Open Systems: A Double Categorical Perspective

Fong developed decorated cospans' to model various kinds of open systems: that is, systems with inputs and outputs. In this framework, open systems are seen as the morphisms of a category and can be composed as such, allowing larger open systems to be built up from smaller ones. Much work has already been done in this direction, but there is a problem: the notion of isomorphism between decorated cospans is often too restrictive. Here we introduce and compare two ways around this problem: structured cospans, and a new version of decorated 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 $L(a)\to x\leftarrow L(b)$. If $\mathsf{A}$ and $\mathsf{X}$ have finite colimits and $L$ is a left adjoint, there is a symmetric monoidal category whose objects are those of $\mathsf{A}$ and whose morphisms are isomorphism classes of structured cospans. However, this category arises from a more fundamental structure: a symmetric monoidal double category. Under certain conditions this symmetric monoidal double category is equivalent to one built using our new version of decorated cospans. We apply these ideas to symmetric monoidal double categories of open electrical circuits, open Markov processes and open Petri nets.