Structured versus Decorated Cospans

Abstract

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.