From Copying to Corelations via Ancestry Partitions
This is an incremental theoretical result for category theorists working on PROPs and hypergraph categories.
The paper identifies the quotient of the free PROP on a binary generator by the kernel of the ancestry functor as equivalent to the PROP for non-counital cocommutative comonoids, and situates this within the cospan/corelation framework. No concrete numbers are provided.
We study the free PROP $\mathrm{Syn}(δ)$ on a single binary generator $δ:1\to 2$. The ancestry functor $Π:\mathrm{Syn}(δ)\to \mathrm{FinCorel}$, 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}(δ)/\ker(Π) ] 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.