From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/3082 Path: news.gmane.org!not-for-mail From: F W Lawvere Newsgroups: gmane.science.mathematics.categories Subject: Re: Undirected graphs Date: Thu, 9 Mar 2006 09:05:13 -0500 (EST) Message-ID: References: <20060308150720.9D5D07375B@chase.mathstat.dal.ca> Reply-To: wlawvere@acsu.buffalo.edu NNTP-Posting-Host: main.gmane.org Mime-Version: 1.0 Content-Type: TEXT/PLAIN; charset=US-ASCII X-Trace: ger.gmane.org 1241019084 7173 80.91.229.2 (29 Apr 2009 15:31:24 GMT) X-Complaints-To: usenet@ger.gmane.org NNTP-Posting-Date: Wed, 29 Apr 2009 15:31:24 +0000 (UTC) To: categories@mta.ca Original-X-From: rrosebru@mta.ca Fri Mar 10 05:51:04 2006 -0400 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Fri, 10 Mar 2006 05:51:04 -0400 Original-Received: from Majordom by mailserv.mta.ca with local (Exim 4.52) id 1FHeBZ-0001lV-UZ for categories-list@mta.ca; Fri, 10 Mar 2006 05:45:25 -0400 X-Sender: wlawvere@hercules.acsu.buffalo.edu In-Reply-To: <20060308150720.9D5D07375B@chase.mathstat.dal.ca> Original-Sender: cat-dist@mta.ca Precedence: bulk X-Keywords: X-UID: 28 Original-Lines: 60 Xref: news.gmane.org gmane.science.mathematics.categories:3082 Archived-At: Dear all, Yes, there are two kinds of loops in the topos of right actions of the four-element monoid A, where A consists of endomaps of the two-element set. Consider for example the concrete structure of the truth-value object in that topos, which is forced to contain a truth-value called "foray". Rather than as "semi"loops, my colleagues and I usually think of them as one-lane in the sense that some other edges are really two lanes, related by the involution operator in the site. My old paper "Qualitative distinctions..." tried to make the point that there are several precise toposes all deserving the rough name of "graph" or "network" and that each of these precise toposes may have a role to play. For example, in any given topos, for any given object L, the category of objects over L, or "L-labelled graphs" (which in practice may serve as a category of networks) is another topos of "graphs". In my experience it is important to consider the whole topos in order to get good exactness properties but, moreover, because the truth-value object and other specific objects which may seem rather far from an initial prejudice about what one wants the objects to mean, nonetheless turn out in a systematic theory to play a key role in representing concepts directly related to the original particular subject matter. A simple example is the representability of gender and moitie in the topos of kinship systems. This example is treated briefly in Conceptual Mathematics and in more detail, (again actually involving several related toposes rather than a single choice) in "Kinship and mathematical categories". Bill ************************************************************ F. William Lawvere Mathematics Department, State University of New York 244 Mathematics Building, Buffalo, N.Y. 14260-2900 USA Tel. 716-645-6284 HOMEPAGE: http://www.acsu.buffalo.edu/~wlawvere ************************************************************ On Wed, 8 Mar 2006 cat-dist@mta.ca wrote: > I've been following the recent posts on undirected graphs > with interest. But I have a question. I think it's being said > that undirected graphs are the same as directed graphs with > involution. (Presheaves on the full subcategory of SET determined > by 1 and 2, or just 2.) Which is nice but what about loops? > The involution might fix a loop or not. So wouldn't we be > getting undirected graphs with two kinds of loops, whole loops > and semiloops? What am I missing? > > Bob > > > >