A bicategory of decorated cospans

Abstract

If C is a category with pullbacks then there is a bicategory with the same objects as 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 C equipped with extra structure. This extra structure arises from a symmetric lax monoidal functor F : C → D; we use this functor to ‘decorate’ each cospan with apex N ɛ C with an element of F(N). Using a result of Shulman, we show that when 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.