Structured Cospans

One goal of applied category theory is to better understand networks appearing throughout science and engineering. Here we introduce “structured cospans” as a way to study networks with inputs and outputs. Given a functor $L \colon $\mathsf{A}$ \to $\mathsf{X}$$,astructuredcospanisadiagramin$$\mathsf{X}$$oftheform$L(a) \rightarrow x \leftarrow L(b)$\mathrm{.}If$$\mathsf{A}$$and$$\mathsf{X}$$havefinitecolimitsand$L$ is a left adjoint, we obtain a symmetric monoidal category whose objects are those of $\mathsf{A}$ and whose morphisms are isomorphism classes of structured cospans. This is a hypergraph category. However, it arises from a more fundamental structure: a symmetric monoidal double category where the horizontal 1-cells are structured cospans. We show how structured cospans solve certain problems in the closely related formalism of “decorated cospans”, and explain how they work in some examples: electrical circuits, Petri nets, and chemical reaction networks.