Jamie Vicary states `the category of graphs is not a topos'. The situation is not so simple, and is discussed for the combinatorially minded reader in 06.04 BROWN, R., MORRIS, I., SHRIMPTON, J. & WENSLEY, C.D. Graphs of morphisms of graphs http://www.informatics.bangor.ac.uk/public/mathematics/research/preprints/06... There are categories of undirected graphs which are not toposes. But ... Ronnie Brown ----- Original Message ----- From: "Jamie Vicary" <jamie.vicary@imperial.ac.uk> To: <categories@mta.ca> Sent: Friday, March 09, 2007 10:01 AM Subject: categories: Re: relations on graphs
Is there any literature which discusses different possible notions for relations on graphs?
In any regular category, and certainly any topos, there is a well defined notion of relation, where a relation between two objects is a subobject of their product. These admit a * operation and compose in a well-behaved way; look towards the end of McLarty's category theory textbook for info on this.
The category of directed graphs is certainly such a category, being regular. The category of graphs is not a topos, I believe, but might still be regular.
Jamie Vicary.