{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2002:QDUVJ5GMOTNOO6SAD7K6KDBI2P","short_pith_number":"pith:QDUVJ5GM","canonical_record":{"source":{"id":"math/0206113","kind":"arxiv","version":2},"metadata":{"license":"","primary_cat":"math.QA","submitted_at":"2002-06-11T08:58:19Z","cross_cats_sorted":["math.CT","math.RA"],"title_canon_sha256":"ccd7120a71c368c5c1f55548a5fdc5c3d0973943bf0062b5ba81dc7899394a4f","abstract_canon_sha256":"dc643937bc7c8dcafac2de59e12b331a8791b1f5ef82d3f311e9fd0c029cbd84"},"schema_version":"1.0"},"canonical_sha256":"80e954f4cc74dae77a401fd5e50c28d3ddda5bbf7ffb6f8b3ab13e3ab1edff21","source":{"kind":"arxiv","id":"math/0206113","version":2},"source_aliases":[{"alias_kind":"arxiv","alias_value":"math/0206113","created_at":"2026-05-17T23:45:55Z"},{"alias_kind":"arxiv_version","alias_value":"math/0206113v2","created_at":"2026-05-17T23:45:55Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.math/0206113","created_at":"2026-05-17T23:45:55Z"},{"alias_kind":"pith_short_12","alias_value":"QDUVJ5GMOTNO","created_at":"2026-05-18T12:25:51Z"},{"alias_kind":"pith_short_16","alias_value":"QDUVJ5GMOTNOO6SA","created_at":"2026-05-18T12:25:51Z"},{"alias_kind":"pith_short_8","alias_value":"QDUVJ5GM","created_at":"2026-05-18T12:25:51Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2002:QDUVJ5GMOTNOO6SAD7K6KDBI2P","target":"record","payload":{"canonical_record":{"source":{"id":"math/0206113","kind":"arxiv","version":2},"metadata":{"license":"","primary_cat":"math.QA","submitted_at":"2002-06-11T08:58:19Z","cross_cats_sorted":["math.CT","math.RA"],"title_canon_sha256":"ccd7120a71c368c5c1f55548a5fdc5c3d0973943bf0062b5ba81dc7899394a4f","abstract_canon_sha256":"dc643937bc7c8dcafac2de59e12b331a8791b1f5ef82d3f311e9fd0c029cbd84"},"schema_version":"1.0"},"canonical_sha256":"80e954f4cc74dae77a401fd5e50c28d3ddda5bbf7ffb6f8b3ab13e3ab1edff21","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-17T23:45:55.728067Z","signature_b64":"n/2JUOROPjEMGbbIko2NUukyqCoVdCBLhb1f1JlQJY1rKkuwiyttlgRd/YwQyP65QnIB3iK73bju3nsGZnDzBw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"80e954f4cc74dae77a401fd5e50c28d3ddda5bbf7ffb6f8b3ab13e3ab1edff21","last_reissued_at":"2026-05-17T23:45:55.727624Z","signature_status":"signed_v1","first_computed_at":"2026-05-17T23:45:55.727624Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"math/0206113","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":"2026-05-17T23:45:55Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"fGuOCr+Qa0gpLYQKI0NQvbmA5bowlZqlLXIeBm+3Z7j1Ap/Gd2ZvcouplqysEa4pJVSdWADQUAzZCW90ns5QBg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-08T02:04:19.043783Z"},"content_sha256":"f643e9daed1ac3d828a5ce6289a4e6fd852c26dc9880dd5f63752616ed88b041","schema_version":"1.0","event_id":"sha256:f643e9daed1ac3d828a5ce6289a4e6fd852c26dc9880dd5f63752616ed88b041"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2002:QDUVJ5GMOTNOO6SAD7K6KDBI2P","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Tannaka-Krein duality for Hopf algebroids","license":"","headline":"","cross_cats":["math.CT","math.RA"],"primary_cat":"math.QA","authors_text":"Phung Ho Hai","submitted_at":"2002-06-11T08:58:19Z","abstract_excerpt":"We develop the Tannaka-Krein duality for monoidal functors with target in the categories of bimodules over a ring. The $\\coend$ of such a functor turns out to be a Hopf algebroid over this ring. Using the result of a previous paper we characterize a small abelian, locally finite rigid monoidal category as the category of rigid comodules over a transitive Hopf algebroid."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"math/0206113","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":"2026-05-17T23:45:55Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"uKoMxE9G1+piItu/YpCpXt2dSh4WNEFIjPnMD973HDWxapOt7RWmIop1nyWkGs7U8NzEwWYTL0YudbUqAU5WBA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-08T02:04:19.044474Z"},"content_sha256":"9a7da6c7760c249687d4c8a576c618edee0f5b54f0d4def6eb888aeea098a2a7","schema_version":"1.0","event_id":"sha256:9a7da6c7760c249687d4c8a576c618edee0f5b54f0d4def6eb888aeea098a2a7"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/QDUVJ5GMOTNOO6SAD7K6KDBI2P/bundle.json","state