From Copying to Corelations via Ancestry Partitions

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.