#
Decorated corelations

##
Brendan Fong

Let $C$ be a category with finite colimits, and let $(E, M)$ be a
factorisation system on $C$ with $M$ stable under
pushout. Writing $C;M^{\op}$ for the symmetric monoidal category
with morphisms cospans of the form $\stackrel{c}\to
\stackrel{m}\leftarrow$, where $c \in C$ and $m \in M$, we give
a method for constructing a category from a symmetric lax monoidal
functor $F : (C; \mc M^{\op},+) \to (Set,\times)$. A morphism in
this category, termed a *decorated corelation*, comprises (i) a
cospan $X \to N \leftarrow Y$ in $C$ such that the canonical
copairing $X+Y \to N$ lies in $E$, together with (ii) an element of
$FN$. Functors between decorated corelation categories can be
constructed from natural transformations between the decorating functors
$F$. This provides a general method for constructing hypergraph
categories - symmetric monoidal categories in which each object is a
special commutative Frobenius monoid in a coherent way - and their
functors. Such categories are useful for modelling network languages,
for example circuit diagrams, and such functors are useful for modelling
their semantics.

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

2010 MSC:
18C10, 18D10

*Theory and Applications of Categories,*
Vol. 33, 2018,
No. 22, pp 608-643.

Published 2018-06-25.

http://www.tac.mta.ca/tac/volumes/33/22/33-22.pdf

TAC Home