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.

After I gave my talk, Blake Pollard gave a closely related talk:

Black-boxing open reaction networks

An open reaction network is a reaction network that interacts with its
environment, a user, or another system. I'll describe a way to treat open
reaction networks as morphisms in a category. Composition in this category
provides a means of building up larger open reaction networks from
smaller ones. For reaction networks obeying mass action kinetics, I'll
describe a functor sending an open reaction network to the relation
between input and output flows that holds in nonequilibrium steady states.
This provides a compositional approach to studying
nonequilibrium steady states.