Decorated Cospans

Let $\mathcal C$ be a category with finite colimits, writing its coproduct $+$, and let $(\mathcal D, \otimes)$ be a braided monoidal category. We describe a method of producing a symmetric monoidal category from a lax braided monoidal functor $F: (\mathcal C,+) \to (\mathcal D, \otimes)$, and of producing a strong monoidal functor between such categories from a monoidal natural transformation between such functors. The objects of these categories, our so-called `decorated cospan categories', are simply the objects of $\mathcal C$, while the morphisms are pairs comprising a cospan $X \rightarrow N \leftarrow Y$ in $\mathcal C$ together with an element $1 \to FN$ in $\mathcal D$. Moreover, decorated cospan categories are multigraph categories---each object is equipped with a special commutative Frobenius monoid---and their functors preserve this structure.

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