_url":"https://pith.science/pith/QDUVJ5GMOTNOO6SAD7K6KDBI2P/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/QDUVJ5GMOTNOO6SAD7K6KDBI2P/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":"2026-06-08T02:04:19Z","links":{"resolver":"https://pith.science/pith/QDUVJ5GMOTNOO6SAD7K6KDBI2P","bundle":"https://pith.science/pith/QDUVJ5GMOTNOO6SAD7K6KDBI2P/bundle.json","state":"https://pith.science/pith/QDUVJ5GMOTNOO6SAD7K6KDBI2P/state.json","well_known_bundle":"https://pith.science/.well-known/pith/QDUVJ5GMOTNOO6SAD7K6KDBI2P/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2002:QDUVJ5GMOTNOO6SAD7K6KDBI2P","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":"dc643937bc7c8dcafac2de59e12b331a8791b1f5ef82d3f311e9fd0c029cbd84","cross_cats_sorted":["math.CT","math.RA"],"license":"","primary_cat":"math.QA","submitted_at":"2002-06-11T08:58:19Z","title_canon_sha256":"ccd7120a71c368c5c1f55548a5fdc5c3d0973943bf0062b5ba81dc7899394a4f"},"schema_version":"1.0","source":{"id":"math/0206113","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"math/0206113","created_at":"2026-05-17T23:45:55Z"},{"alias_kind":"arxiv_version","alias_value":"math/0206113v2","created_at":"2026-05-17T23:45:55Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.math/0206113","created_at":"2026-05-17T23:45:55Z"},{"alias_kind":"pith_short_12","alias_value":"QDUVJ5GMOTNO","created_at":"2026-05-18T12:25:51Z"},{"alias_kind":"pith_short_16","alias_value":"QDUVJ5GMOTNOO6SA","created_at":"2026-05-18T12:25:51Z"},{"alias_kind":"pith_short_8","alias_value":"QDUVJ5GM","created_at":"2026-05-18T12:25:51Z"}],"graph_snapshots":[{"event_id":"sha256:9a7da6c7760c249687d4c8a576c618edee0f5b54f0d4def6eb888aeea098a2a7","target":"graph","created_at":"2026-05-17T23:45:55Z","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":"We develop the Tannaka-Krein duality for monoidal functors with target in the categories of bimodules over a ring. The $\\coend$ of such a functor turns out to be a Hopf algebroid over this ring. Using the result of a previous paper we characterize a small abelian, locally finite rigid monoidal category as the category of rigid comodules over a transitive Hopf algebroid.","authors_text":"Phung Ho Hai","cross_cats":["math.CT","math.RA"],"headline":"","license":"","primary_cat":"math.QA","submitted_at":"2002-06-11T08:58:19Z","title":"Tannaka-Krein duality for Hopf algebroids"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"math/0206113","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:f643e9daed1ac3d828a5ce6289a4e6fd852c26dc9880dd5f63752616ed88b041","target":"record","created_at":"2026-05-17T23:45:55Z","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":"dc643937bc7c8dcafac2de59e12b331a8791b1f5ef82d3f311e9fd0c029cbd84","cross_cats_sorted":["math.CT","math.RA"],"license":"","primary_cat":"math.QA","submitted_at":"2002-06-11T08:58:19Z","title_canon_sha256":"ccd7120a71c368c5c1f55548a5fdc5c3d0973943bf0062b5ba81dc7899394a4f"},"schema_version":"1.0","source":{"id":"math/0206113","kind":"arxiv","version":2}},"canonical_sha256":"80e954f4cc74dae77a401fd5e50c28d3ddda5bbf7ffb6f8b3ab13e3ab1edff21","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"80e954f4cc74dae77a401fd5e50c28d3ddda5bbf7ffb6f8b3ab13e3ab1edff21","first_computed_at":"2026-05-17T23:45:55.727624Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-17T23:45:55.727624Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"n/2JUOROPjEMGbbIko2NUukyqCoVdCBLhb1f1JlQJY1rKkuwiyttlgRd/YwQyP65QnIB3iK73bju3nsGZnDzBw==","signature_status":"signed_v1","signed_at":"2026-05-17T23:45:55.728067Z","signed_message":"canonical_sha256_bytes"},"source_id":"math/0206113","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:f643e9daed1ac3d828a5ce6289a4e6fd852c26dc9880dd5f63752616ed88b041","sha256:9a7da6c7760c249687d4c8a576c618edee0f5b54f0d4def6eb888aeea098a2a7"],"state_sha256":"d6a6e2fab93e93cfb08a8429adf5a338d60b61db7bfc6452c01b1318dc2dbdcd"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"iQRXp3tFBPp7dm1JLbBetKfv7tDe/RqGH+E4+d7TeWetCdh6xXZdQcoTlHPzEo76l1fuRbTmQv/xsLRqrXnlAw==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-08T02:04:19.048195Z","bundle_sha256":"2f8db1e3ca55e66a954499e336740c272bc1776041cd8d5463de3baad8614fcb"}}