Category of directed multigraphs with loops

Category of directed multigraphs with loops

[Galchin Vasili]

Hello,

      Given two graphs, A and B, I am trying figure out how to construct
the product AxB. I have been rereading "Conceptual Mathemtatics" by
Lawvere. The category of irreflexive graphs is one of the running examples
throughout the book. I have been concentrating on the chapters concerned
with the product of objects, but I don't see any details of how to
construct AxB. Have I skipped over something?

Regards, Bill Halchin

Re: Category of directed multigraphs with loops

[Francois Magnan]

Hi,

	The product in the category of directed irreflexive multigraphs is 
simple to compute. In fact, if you learn further about topos theory you 
will learn that all finite limits in a presheaf topos are computed 
pointwise. The category that interests you is a presheaf category since 
it is the category of all functors from the category

                      ------------>
C^op  =  A                     V
                      ------------>

to the category of sets.


In simpler words, it means that if you take two graphs: G_1 and G_2 
with vertices sets V_1 and  V_2 and arrows sets A_1 and A_2 
respectively. The product graph P=G_1 x G_2 will have V_P=V_1xV_2 
(normal catesian product in sets) as vertices and A_P=A_1xA_2 as 
arrows. The incidence relation are the expected ones.

For the reflexive directed multigraphs as all presheafs the recipe is 
the same. I would suggest you to read the book:

M.La Palme Reyes, G. Reyes, Generic Figures and their glueings: A 
constructive approach to functor categories.

Which is still unpublished as of now but I can send you a PDF.

Hope this helps,
Francois Magnan


On Monday, Feb 17, 2003, at 16:09 America/Montreal, Galchin Vasili 
wrote:

>
> Hello,
>
>       Given two graphs, A and B, I am trying figure out how to 
> construct
> the product AxB. I have been rereading "Conceptual Mathemtatics" by
> Lawvere. The category of irreflexive graphs is one of the running 
> examples
> throughout the book. I have been concentrating on the chapters 
> concerned
> with the product of objects, but I don't see any details of how to
> construct AxB. Have I skipped over something?
>
> Regards, Bill Halchin
>
>
>
>
- --------------------------------------------
Francois Magnan
Recherche & Développement
Cogniscience Editeurs Inc.
fmagnan@cogniscienceinc.com