Yeah. We have a bit of a cart before the horse issue here. The software is already working but a coherent mathematical model is evasive (and I generally believe that very few projects do have a good model) and so we can’t know its exact properties until we find that model. One of the best ways to solve is a problem is to grow a theory around it and find your problem as a special case. That’s the goal with sheaves.
Petri nets are a bottom-up approach to finding sheaves in this problem. What I suggest in this article is a top-down approach. They both end at sheaves, and ought to agree to some exactly determinable extend.