{"paper":{"title":"Categorical (Co)Limits of Quantum Graphs","license":"http://creativecommons.org/licenses/by/4.0/","headline":"Quantum graphs are identified with left ideals in the extended Haagerup tensor product to give representation-free morphisms and categorical colimits.","cross_cats":["math.CT"],"primary_cat":"math.OA","authors_text":"Jennifer Zhu","submitted_at":"2026-05-13T05:21:07Z","abstract_excerpt":"We begin with the characterization of quantum graphs as left ideals in $\\mathcal M \\otimes_{eh} \\mathcal M$ (the extended Haagerup tensor product of $\\mathcal M$ with itself) to avoid technicalities surrounding representation dependence of quantum graphs. These left ideals roughly correspond to a canonical complement of a quantum graph. Using these left ideals and some operator space theory, we find a new, representation-free characterization of a morphism of quantum graphs compatible with previous representation-dependent morphisms. A notion of categorical (co)limit of quantum graphs follows."},"claims":{"count":4,"items":[{"kind":"strongest_claim","text":"Using these left ideals and some operator space theory, we find a new, representation-free characterization of a morphism of quantum graphs compatible with previous representation-dependent morphisms. A notion of categorical (co)limit of quantum graphs follows.","source":"verdict.strongest_claim","status":"machine_extracted","claim_id":"C1","attestation":"unclaimed"},{"kind":"weakest_assumption","text":"That left ideals in the extended Haagerup tensor product M ⊗_eh M provide a canonical complement to a quantum graph and that this correspondence yields morphisms compatible with all prior representation-dependent definitions.","source":"verdict.weakest_assumption","status":"machine_extracted","claim_id":"C2","attestation":"unclaimed"},{"kind":"one_line_summary","text":"Quantum graphs are redefined as left ideals in the extended Haagerup tensor product, enabling representation-independent morphisms and categorical (co)limits.","source":"verdict.one_line_summary","status":"machine_extracted","claim_id":"C3","attestation":"unclaimed"},{"kind":"headline","text":"Quantum graphs are identified with left ideals in the extended Haagerup tensor product to give representation-free morphisms and categorical colimits.","source":"verdict.pith_extraction.headline","status":"machine_extracted","claim_id":"C4","attestation":"unclaimed"}],"snapshot_sha256":"a8b14b035e387a8d34d172483d1a314e563453d1821b94a7fe7954fc203840ad"},"source":{"id":"2605.13019","kind":"arxiv","version":1},"verdict":{"id":"3a66167b-f0e3-4151-bc44-756e9a24b8ff","model_set":{"reader":"grok-4.3"},"created_at":"2026-05-14T02:02:35.720291Z","strongest_claim":"Using these left ideals and some operator space theory, we find a new, representation-free characterization of a morphism of quantum graphs compatible with previous representation-dependent morphisms. A notion of categorical (co)limit of quantum graphs follows.","one_line_summary":"Quantum graphs are redefined as left ideals in the extended Haagerup tensor product, enabling representation-independent morphisms and categorical (co)limits.","pipeline_version":"pith-pipeline@v0.9.0","weakest_assumption":"That left ideals in the extended Haagerup tensor product M ⊗_eh M provide a canonical complement to a quantum graph and that this correspondence yields morphisms compatible with all prior representation-dependent definitions.","pith_extraction_headline":"Quantum graphs are identified with left ideals in the extended Haagerup tensor product to give representation-free morphisms and categorical colimits."},"references":{"count":15,"sample":[{"doi":"10.1063/5.0072288(cit","year":2022,"title":"The quantum-to- classical graph homomorphism game","work_id":"892b9f9c-0846-48a3-9591-0a733581fdef","ref_index":1,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":2004,"title":"The Dual of the Haagerup Tensor Prod- uct","work_id":"168b8627-4fda-4fda-9f54-9982bcf66c3b","ref_index":2,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"10.1112/jlms/s2-45.1.126(cit","year":null,"title":"onlinelibrary","work_id":"fd69837b-9b9b-4377-9de2-7a5fd9478403","ref_index":3,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"10.1007/978-3-031-","year":2025,"title":"Connectivity for Quantum Graphs via Quantum Adjacency Operators","work_id":"4a7d968d-ee43-47f9-9cca-832cf129af59","ref_index":4,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"10.1109/tit.2012.2221677.url:http://dx.doi","year":2013,"title":"Zero-Error Communica- tion via Quantum Channels, Noncommutative Graphs, and a Quantum Lov´ asz Number","work_id":"d91bb95a-3662-40fd-94dd-cbb7c6cbd123","ref_index":5,"cited_arxiv_id":"","is_internal_anchor":false}],"resolved_work":15,"snapshot_sha256":"7ea1fdfc7831c98e99f81d2cf43a135b3965e2a9a0e484e8783469d39f35b8b3","internal_anchors":1},"formal_canon":{"evidence_count":2,"snapshot_sha256":"43160ce1ee8229fcf8de8d38b66e16ee24f8fe666ad0851d3ffd555f063aa03b"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}