Decorated Cospans

Abstract

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 $X\to N\leftarrow Y$ 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.

25 pages

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