From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/4019 Path: news.gmane.org!not-for-mail From: "Marta Bunge" Newsgroups: gmane.science.mathematics.categories Subject: RE: connectedness Date: Mon, 15 Oct 2007 09:29:32 -0400 Message-ID: NNTP-Posting-Host: main.gmane.org Mime-Version: 1.0 Content-Type: text/plain; format=flowed X-Trace: ger.gmane.org 1241019669 11354 80.91.229.2 (29 Apr 2009 15:41:09 GMT) X-Complaints-To: usenet@ger.gmane.org NNTP-Posting-Date: Wed, 29 Apr 2009 15:41:09 +0000 (UTC) To: categories@mta.ca Original-X-From: rrosebru@mta.ca Mon Oct 15 21:44:11 2007 -0300 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Mon, 15 Oct 2007 21:44:11 -0300 Original-Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1IhaKX-0005tE-Nm for categories-list@mta.ca; Mon, 15 Oct 2007 21:30:41 -0300 Original-Sender: cat-dist@mta.ca Precedence: bulk X-Keywords: X-UID: 76 Original-Lines: 71 Xref: news.gmane.org gmane.science.mathematics.categories:4019 Archived-At: Dear Paul, >First, though, I would like to underline something that Steve Lack >(almost) said, namely that the category in which you index your >components, and therefore also the one in which you define >connectedness, need to be EXTENSIVE, ie their coproducts should >be disjoint, and stable under pullback, and the initial object strict. > >Maybe we've over-done philology recently, but "component" means >"putting together", where we expect the parts to cover the whole >(coproduct), without overlapping (disjoint), to be distinguishable >(like disjoint union, but unlike addition and disjunction). >The modern notion of extensivity, in which Steve had a part, >captures this idea very neatly. The equivalence between definitions >of connectedness based on 1+1 and on X+Y surely depends on stability >under pullback, and the requirement that the choice between left >and right be unique surely requires disjointness. Maybe a close >study of Marta Bunge's work on abstract connectedness would clarify >this. > In my now obsolete 1966 thesis, the context was that of a category with finite limits and finite coproducts, but I had not assumed therein that coproducts should be disjoint and universal. With these assumptions, the definitions of `abstractly unary' for arbitrary binary products (factors through AT LEAST one of the injections) and of `abstractly exclusively unary' (factors through exactly one of the injections) are equivalent by the disjointness part, and are equivalent also to the same notions with binary coproducts of 1 instead of arbitrary binary coproducts (by the universal or stability part). So, *any* of those in this context should mean `connected'. Without those conditions, but with just a terminal object and binary coproducts, then the `at least' part does not reduce to that of coproducts of 1, but the `at most' part does. In that case, the notions correspond to `abstractly unary' and `abstractly exclusively unary', and they are not equivalent. So it all depends on the ambient category. > >So far, I have only mentioned BINARY notions of connectedness, >but if we want to talk about families of connected COMPONENTS >then we must also consider INFINITARY connectedness (as Marta >stressed). Here the results for the constructive real line are >somewhat surprising. > > I still have to digest your disquisitions on constructive analysis, which seem most interesting, but on the above point, I emphasize that inded one must keep the disctinction between connectedness wirt binary coproducts and connectedness wrt arbitrary coproducts (indexed externally, e.g. by a set in a Grothendieck topos E-->SET, or by an object of S in the cae of a bounded topos E-->S. Whether the terminology must make that distinction I am not sure of, or maybe we could say `connected' for the binary case (which is also intrinsic), and `S-connected' for the case of S-indexed coproducts. Once again, under enough hypotheses as I szaid above and was also mentioned by Steve Lack. I am not sure of which hypotheses Vaughan wants to make but, if the minimal possible, then he might need the detailed analysis that I proposed and, in that case, reserve `connected' in the binary case for `abstractly exclusively unary', not simply `abstractly unary', and `connected' when only the binary coproduct considered is 1+1. But if his categories ae categories of graphs, I don't see his problem. With best regards, Marta