2025/05/28 by Andreu Ballus Santacana, Santacana, Andreu Ballus
Computer Science · Mathematics · #03B45 #03G12 #03G30 #18C10 #18F20 #Category Theory (math.CT) #F.1.1 #F.3.2 #F.4.1 #FOS: Computer and information sciences #FOS: Mathematics #Homotopy and Cohomology in Algebraic Topology #Logic in Computer Science (cs.LO) #Logic, Reasoning, and Knowledge #Logic, programming, and type systems
paper · pdf · doi:10.48550/arxiv.2505.22931
openalex publication_date 2025/05/28 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We study the free PROP Syn(δ) on a single binary generator δ:1→ 2. The ancestry functor Π:Syn(δ)→ FinCorel, defined by connected components of the underlying undirected string diagram, has image the sub-PROP FinCorel∘ of finite corelations whose equivalence classes contain exactly one input and at least one output. The induced quotient [ AncQ:=Syn(δ)/ker(Π) ] is equivalent as a PROP to Cocom, the PROP for non-counital cocommutative comonoids. We then locate this primitive construction inside the standard cospan/corelation framework: Cospan(\mathcal B) realizes pushout-style gluing as a free hypergraph category; Cospan(FinSet) collapses under jointly epic corestriction to FinCorel, the PROP for extraspecial commutative Frobenius monoids; and the Yoneda envelope [ \mathcal W=Fun(FinCorelop,Spc) ] is a presheaf ∞-topos carrying the standard subobject, modality, and monotone fixed-point apparatus. The PROP-level identification AncQ≃ Cocom is the only result claimed as new; the remaining material is organizational and reduces explicitly to cited classical results.