From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/4000 Path: news.gmane.org!not-for-mail From: Vaughan Pratt Newsgroups: gmane.science.mathematics.categories Subject: Re: What is the right abstract definition of "connected"? Date: Wed, 10 Oct 2007 13:43:38 -0700 Message-ID: NNTP-Posting-Host: main.gmane.org Mime-Version: 1.0 Content-Type: text/plain; charset=ISO-8859-1; format=flowed Content-Transfer-Encoding: 7bit X-Trace: ger.gmane.org 1241019653 11249 80.91.229.2 (29 Apr 2009 15:40:53 GMT) X-Complaints-To: usenet@ger.gmane.org NNTP-Posting-Date: Wed, 29 Apr 2009 15:40:53 +0000 (UTC) To: categories list Original-X-From: rrosebru@mta.ca Thu Oct 11 12:38:23 2007 -0300 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Thu, 11 Oct 2007 12:38:23 -0300 Original-Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1Ifzu3-00023k-9t for categories-list@mta.ca; Thu, 11 Oct 2007 12:24:47 -0300 Original-Sender: cat-dist@mta.ca Precedence: bulk X-Keywords: X-UID: 57 Original-Lines: 84 Xref: news.gmane.org gmane.science.mathematics.categories:4000 Archived-At: Steve Vickers wrote: > The topological condition is often stated differently: that every map > X -> 1+1 factors via 1. Thus C(X,1+1) <= C(1,1+1). I think in most > contexts you would want to say that, if anything is connected, 1 is, > but you can easily find C(1,1+1) > 2. Thanks, Steve, this is great. I didn't want to go out on a limb with C(X,1+1) <= 2 (or = 2) if it was buggy, good to know about the C(1,1+1) > 2 problem. This also takes care of my concern about situations where 1+1 = 1, since your definition as stated makes Boolean algebras etc. connected. Presumably my taking the anarchist side (no unity) in the definition of locally cartesian closed obligates me to ask for the right formulation of "connected" in the absence of 1. How about the following? ================================================================= An object of a category is *connected* when its every morphism to a nonempty coproduct factors through an inclusion thereof. ================================================================= This eliminates all assumptions about the category -- if there are no nontrivial coproducts every object is connected by default (any morphism to a trivial coproduct factors through its one inclusion), reasonable when there is no recognizable (by the coproduct test) example of disconnectedness in the category to compare with. It also accomodates: > In constructive locale theory the standard definition is stronger and > requires that for every discrete I, every map X -> I must factor via > 1. This allows "infinite n". with the same benefits - constructive I suppose (how is that judged exactly?), and allows infinite comparisons. If necessary one could qualify "coproduct" with "small" but methodologically it would seem preferable to let such size limits be set by a larger context. The effect of > The little reasons I alluded to are that it is often useful to > require every map to 0 also to factor via 1. That excludes 0 itself > from connectedness. can be had by omitting "nonempty" from the definition. While this might seem a very natural omission, my concern with it is not so much 0 itself as the objects with morphisms to 0, e.g. all Boolean algebras except 1, which this definition would therefore make not connected. Stone spaces being totally disconnected, it just seems plain wrong to have their duals not connected either when they are so obviously connected, like totally (except 1, which is, like, connected but not totally, being dual to the empty Stone space, which is, like, disconnected but not totally). In the geometric duality of points and lines in the plane, two points are disconnected unless they coincide, while two lines are connected unless they are parallel. And an undirected graph and its complement either both contain an N or neither do, and in the latter case you can ask Google the following. Is an N-free graph connected if and only if its complement is disconnected? Google will confirm that it is, no need to click on any of the links it returns. (You may have to read several of Google's "answers" though since Google isn't yet smart enough to just say yes, or even to give the most direct "answer" first.) Graphs with an N are the undirected graph counterpart of the empty Stone space and the one-element Boolean algebra, being neither totally connected nor totally disconnected. Incidentally it's amazing just how many questions Google "knows" the answer to. Like all oracles though it tends to be a little erratic on questions involving future events. Google's staggering R&D budget notwithstanding, asking it whether Hillary will win the election is about as useful as asking the 8-ball: you're way better off asking the people who place sub-Google-sized ($100) bets on such questions. And asking NSF for funding for your research into questions you propose to answer by asking Google has even lower odds than asking Google. Vaughan