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.