Corelations are the prop for extraspecial commutative Frobenius monoids

TL;DR

This paper establishes a categorical framework linking corelations to extraspecial commutative Frobenius monoids, providing a mathematical foundation for modeling interconnections.

Contribution

It introduces corelations as a categorical concept and proves their equivalence to the prop for extraspecial commutative Frobenius monoids, extending the understanding of interconnection modeling.

Findings

Category of corelations is equivalent to the prop for extraspecial commutative Frobenius monoids.

Relations are equivalent to the prop for special commutative bimonoids.

Corelations effectively model interconnection structures.

Abstract

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 \emph{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.