2022/09/01 by John C. Baez, Kenny Courser, Christina Vasilakopoulou · 5 citations
Mathematics · #Homotopy and Cohomology in Algebraic Topology
paper · pdf · doi:10.32408/compositionality-4-3
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 \colon 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 preserves them, it is known that 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 \colon 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 \colon (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.