From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/3988 Path: news.gmane.org!not-for-mail From: "Jonathon Funk" Newsgroups: gmane.science.mathematics.categories Subject: Re: What is the right abstract definition of "connected"? Date: Tue, 9 Oct 2007 10:43:12 -0400 Message-ID: NNTP-Posting-Host: main.gmane.org Mime-Version: 1.0 Content-Type: text/plain;charset="iso-8859-1" Content-Transfer-Encoding: 7bit X-Trace: ger.gmane.org 1241019645 11203 80.91.229.2 (29 Apr 2009 15:40:45 GMT) X-Complaints-To: usenet@ger.gmane.org NNTP-Posting-Date: Wed, 29 Apr 2009 15:40:45 +0000 (UTC) To: "categories list" Original-X-From: rrosebru@mta.ca Tue Oct 9 22:00:10 2007 -0300 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Tue, 09 Oct 2007 22:00:10 -0300 Original-Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1IfPqQ-0001MR-VR for categories-list@mta.ca; Tue, 09 Oct 2007 21:54:39 -0300 Original-Sender: cat-dist@mta.ca Precedence: bulk X-Keywords: X-UID: 45 Original-Lines: 93 Xref: news.gmane.org gmane.science.mathematics.categories:3988 Archived-At: One suggestion is to say that an object X in a category C (with products) is connected relative to a functor F:B-->C if passing from maps m:b-->b' in B to maps XxF(b)-->F(b') (by composing the projection XxF(b)-->F(b) with F(m) ) is a bijection for every b,b' (or possibly just onto, not bijection, could be stipulated, but I don't know how inappropriate that would be). If pullbacks exist X*: C-->C/X, then this is equivalent to X*F full and faithful (or just full). If say b=1=terminal of B (and F(1)=1), then it is as if to say that if X is connected (relative to F), then elements of any b' are in bijection with (or at least onto) maps X --> F(b'): every such map is thus `constant'. For example, in this sense we may speak of a connected object X in a topos E-->S relative to Delta: S--->E. Jonathon ----- Original Message ----- From: "Vaughan Pratt" To: "categories list" Sent: Monday, October 08, 2007 4:18 PM Subject: categories: What is the right abstract definition of "connected"? > I'd like to say that "connected" is defined on objects of any category C > having an object 1+1 (coproduct of two final objects). X is connected > just when C(X,1+1) <= 2. > > If this definition appears in print somewhere I can just cite it. If > not is there a better or more standard generally applicable definition I > can use? > > If C(X,1+1) = 2 is citable but not <= 2, have the proponents of =2 > taken into account that no Boolean algebra is connected according to the > =2 definition? This is because 1+1 ~ 1 in Bool, CABA, DLat, StoneDLat, > etc. (dual to 0x0 ~ 0 in Set, Pos, etc.), forcing C(X,1+1) = 1. Boolean > algebras and distributive lattices fail the =2 test not because they are > disconnected in any natural sense but rather because they are > hyperconnected. It seems unreasonable to say that hyperconnected > objects are not connected. > > There is also the question of the object of connected components of an > object. In Set and Grph, if X has k connected components then C(X,1+1) > = 2^k for all X, a set (C being ordinary, i.e. enriched in Set). In > Stone (Stone spaces) however this only holds for finite X, with k = X. > For infinite X Stone(X,1+1) is the set of clopen sets of X, which can be > countably infinite and hence not 2^k for any k. > > If we read 2^k as Stone(k,2), taking k = X and 2 the Sierpinski space > this doesn't help. However Stone(k,1+1) is ideal: instead of treating > the object of connected components of a Stone space k = X as a set we > can treat them as a Boolean algebra, namely that of the clopen sets of X. > > These examples are worth bearing in mind when considering the > appropriate general definition of number of connected components of an > object, and whether even to treat it as a number (cardinal) or a more > general object. > > Connectedness seems somehow more basic than finiteness because we can > easily draw examples of connected and disconnected objects, whereas it > requires a vivid imagination to see the boundary between finite and > infinite objects one might try to draw on paper. > > This motivates making connectedness prior to finiteness. > > Another familiar and easily visualized notion with small examples is > that of path. Define a *path* to be a connected directed graph having > one vertex each of degree (0,1) and (1,0), and all others (1,1). (The > degree (m,n) specifies the in-degree as m and the out-degree as n.) > > We can then define a finite set to be one in bijection with the set of > vertices of some path. This seems more natural than defining it to be > one such that every injection on itself is a surjection, because there > are a lot of injections to worry about and how do you convince yourself > that surjective injections don't kick in until omega? > > Those who are already wedded to some other definition of finite will > want to check that this path-based definition draws the boundary in the > same place as theirs. For what definitions of "finite" can this not be > shown? And are any of them more palatable than the path-based definition? > > Vaughan > > >