{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:1993:O6DNFQZBLQ23ZWPBEJNUYR2Y7F","short_pith_number":"pith:O6DNFQZB","canonical_record":{"source":{"id":"hep-th/9310164","kind":"arxiv","version":2},"metadata":{"license":"","primary_cat":"hep-th","submitted_at":"1993-10-25T15:45:14Z","cross_cats_sorted":["math.QA"],"title_canon_sha256":"7fa93603af994110bf0c13305a0186cdff5de3bf545c992a06d5cf13ef65a1c8","abstract_canon_sha256":"b4a0f3c71b103c29c7caed2e2124de9b8ab5a8af8ec8c2bfa36eb08c749ffa48"},"schema_version":"1.0"},"canonical_sha256":"7786d2c3215c35bcd9e1225b4c4758f95c6d8c8d62a34875eb7169ead1305f95","source":{"kind":"arxiv","id":"hep-th/9310164","version":2},"source_aliases":[{"alias_kind":"arxiv","alias_value":"hep-th/9310164","created_at":"2026-07-04T15:21:35Z"},{"alias_kind":"arxiv_version","alias_value":"hep-th/9310164v2","created_at":"2026-07-04T15:21:35Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.hep-th/9310164","created_at":"2026-07-04T15:21:35Z"},{"alias_kind":"pith_short_12","alias_value":"O6DNFQZBLQ23","created_at":"2026-07-04T15:21:35Z"},{"alias_kind":"pith_short_16","alias_value":"O6DNFQZBLQ23ZWPB","created_at":"2026-07-04T15:21:35Z"},{"alias_kind":"pith_short_8","alias_value":"O6DNFQZB","created_at":"2026-07-04T15:21:35Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:1993:O6DNFQZBLQ23ZWPBEJNUYR2Y7F","target":"record","payload":{"canonical_record":{"source":{"id":"hep-th/9310164","kind":"arxiv","version":2},"metadata":{"license":"","primary_cat":"hep-th","submitted_at":"1993-10-25T15:45:14Z","cross_cats_sorted":["math.QA"],"title_canon_sha256":"7fa93603af994110bf0c13305a0186cdff5de3bf545c992a06d5cf13ef65a1c8","abstract_canon_sha256":"b4a0f3c71b103c29c7caed2e2124de9b8ab5a8af8ec8c2bfa36eb08c749ffa48"},"schema_version":"1.0"},"canonical_sha256":"7786d2c3215c35bcd9e1225b4c4758f95c6d8c8d62a34875eb7169ead1305f95","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-04T15:21:35.822897Z","signature_b64":"tMY+sN4Da68Tsk3E86J9XHrjfx+e+AsNICOLYF56nduLYsHTW+nkT8oXscp50ZBqIgvCOJ/H0qM/IBeTD666Bw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"7786d2c3215c35bcd9e1225b4c4758f95c6d8c8d62a34875eb7169ead1305f95","last_reissued_at":"2026-07-04T15:21:35.822501Z","signature_status":"signed_v1","first_computed_at":"2026-07-04T15:21:35.822501Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"hep-th/9310164","source_version":2,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-07-04T15:21:35Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"i9EQEifSVZNGJHTytXJgW+No4KCtzb26t2dX9gqXa6IbwG5Qm3u51XOjKJISqZZ0CO7DaXGqC/xD/FztNA9MAA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-10T01:41:40.557083Z"},"content_sha256":"f7ef62c7a6711d95974d388e3dbef0f3b340997800d67e77cbc033d0c0e5eaf2","schema_version":"1.0","event_id":"sha256:f7ef62c7a6711d95974d388e3dbef0f3b340997800d67e77cbc033d0c0e5eaf2"},{"event_type":"graph_snapshot","subject_pith_number":"pith:1993:O6DNFQZBLQ23ZWPBEJNUYR2Y7F","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Spherical Categories","license":"","headline":"","cross_cats":["math.QA"],"primary_cat":"hep-th","authors_text":"Bruce W. Westbury, John W. Barrett","submitted_at":"1993-10-25T15:45:14Z","abstract_excerpt":"This paper is a study of monoidal categories with duals where the tensor product need not be commutative. The motivating examples are categories of representations of Hopf algebras and the motivating application is the definition of 6j-symbols as used in topological field theories.