The algebra of processes X
(There is a mistake in the next page, pointed out to me by Matias Menni. I should have said connected components preserves products of reflexive graphs - but the graphs I am interested in are graph objects in the category of labelled graphs - i.e. G, H, K are labelled graphs, and G • H is the product in labelled graphs - a pullback there .)
What is the meaning of this second example? Remember that in the notion of process G, H, K, L are actually graphs, not just objects of a category. So we are considering the parallel of two processes each of which is a sequence of two processes. Consider a behaviour of such a system. Notionally we begin parallel behaviours in the automata G and K. Now there are three different things that can happen: (i) we remain in G while K exits and passes to L, (ii) we exit from both processes and pass to H in parallel with L, (iii) G exits and passes to H while the second process remains in K. And so on.
The whole system is a sequential structure of parallel processes.
I notice that the programme for Category Theory 2014 in Cambridge has been published. It includes my lecture on the algebra of processes. I must say that looking at the other talks I feel a little bit like a shag on a rock. The conference is very much concerned with category theory in mathematics and logic, and nothing resembling my talk.
As I have mentioned earlier I would like very much to attend, but am still not sure whether I will be able to attend as I have health problems.
Regarding the changed form of the post: I am sure if you are really interested you will manage to decipher it. It was produced by the high tech means of writing on paper and taking a photograph with my phone.