What is the future of causal structure: Domains or wiring diagrams?

Question

Hi,

To underpin all of physics, we are going to have to decide on a type of discrete causal structure.  Panangaden and Martin made a very good case for Domains as a discrete causal structure.  There has been a lot of work done on generalizing wiring diagrams and it looks like operads seem to capture these and also decorated spans seem to capture  wiring diagrams too.  I get the feeling, at some point we are going to want to consider our causal backgrounds in the more general sense of wiring diagrams like those in the articles I gave. 

Does anyone know reasons why one would choose one approach over the other?  In other words, a mechanical step in the evolution of a universe is a Domain map.  But we have generalizations of causal structures (ie wiring diagrams) and their maps (in the way that a span is a generalization of just a morphism).  Why choose wiring diagrams and their maps over domains and their maps?  Baez has also shown that cospans give a more powerful tool for looking at cobordisms as our background spacetimes.  I see domains as the discrete causal version of cobordisms.  Perhaps, cospans of domains would be fruitful as a direction of research.  Does anyone have an opinion on that?

Comments

Many domains classes are well casted in words of categories, specially when an order relation is defined and total. The continuity/discretion duality in categories constructions of topologies is probably a powerful factor of interest. A better link to find precise answers is Grothendieck topology and the pages of the site. The Baez document deserves a separate discussion.