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?