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Fong developed `decorated cospans' to model various kinds of open systems:that is, systems with inputs and outputs. In this framework, open systems areseen as the morphisms of a category and can be composed as such, allowinglarger open systems to be built up from smaller ones. Much work has alreadybeen done in this direction, but there is a problem: the notion of isomorphismbetween decorated cospans is often too restrictive. Here we introduce andcompare two ways around this problem: structured cospans, and a new version ofdecorated cospans. Structured cospans are very simple: given a functor $L\colon \mathsf{A} \to \mathsf{X}$, a `structured cospan' is a diagram in$\mathsf{X}$ of the form $L(a) \rightarrow x \leftarrow L(b)$. If $\mathsf{A}$and $\mathsf{X}$ have finite colimits and $L$ is a left adjoint, there is asymmetric monoidal category whose objects are those of $\mathsf{A}$ and whosemorphisms are isomorphism classes of structured cospans. However, this categoryarises from a more fundamental structure: a symmetric monoidal double category.Under certain conditions this symmetric monoidal double category is equivalentto one built using our new version of decorated cospans. We apply these ideasto symmetric monoidal double categories of open electrical circuits, openMarkov processes and open Petri nets.

Open Systems: A Double Categorical Perspective