#
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.

http://www.tac.mta.ca/tac/volumes/30/33/30-33.pdf

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