{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2008:7WK5TS3VJ4FF4L3XNA7WJU6VNK","short_pith_number":"pith:7WK5TS3V","canonical_record":{"source":{"id":"0804.1820","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CT","submitted_at":"2008-04-11T01:08:52Z","cross_cats_sorted":[],"title_canon_sha256":"b95e483068922062158fa0153e1418b35b04669cf8de6ebf7873a80ae441f757","abstract_canon_sha256":"05264331e6c85acbd75f1eced714dc774de52b4180f403e065d34ae4f6a5d82c"},"schema_version":"1.0"},"canonical_sha256":"fd95d9cb754f0a5e2f77683f64d3d56a8e8c23d49ec6a749dc0d47f24e202fad","source":{"kind":"arxiv","id":"0804.1820","version":2},"source_aliases":[{"alias_kind":"arxiv","alias_value":"0804.1820","created_at":"2026-05-18T03:13:22Z"},{"alias_kind":"arxiv_version","alias_value":"0804.1820v2","created_at":"2026-05-18T03:13:22Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.0804.1820","created_at":"2026-05-18T03:13:22Z"},{"alias_kind":"pith_short_12","alias_value":"7WK5TS3VJ4FF","created_at":"2026-05-18T12:25:56Z"},{"alias_kind":"pith_short_16","alias_value":"7WK5TS3VJ4FF4L3X","created_at":"2026-05-18T12:25:56Z"},{"alias_kind":"pith_short_8","alias_value":"7WK5TS3V","created_at":"2026-05-18T12:25:56Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2008:7WK5TS3VJ4FF4L3XNA7WJU6VNK","target":"record","payload":{"canonical_record":{"source":{"id":"0804.1820","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CT","submitted_at":"2008-04-11T01:08:52Z","cross_cats_sorted":[],"title_canon_sha256":"b95e483068922062158fa0153e1418b35b04669cf8de6ebf7873a80ae441f757","abstract_canon_sha256":"05264331e6c85acbd75f1eced714dc774de52b4180f403e065d34ae4f6a5d82c"},"schema_version":"1.0"},"canonical_sha256":"fd95d9cb754f0a5e2f77683f64d3d56a8e8c23d49ec6a749dc0d47f24e202fad","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T03:13:22.249364Z","signature_b64":"KfuBDhfT9krLUbMazQYc3SMCoq14O16+A9cp4BVQ6xiEGaYYoFjl2HIqe1pGLVt0pUykh+H4/f8vWIfttobVBw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"fd95d9cb754f0a5e2f77683f64d3d56a8e8c23d49ec6a749dc0d47f24e202fad","last_reissued_at":"2026-05-18T03:13:22.248662Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T03:13:22.248662Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"0804.1820","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":"2026-05-18T03:13:22Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"Dd6ZJyTJ+duB5dbTrpJ+8CThJp9g4KgGepXdCDp2MRU9Ch51J7L/SKmmVlvPFd/XeGhYvv/yx59cjR34XcYPDQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-24T23:18:25.298871Z"},"content_sha256":"663e4052873214ed682be7ee7fe9027b6fae28764ffaaa37c258f6dbb491a292","schema_version":"1.0","event_id":"sha256:663e4052873214ed682be7ee7fe9027b6fae28764ffaaa37c258f6dbb491a292"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2008:7WK5TS3VJ4FF4L3XNA7WJU6VNK","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Structure of Ann-Categories","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CT","authors_text":"Nguyen Tien Quang","submitted_at":"2008-04-11T01:08:52Z","abstract_excerpt":"This paper presents the structure conversion by which from an Ann-category $\\A,$ we can obtain its reduced Ann-category of the type $(R,M)$ whose structure is a family of five functions $k=(\\xi,\\eta,\\alpha,\\lambda,\\rho)$. Then we will show that each Ann-category is determined by three invariants:\n  1. The ring $\\Pi_0(\\A)$ of the isomorphic classes of objects of $\\A$,\n  2. $\\Pi_0(\\A)$-bimodule $\\Pi_1(\\A) = \\Aut_{\\A}(0),$\n  3. The element $ \\bar{k}\\in H^{3}_{M}(\\Pi_0(\\A), \\Pi_1(\\A))$ (the ring cohomology due to MacLane)."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"0804.1820","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":""},"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":"2026-05-18T03:13:22Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"iSLaVSU2/iv14xynnxMCmsbZrkMrzPtAftnS4dM9NlUs6u/Jn5uW465qObuvlpgJ1jjtvRxUsYsNWl5px25cBQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-24T23:18:25.299211Z"},"content_sha256":"ffb7ad6f4f1af6d5cf55f7ea6e8d937647fc45afd176a744fb08c914d71fe127","schema_version":"1.0","event_id":"sha256:ffb7ad6f4f1af6d5cf55f7ea6e8d937647fc45afd176a744fb08c914d71fe127"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/7WK5TS3VJ4FF4L3XNA7WJU6VNK/bundle.json","state_url":"https