# Decorated cospans

## Brendan Fong

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

Keywords: cospan, decorated cospan, hypergraph category, well-supported compact closed category, separable algebra, Frobenius algebra, Frobenius monoid

2010 MSC: 18C10, 18D10

Theory and Applications of Categories, Vol. 30, 2015, No. 33, pp 1096-1120.

Published 2015-08-06.

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