is the classifier for Boolean algebras. T= hey must preserve filtered colimits (automatic for maps) and finite limits.= NX would exist for arbitrary X, and classify those maps.=20 Obviously there's lots to go wrong there, but do you think your coherent mo= nad fits any of those points for coherent X? By the way, although I haven't mention the effective lax descent and relati= vely tidy maps, I am interested in them. They are connected with stable com= pactness and Priestley duality. All the best, [For admin and other information see: http://www.mta.ca/~cat-dist/ ]