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

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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?

asked Nov 24, 2017 in Open problems by bensprott (35 points) [ revision history ]

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.

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