Re: Undirected graphs

[F W Lawvere <wlawvere@buffalo.edu>]

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
>
>
>
>