{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2024:E53JEVGQOK3QSN5RASQCXQPG3I","short_pith_number":"pith:E53JEVGQ","schema_version":"1.0","canonical_sha256":"27769254d072b70937b104a02bc1e6da213a1a1775d44bb31d5134391f5d0883","source":{"kind":"arxiv","id":"2411.01678","version":1},"attestation_state":"computed","paper":{"title":"Complete W*-categories","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":["math.CT"],"primary_cat":"math.OA","authors_text":"Andr\\'e Henriques, David Penneys, Nivedita","submitted_at":"2024-11-03T20:36:37Z","abstract_excerpt":"We study $\\mathrm{W}^*$-categories, and explain the ways in which complete $\\mathrm{W}^*$-categories behave like categorified Hilbert spaces. Every $\\mathrm{W}^*$-category $C$ admits a canonical categorified inner product $\\langle\\,\\,,\\,\\rangle_{\\mathrm{Hilb}}\\,:\\,\\overline C\\times C\\,\\to\\, \\mathrm{Hilb}$. Moreover, if $C$ and $D$ are complete $\\mathrm{W}^*$-categories there is an antilinear equivalence $$\\dagger:\\mathrm{Func}(C,D) \\leftrightarrow \\mathrm{Func}(D,C)$$ characterised by $\\langle c,F^\\dagger(d)\\rangle_{\\mathrm{Hilb}} \\simeq \\langle F(c),d\\rangle_{\\mathrm{Hilb}}$, for $c\\in C$ and"},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"2411.01678","kind":"arxiv","version":1},"metadata":{"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.OA","submitted_at":"2024-11-03T20:36:37Z","cross_cats_sorted":["math.CT"],"title_canon_sha256":"9f3e1690c9080175e0eeb5e3f49a8685d8361b2406a468242c872062a5eb3417","abstract_canon_sha256":"72d74dcc4f092d3ad070b938cedbff8ac0d6c3132e38cd5962906d0a7800a0c9"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T09:30:34.449613Z","signature_b64":"JCUCGScVqD3xo4HczKy2a3VwRR4aZsC0iI6YYQwtG7Y6SV10Ds2vASARuMs3wZyjGY476jPmPE9lqDPHw44fCA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"27769254d072b70937b104a02bc1e6da213a1a1775d44bb31d5134391f5d0883","last_reissued_at":"2026-07-05T09:30:34.449165Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T09:30:34.449165Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Complete W*-categories","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":["math.CT"],"primary_cat":"math.OA","authors_text":"Andr\\'e Henriques, David Penneys, Nivedita","submitted_at":"2024-11-03T20:36:37Z","abstract_excerpt":"We study $\\mathrm{W}^*$-categories, and explain the ways in which complete $\\mathrm{W}^*$-categories behave like categorified Hilbert spaces. Every $\\mathrm{W}^*$-category $C$ admits a canonical categorified inner product $\\langle\\,\\,,\\,\\rangle_{\\mathrm{Hilb}}\\,:\\,\\overline C\\times C\\,\\to\\, \\mathrm{Hilb}$. Moreover, if $C$ and $D$ are complete $\\mathrm{W}^*$-categories there is an antilinear equivalence $$\\dagger:\\mathrm{Func}(C,D) \\leftrightarrow \\mathrm{Func}(D,C)$$ characterised by $\\langle c,F^\\dagger(d)\\rangle_{\\mathrm{Hilb}} \\simeq \\langle F(c),d\\rangle_{\\mathrm{Hilb}}$, for $c\\in C$ and"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2411.01678","kind":"arxiv","version":1},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2411.01678/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"},"aliases":[{"alias_kind":"arxiv","alias_value":"2411.01678","created_at":"2026-07-05T09:30:34.449223+00:00"},{"alias_kind":"arxiv_version","alias_value":"2411.01678v1","created_at":"2026-07-05T09:30:34.449223+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2411.01678","created_at":"2026-07-05T09:30:34.449223+00:00"},{"alias_kind":"pith_short_12","alias_value":"E53JEVGQOK3Q","created_at":"2026-07-05T09:30:34.449223+00:00"},{"alias_kind":"pith_short_16","alias_value":"E53JEVGQOK3QSN5R","created_at":"2026-07-05T09:30:34.449223+00:00"},{"alias_kind":"pith_short_8","alias_value":"E53JEVGQ","created_at":"2026-07-05T09:30:34.449223+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":5,"internal_anchor_count":0,"sample":[{"citing_arxiv_id":"2607.01322","citing_title":"Wormholes as red herrings: reflection positivity and the reconstruction of unitary quantum field theories","ref_index":75,"is_internal_anchor":false},{"citing_arxiv_id":"2606.06402","citing_title":"Balanced tensor categories of representations of fixed-points conformal nets","ref_index":18,"is_internal_anchor":false},{"citing_arxiv_id":"2606.03623","citing_title":"Twisted representations of conformal nets and crossed balanced tensor categories","ref_index":22,"is_internal_anchor":false},{"citing_arxiv_id":"2605.18446","citing_title":"The balanced structure on the category of representations of a conformal net","ref_index":1,"is_internal_anchor":false},{"citing_arxiv_id":"2605.18446","citing_title":"The balanced structure on the category of representations of a conformal net","ref_index":16,"is_internal_anchor":false}]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/E53JEVGQOK3QSN5RASQCXQPG3I","json":"https://pith.science/pith/E53JEVGQOK3QSN5RASQCXQPG3I.json","graph_json":"https://pith.science/api/pith-number/E53JEVGQOK3QSN5RASQCXQPG3I/graph.json","events_json":"https://pith.science/api/pith-number/E53JEVGQOK3QSN5RASQCXQPG3I/events.json","paper":"https://pith.science/paper/E53JEVGQ"},"agent_actions":{"view_html":"https://pith.science/pith/E53JEVGQOK3QSN5RASQCXQPG3I","download_json":"https://pith.science/pith/E53JEVGQOK3QSN5RASQCXQPG3I.json","view_paper":"https://pith.science/paper/E53JEVGQ","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=2411.01678&json=true","fetch_graph":"https://pith.science/api/pith-number/E53JEVGQOK3QSN5RASQCXQPG3I/graph.json","fetch_events":"https://pith.science/api/pith-number/E53JEVGQOK3QSN5RASQCXQPG3I/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/E53JEVGQOK3QSN5RASQCXQPG3I/action/timestamp_anchor","attest_storage":"https://pith.science/pith/E53JEVGQOK3QSN5RASQCXQPG3I/action/storage_attestation","attest_author":"https://pith.science/pith/E53JEVGQOK3QSN5RASQCXQPG3I/action/author_attestation","sign_citation":"https://pith.science/pith/E53JEVGQOK3QSN5RASQCXQPG3I/action/citation_signature","submit_replication":"https://pith.science/pith/E53JEVGQOK3QSN5RASQCXQPG3I/action/replication_record"}},"created_at":"2026-07-05T09:30:34.449223+00:00","updated_at":"2026-07-05T09:30:34.449223+00:00"}