Interestingly, one can formalize (open) petri nets using category theory. This is very nicely described by John Baez [here](https://math.ucr.edu/home/baez/petri.pdf), and naturally gives compositions of petri nets.

There is also a [Julia package](https://algebraicjulia.github.io/AlgebraicPetri.jl/dev/) to simulate them!