Structured versus Decorated Cospans

TL;DR

This paper compares structured and decorated cospans in applied category theory, showing conditions under which they are equivalent and illustrating their use in modeling various open systems like circuits and epidemiological models.

Contribution

It establishes an isomorphism between structured and decorated cospans via Grothendieck categories, extending the framework for modeling open systems.

Findings

Structured and decorated cospans can be made equivalent under certain conditions.

The paper generalizes Fong's decorated cospan construction to symmetric lax monoidal functors.

Applications include modeling electrical circuits, Petri nets, and epidemiological systems.

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 $L \colon \mathsf{A} \to \mathsf{X}$, 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 $L$ preserves them, it is known that there is a symmetric monoidal double category whose objects are those of $\mathsf{A}$ and whose horizontal 1-cells are structured cospans. Second, given a pseudofunctor $F \colon \mathsf{A} \to \mathbf{Cat}$, a "decorated cospan" is a diagram in $\mathsf{A}$ of the form $a \rightarrow m \leftarrow b$ together with an object of $F(m)$. Generalizing the work of Fong, we show that if $\mathsf{A}$ has finite colimits and $F \colon (\mathsf{A},+) \to (\mathsf{Cat},\times)$ is symmetric lax monoidal, there is a symmetric monoidal double category whose objects are those of $\mathsf{A}$ and whose horizontal 1-cells are decorated cospans. We prove that under certain conditions, these two constructions become isomorphic when we take $\mathsf{X} = \int F$ to be the Grothendieck category of $F$. We illustrate these ideas with applications to electrical circuits, Petri nets, dynamical systems and epidemiological modeling.