From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/9536 Path: news.gmane.org!.POSTED!not-for-mail From: Steve Vickers Newsgroups: gmane.science.mathematics.categories Subject: Re: Topos theory for spaces of connected components Date: Tue, 06 Feb 2018 10:26:26 +0000 Message-ID: References: <244986425.357598.1517854022932.JavaMail.zimbra@math.mcgill.ca> Reply-To: Steve Vickers NNTP-Posting-Host: blaine.gmane.org Mime-Version: 1.0 Content-Type: text/plain; charset=UTF-8 Content-Transfer-Encoding: 8bit X-Trace: blaine.gmane.org 1517922772 28291 195.159.176.226 (6 Feb 2018 13:12:52 GMT) X-Complaints-To: usenet@blaine.gmane.org NNTP-Posting-Date: Tue, 6 Feb 2018 13:12:52 +0000 (UTC) Cc: categories@mta.ca To: bunge@math.mcgill.ca Original-X-From: majordomo@mlist.mta.ca Tue Feb 06 14:12:47 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 1ej334-0006GP-E9 for gsmc-categories@m.gmane.org; Tue, 06 Feb 2018 14:12:34 +0100 Original-Received: from mlist.mta.ca ([138.73.1.63]:38241) by smtp2.mta.ca with esmtp (Exim 4.80) (envelope-from ) id 1ej34u-0000ZF-KQ; Tue, 06 Feb 2018 09:14:28 -0400 Original-Received: from majordomo by mlist.mta.ca with local (Exim 4.71) (envelope-from ) id 1ej340-0004yl-5A for categories-list@mlist.mta.ca; Tue, 06 Feb 2018 09:13:32 -0400 In-Reply-To: <244986425.357598.1517854022932.JavaMail.zimbra@math.mcgill.ca> Precedence: bulk Xref: news.gmane.org gmane.science.mathematics.categories:9536 Archived-At: Correction - In Conjecture 1 I mistakenly wrote "local compactness" for "local connectedness". ------ 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 distribution) is to take locally connected space Y with map p: Y -> X, and then to each sheaf U over X assign the set of connected components of p*U. This 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 the 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 sense: 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 cosheaf to the terminal one. If X is exponentiable, then (always? In favourable cases?) the cosheaf as described 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 analogy with integration, where R would be the real line. Then just as Riesz picks out the linear functionals as the distributions, we are interested in 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 happens 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 (strongly compact?) as analogue of local connectedness. (By Stone duality that could 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 to say, by Stone duality and contravariantly, a sheaf of Boolean algebras. 3. For generalized X that will provide our Stone notion of cosheaf. The assignment from entire bundles to Stone spaces should preserve finite colimits and cofiltered limits. There's an obvious technical hurdle of how to express that directly in terms of sheaves instead of entire bundles. 4. If X is exponentiable then this time, by Stone duality, we are looking for maps [BA]^X -> [BA] where [BA] is the classifier for Boolean algebras. They must preserve filtered colimits (automatic for maps) and finite limits. NX would exist for arbitrary X, and classify those maps. Obviously there's lots to go wrong there, but do you think your coherent monad fits any of those points for coherent X? By the way, although I haven't mention the effective lax descent and relatively tidy maps, I am interested in them. They are connected with stable compactness and Priestley duality. All the best, Steve. [For admin and other information see: http://www.mta.ca/~cat-dist/ ]