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: A -> X, a structured cospan is a diagram in X of the form L(a) -> x <- L(b). If A and X have finite colimits and L is a left adjoint, we obtain a symmetric monoidal category whose objects are those of 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.
Keywords: bicategory, cospan, double category, monoidal category, network
2020 MSC: 18B10, 18M35, 18N10
Theory and Applications of Categories, Vol. 35, 2020, No. 48, pp 1771-1822.