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Corelations are the prop for extraspecial commutative Frobenius monoids

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Brandon Coya and Brendan Fong

Just as binary relations between sets may be understood as jointly monic
spans, so too may equivalence relations on the disjoint union of sets be
understood as jointly epic cospans. With the ensuing notion of composition
inherited from the pushout of cospans, we call these equivalence relations
*corelations*. We define the category of corelations between finite
sets and prove that it is equivalent to the prop for extraspecial
commutative Frobenius monoids. Dually, we show that the category of
relations is equivalent to the prop for special commutative bimonoids.
Throughout, we emphasise how corelations model interconnection.

Keywords:
corelation, extra law, Frobenius monoid, prop, PROP

2010 MSC:
18C10, 18D10

*Theory and Applications of Categories,*
Vol. 32, 2017,
No. 11, pp 380-395.

Published 2017-02-27.

http://www.tac.mta.ca/tac/volumes/32/11/32-11.pdf

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