One goal of applied category theory is to understand open systems. We compare two ways of describing open systems as cospans equipped with extra data. First, given a functor L:A→X, a `structured cospan' is a diagram in X of the form L(a)→x←L(b). We give a new proof that if A and X have finite colimits and L preserves them, there is a symmetric monoidal double category whose objects are those of A and whose horizontal 1-cells are structured cospans. Second, given a pseudofunctor F:A→Cat, a `decorated cospan' is a diagram in A of the form a→m←b together with an object of F(m). Generalizing the work of Fong, we show that if A has finite colimits and F:(A,+)→(Cat,×) is symmetric lax monoidal, there is a symmetric monoidal double category whose objects are those of A and whose horizontal 1-cells are decorated cospans. We prove that under certain conditions, these two constructions become isomorphic when we take X=∫F to be the Grothendieck category of F. We illustrate these ideas with applications to electrical circuits, Petri nets, dynamical systems and epidemiological modeling.
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