From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/4006 Path: news.gmane.org!not-for-mail From: "Marta Bunge" Newsgroups: gmane.science.mathematics.categories Subject: Re: What is the right abstract definition of "connected"? Date: Thu, 11 Oct 2007 14:48:09 -0400 Message-ID: NNTP-Posting-Host: main.gmane.org Mime-Version: 1.0 Content-Type: text/plain; format=flowed X-Trace: ger.gmane.org 1241019656 11273 80.91.229.2 (29 Apr 2009 15:40:56 GMT) X-Complaints-To: usenet@ger.gmane.org NNTP-Posting-Date: Wed, 29 Apr 2009 15:40:56 +0000 (UTC) To: categories@mta.ca Original-X-From: rrosebru@mta.ca Thu Oct 11 22:24:12 2007 -0300 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Thu, 11 Oct 2007 22:24:12 -0300 Original-Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1Ig99x-0001qN-A5 for categories-list@mta.ca; Thu, 11 Oct 2007 22:17:49 -0300 Original-Sender: cat-dist@mta.ca Precedence: bulk X-Keywords: X-UID: 63 Original-Lines: 31 Xref: news.gmane.org gmane.science.mathematics.categories:4006 Archived-At: Dear Vaughan ============================================================== An object of a category is *connected* when its every morphism to a nonempty coproduct factors through an inclusion thereof. ============================================================== Your proposed definition above is precisely the notion of *abstractly unary* from my J.Algebra '69 paper. It was so termed (instead of *connected*) since it does not need a terminal object to state it (precisely your motivation) and since one does not want to restrict to binary coproducts. When there is a terminal object, and when the coproducts considered are just the binary ones, it is enough to consider morphisms into the coproducts 1+1 (as I show in my thesis) and, in that case, it should be simply called *connected*. In another guise, this is the definition of *connected* given in Cats and Alligators, and it is the one directly inspired by topology. I see no reason to change the terminology. In short, your connected objects I have called abstractly unary. They came about in connection with atoms. An object A in a cocomplete (concrete) category E is an *atom* if HOM(A,-):E--->Set preserves colimits. More objectively, if E has exponentiation, Lawvere uses the notion of an *A.T.O.* instead, meaning that the functor (-)^A : E---> E has a right adjoint (the "amazing right adjoint"). I hope this helps, Cordially, Marta