://pith.science/pith/7WK5TS3VJ4FF4L3XNA7WJU6VNK/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/7WK5TS3VJ4FF4L3XNA7WJU6VNK/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":"2026-06-24T23:18:25Z","links":{"resolver":"https://pith.science/pith/7WK5TS3VJ4FF4L3XNA7WJU6VNK","bundle":"https://pith.science/pith/7WK5TS3VJ4FF4L3XNA7WJU6VNK/bundle.json","state":"https://pith.science/pith/7WK5TS3VJ4FF4L3XNA7WJU6VNK/state.json","well_known_bundle":"https://pith.science/.well-known/pith/7WK5TS3VJ4FF4L3XNA7WJU6VNK/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2008:7WK5TS3VJ4FF4L3XNA7WJU6VNK","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":"05264331e6c85acbd75f1eced714dc774de52b4180f403e065d34ae4f6a5d82c","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CT","submitted_at":"2008-04-11T01:08:52Z","title_canon_sha256":"b95e483068922062158fa0153e1418b35b04669cf8de6ebf7873a80ae441f757"},"schema_version":"1.0","source":{"id":"0804.1820","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"0804.1820","created_at":"2026-05-18T03:13:22Z"},{"alias_kind":"arxiv_version","alias_value":"0804.1820v2","created_at":"2026-05-18T03:13:22Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.0804.1820","created_at":"2026-05-18T03:13:22Z"},{"alias_kind":"pith_short_12","alias_value":"7WK5TS3VJ4FF","created_at":"2026-05-18T12:25:56Z"},{"alias_kind":"pith_short_16","alias_value":"7WK5TS3VJ4FF4L3X","created_at":"2026-05-18T12:25:56Z"},{"alias_kind":"pith_short_8","alias_value":"7WK5TS3V","created_at":"2026-05-18T12:25:56Z"}],"graph_snapshots":[{"event_id":"sha256:ffb7ad6f4f1af6d5cf55f7ea6e8d937647fc45afd176a744fb08c914d71fe127","target":"graph","created_at":"2026-05-18T03:13:22Z","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"},"paper":{"abstract_excerpt":"This paper presents the structure conversion by which from an Ann-category $\\A,$ we can obtain its reduced Ann-category of the type $(R,M)$ whose structure is a family of five functions $k=(\\xi,\\eta,\\alpha,\\lambda,\\rho)$. Then we will show that each Ann-category is determined by three invariants:\n  1. The ring $\\Pi_0(\\A)$ of the isomorphic classes of objects of $\\A$,\n  2. $\\Pi_0(\\A)$-bimodule $\\Pi_1(\\A) = \\Aut_{\\A}(0),$\n  3. The element $ \\bar{k}\\in H^{3}_{M}(\\Pi_0(\\A), \\Pi_1(\\A))$ (the ring cohomology due to MacLane).","authors_text":"Nguyen Tien Quang","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CT","submitted_at":"2008-04-11T01:08:52Z","title":"Structure of Ann-Categories"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"0804.1820","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":"sha256:663e4052873214ed682be7ee7fe9027b6fae28764ffaaa37c258f6dbb491a292","target":"record","created_at":"2026-05-18T03:13:22Z","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":"05264331e6c85acbd75f1eced714dc774de52b4180f403e065d34ae4f6a5d82c","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CT","submitted_at":"2008-04-11T01:08:52Z","title_canon_sha256":"b95e483068922062158fa0153e1418b35b04669cf8de6ebf7873a80ae441f757"},"schema_version":"1.0","source":{"id":"0804.1820","kind":"arxiv","version":2}},"canonical_sha256":"fd95d9cb754f0a5e2f77683f64d3d56a8e8c23d49ec6a749dc0d47f24e202fad","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"fd95d9cb754f0a5e2f77683f64d3d56a8e8c23d49ec6a749dc0d47f24e202fad","first_computed_at":"2026-05-18T03:13:22.248662Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T03:13:22.248662Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"KfuBDhfT9krLUbMazQYc3SMCoq14O16+A9cp4BVQ6xiEGaYYoFjl2HIqe1pGLVt0pUykh+H4/f8vWIfttobVBw==","signature_status":"signed_v1","signed_at":"2026-05-18T03:13:22.249364Z","signed_message":"canonical_sha256_bytes"},"source_id":"0804.1820","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:663e4052873214ed682be7ee7fe9027b6fae28764ffaaa37c258f6dbb491a292","sha256:ffb7ad6f4f1af6d5cf55f7ea6e8d937647fc45afd176a744fb08c914d71fe127"],"state_sha256":"70a8211eb53e1d15885c0228a1f96d8659fe71b839422780ce79ffe2e265b81f"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"u8fr3G2ZpKjVFHGNQcpHX/AzOyB8C2mgu+I3dW8nvjzsRGxYizy/sm4A2csMSMnq15qmVoEocEzx9hI/P6EHCg==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-24T23:18:25.301183Z","bundle_sha256":"81a231b9e5f80a8294d0f5f5277de1fc8d4098db86d38256f2341afec8565487"}}