A bicategory of decorated cospans

Abstract

If $\mathbf{C}$ is a category with pullbacks then there is a bicategory with the same objects as $\mathbf{C}$, spans as morphisms, and maps of spans as 2-morphisms, as shown by Benabou. Fong has developed a theory of "decorated" cospans, which are cospans in $\mathbf{C}$ equipped with extra structure. This extra structure arises from a lax symmetric monoidal functor $F \colon \mathbf{C} \to \mathbf{D}$; we use this functor to "decorate" each cospan with apex $N \in \mathbf{C}$ with an element of $F(N)$. Using a result of Shulman, we show that when $\mathbf{C}$ has finite colimits, decorated cospans are morphisms in a symmetric monoidal bicategory. We illustrate our construction with examples from electrical engineering and the theory of chemical reaction networks.