\n  We introduce the new notion of a spherical category. In the first section we prove a coherence theorem for a monoidal category with duals following MacLane (1963). In the second section we give the definition of a spherical category, and construct a natural quotient which is also spherical.\n  In the third section "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"hep-th/9310164","kind":"arxiv","version":2},"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/hep-th/9310164/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"},"verdict_id":null},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-07-04T15:21:35Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"1OF4OeUouYVoMB1uIfRHCOUbfBRPGpsqlK3qSqjOS9DgKDtPia/nQIslQf9wD+NAWAtcOLAqqAIgc3VyyqC2Cw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-10T01:41:40.557587Z"},"content_sha256":"a668c8eab36bb1b8dbfdc838f4e69d139403091dd7ccdc75d9be36b2afc661fe","schema_version":"1.0","event_id":"sha256:a668c8eab36bb1b8dbfdc838f4e69d139403091dd7ccdc75d9be36b2afc661fe"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/O6DNFQZBLQ23ZWPBEJNUYR2Y7F/bundle.json","state_url":"https://pith.science/pith/O6DNFQZBLQ23ZWPBEJNUYR2Y7F/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/O6DNFQZBLQ23ZWPBEJNUYR2Y7F/bundle.json","status":"primary"}],"public_keys":[{"key_id":"pith-v1-2026-05","algorithm":"ed25519","format":"raw","public_key_b64":"stVStoiQhXFxp4s2pdzPNoqVNBMojDU/fJ2db5S3CbM=","public_key_hex":"b2d552b68890857171a78b36a5dccf368a953413288c353f7c9d9d6f94b709b3","fingerprint_sha256_b32_first128bits":"RVFV5Z2OI2J3ZUO7ERDEBCYNKS","fingerprint_sha256_hex":"8d4b5ee74e4693bcd1df2446408b0d54","rotates_at":null,"url":"https://pith.science/pith-signing-key.json","notes":"Pith uses this Ed25519 key to sign canonical record SHA-256 digests. Verify with: ed25519_verify(public_key, message=canonical_sha256_bytes, signature=base64decode(signature_b64))."}],"merge_version":"pith-open-graph-merge-v1","built_at":"2026-08-10T01:41:40Z","links":{"resolver":"https://pith.science/pith/O6DNFQZBLQ23ZWPBEJNUYR2Y7F","bundle":"https://pith.science/pith/O6DNFQZBLQ23ZWPBEJNUYR2Y7F/bundle.json","state":"https://pith.science/pith/O6DNFQZBLQ23ZWPBEJNUYR2Y7F/state.json","well_known_bundle":"https://pith.science/.well-known/pith/O6DNFQZBLQ23ZWPBEJNUYR2Y7F/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:1993:O6DNFQZBLQ23ZWPBEJNUYR2Y7F","merge_version":"pith-open-graph-merge-v1","event_count":2,"valid_event_count":2,"invalid_event_count":0,"equivocation_count":0,"current":{"canonical_record":{"metadata":{"abstract_canon_sha256":"b4a0f3c71b103c29c7caed2e2124de9b8ab5a8af8ec8c2bfa36eb08c749ffa48","cross_cats_sorted":["math.QA"],"license":"","primary_cat":"hep-th","submitted_at":"1993-10-25T15:45:14Z","title_canon_sha256":"7fa93603af994110bf0c13305a0186cdff5de3bf545c992a06d5cf13ef65a1c8"},"schema_version":"1.0","source":{"id":"hep-th/9310164","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"hep-th/9310164","created_at":"2026-07-04T15:21:35Z"},{"alias_kind":"arxiv_version","alias_value":"hep-th/9310164v2","created_at":"2026-07-04T15:21:35Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.hep-th/9310164","created_at":"2026-07-04T15:21:35Z"},{"alias_kind":"pith_short_12","alias_value":"O6DNFQZBLQ23","created_at":"2026-07-04T15:21:35Z"},{"alias_kind":"pith_short_16","alias_value":"O6DNFQZBLQ23ZWPB","created_at":"2026-07-04T15:21:35Z"},{"alias_kind":"pith_short_8","alias_value":"O6DNFQZB","created_at":"2026-07-04T15:21:35Z"}],"graph_snapshots":[{"event_id":"sha256:a668c8eab36bb1b8dbfdc838f4e69d139403091dd7ccdc75d9be36b2afc661fe","target":"graph","created_at":"2026-07-04T15:21:35Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"graph_snapshot":{"author_claims":{"count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","strong_count":0},"builder_version":"pith-number-builder-2026-05-17-v1","claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"integrity":{"available":true,"clean":true,"detectors_run":[],"endpoint":"/pith/hep-th/9310164/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"This