From Copying to Corelations via Ancestry Partitions

TL;DR

This paper explores the structure of a specific PROP related to binary generators, connecting it to corelations, cocommutative comonoids, and hypergraph categories, with new identification results at the PROP level.

Contribution

It provides a new identification of a quotient PROP with cocommutative comonoids, embedding it within the cospan framework and relating it to presheaf topos structures.

Findings

The quotient PROP is equivalent to cocommutative comonoids.

Connections established between PROP structures and hypergraph categories.

Organizational reduction to classical results with a novel PROP-level identification.

Abstract

We study the free PROP $\mathrm{Syn}(\delta)$ on a single binary generator $\delta:1\to 2$. The ancestry functor $\Pi:\mathrm{Syn}(\delta)\to \mathrm{FinCorel}$, defined by connected components of the underlying undirected string diagram, has image the sub-PROP $\mathrm{FinCorel}^{\circ}$ of finite corelations whose equivalence classes contain exactly one input and at least one output. The induced quotient [ \mathrm{AncQ}:=\mathrm{Syn}(\delta)/\ker(\Pi) ] is equivalent as a PROP to $\mathrm{Cocom}$, the PROP for non-counital cocommutative comonoids. We then locate this primitive construction inside the standard cospan/corelation framework: $\mathrm{Cospan}(\mathcal B)$ realizes pushout-style gluing as a free hypergraph category; $\mathrm{Cospan}(\mathrm{FinSet})$ collapses under jointly epic corestriction to $\mathrm{FinCorel}$, the PROP for extraspecial commutative Frobenius monoids; and the Yoneda envelope [ \mathcal W=\mathrm{Fun}(\mathrm{FinCorel}^{op},\mathrm{Spc}) ] is a presheaf $\infty$-topos carrying the standard subobject, modality, and monotone fixed-point apparatus. The PROP-level identification $\mathrm{AncQ}\simeq \mathrm{Cocom}$ is the only result claimed as new; the remaining material is organizational and reduces explicitly to cited classical results.