To describe systems composed of interacting parts, scientists and
engineers draw diagrams of networks: flow charts, electrical circuit
diagrams, signal-flow graphs, Feynman diagrams and the like. In
principle all these different diagrams fit into a common framework:
the mathematics of monoidal categories. This has been known
for some time. However, the details are more challenging, and
ultimately more rewarding, than this basic insight. Here we explain
how various applications of reaction networks and Petri nets fit into
this framework.