paper is a study of monoidal categories with duals where the tensor product need not be commutative. The motivating examples are categories of representations of Hopf algebras and the motivating application is the definition of 6j-symbols as used in topological field theories.\n  We introduce the new notion of a spherical category. In the first section we prove a coherence theorem for a monoidal category with duals following MacLane (1963). In the second section we give the definition of a spherical category, and construct a natural quotient which is also spherical.\n  In the third section ","authors_text":"Bruce W. Westbury, John W. Barrett","cross_cats":["math.QA"],"headline":"","license":"","primary_cat":"hep-th","submitted_at":"1993-10-25T15:45:14Z","title":"Spherical Categories"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"hep-th/9310164","kind":"arxiv","version":2},"verdict":{"created_at":null,"id":null,"model_set":{},"one_line_summary":"","pipeline_version":null,"pith_extraction_headline":"","strongest_claim":"","weakest_assumption":""}},"verdict_id":null}}],"author_attestations":[],"timestamp_anchors":[],"storage_attestations":[],"citation_signatures":[],"replication_records":[],"corrections":[],"mirror_hints":[],"record_created":{"event_id":"sha256:f7ef62c7a6711d95974d388e3dbef0f3b340997800d67e77cbc033d0c0e5eaf2","target":"record","created_at":"2026-07-04T15:21:35Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"attestation_state":"computed","canonical_record":{"metadata":{"abstract_canon_sha256":"b4a0f3c71b103c29c7caed2e2124de9b8ab5a8af8ec8c2bfa36eb08c749ffa48","cross_cats_sorted":["math.QA"],"license":"","primary_cat":"hep-th","submitted_at":"1993-10-25T15:45:14Z","title_canon_sha256":"7fa93603af994110bf0c13305a0186cdff5de3bf545c992a06d5cf13ef65a1c8"},"schema_version":"1.0","source":{"id":"hep-th/9310164","kind":"arxiv","version":2}},"canonical_sha256":"7786d2c3215c35bcd9e1225b4c4758f95c6d8c8d62a34875eb7169ead1305f95","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"7786d2c3215c35bcd9e1225b4c4758f95c6d8c8d62a34875eb7169ead1305f95","first_computed_at":"2026-07-04T15:21:35.822501Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-07-04T15:21:35.822501Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"tMY+sN4Da68Tsk3E86J9XHrjfx+e+AsNICOLYF56nduLYsHTW+nkT8oXscp50ZBqIgvCOJ/H0qM/IBeTD666Bw==","signature_status":"signed_v1","signed_at":"2026-07-04T15:21:35.822897Z","signed_message":"canonical_sha256_bytes"},"source_id":"hep-th/9310164","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:f7ef62c7a6711d95974d388e3dbef0f3b340997800d67e77cbc033d0c0e5eaf2","sha256:a668c8eab36bb1b8dbfdc838f4e69d139403091dd7ccdc75d9be36b2afc661fe"],"state_sha256":"0212c5ff9885e5e2585408df3a4e75af907a5f4f58633d150b1118a7b6f2b1c9"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"4shrGppml4Wb3a76p6aOe9G8yyUL/1J/Kwn37kkMvgc0kDjZ+uARWeiyLnY3f1nCFvpAWnc4z+r65szgosBODg==","signed_message":"bundle_sha256_bytes","signed_at":"2026-08-10T01:41:40.564099Z","bundle_sha256":"aba43e49a673c8237180bf6f42b0d7977ef7d7293d4a792e27984ea7d03c48b0"}}