{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2009:RUQN2VZQJ7KUH7SBAUPY2LXEDX","short_pith_number":"pith:RUQN2VZQ","canonical_record":{"source":{"id":"0902.4197","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CT","submitted_at":"2009-02-24T17:17:38Z","cross_cats_sorted":["math.RA"],"title_canon_sha256":"ef34571f08043e671b4b408093b1d8fd1204f446c2aa5117d83863757fbb4d73","abstract_canon_sha256":"a405bc6bba937ca565b077b8269c12bc5d20bcdf46c8bb6779236d51abf505d0"},"schema_version":"1.0"},"canonical_sha256":"8d20dd57304fd543fe41051f8d2ee41dcb052b8c96e0d4b4242ed92a5c02308d","source":{"kind":"arxiv","id":"0902.4197","version":2},"source_aliases":[{"alias_kind":"arxiv","alias_value":"0902.4197","created_at":"2026-05-18T04:40:45Z"},{"alias_kind":"arxiv_version","alias_value":"0902.4197v2","created_at":"2026-05-18T04:40:45Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.0902.4197","created_at":"2026-05-18T04:40:45Z"},{"alias_kind":"pith_short_12","alias_value":"RUQN2VZQJ7KU","created_at":"2026-05-18T12:26:01Z"},{"alias_kind":"pith_short_16","alias_value":"RUQN2VZQJ7KUH7SB","created_at":"2026-05-18T12:26:01Z"},{"alias_kind":"pith_short_8","alias_value":"RUQN2VZQ","created_at":"2026-05-18T12:26:01Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2009:RUQN2VZQJ7KUH7SBAUPY2LXEDX","target":"record","payload":{"canonical_record":{"source":{"id":"0902.4197","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CT","submitted_at":"2009-02-24T17:17:38Z","cross_cats_sorted":["math.RA"],"title_canon_sha256":"ef34571f08043e671b4b408093b1d8fd1204f446c2aa5117d83863757fbb4d73","abstract_canon_sha256":"a405bc6bba937ca565b077b8269c12bc5d20bcdf46c8bb6779236d51abf505d0"},"schema_version":"1.0"},"canonical_sha256":"8d20dd57304fd543fe41051f8d2ee41dcb052b8c96e0d4b4242ed92a5c02308d","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T04:40:45.397449Z","signature_b64":"ip5rXITmxSWEnrxb3X1EzjlPHC9jqgWs4533P5UnKShJNvcRSPh/QjQ2ics/32YK4WvxkiemJmcOldBtUZyxDA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"8d20dd57304fd543fe41051f8d2ee41dcb052b8c96e0d4b4242ed92a5c02308d","last_reissued_at":"2026-05-18T04:40:45.396908Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T04:40:45.396908Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"0902.4197","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":"2026-05-18T04:40:45Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"N+sKJwf0fzV40uHwGEw+T+K0sSOxm6dLC/tzsBzxF9HvC6J8lMAn3ZMTSS8m5oRqm0I/JVjLEi50bJjaDxGuCQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-25T07:39:30.190341Z"},"content_sha256":"2a43d4c5b2c34f85c0e7de2816e19be6b1e098f466d51c3edeb1a314b992e4bc","schema_version":"1.0","event_id":"sha256:2a43d4c5b2c34f85c0e7de2816e19be6b1e098f466d51c3edeb1a314b992e4bc"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2009:RUQN2VZQJ7KUH7SBAUPY2LXEDX","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"The 2-category of weak entwining structures","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.RA"],"primary_cat":"math.CT","authors_text":"Gabriella B\\\"ohm","submitted_at":"2009-02-24T17:17:38Z","abstract_excerpt":"A weak entwining structure in a 2-category K consists of a monad t and a comonad c, together with a 2-cell relating both structures in a way that generalizes a mixed distributive law.A weak entwining structure can be characterized as a compatible pair of a monad and a comonad, in 2-categories generalizing the 2-category of comonads and the 2-category of monads in K, respectively. This observation is used to define a 2-category Entw^w(K) of weak entwining structures in K. If the 2-category K admits Eilenberg-Moore constructions for both monads and comonads and idempotent 2-cells in K split, the"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"0902.4197","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":"2026-05-18T04:40:45Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"Xygq4ZFxroN2kuSb10AqGI6axWer2FNk9JimxrF9f531YIow3Y2UVmwycqWQLgQDCI2RDXvS0eEye9U5ctiZAA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-25T07:39:30.190693Z"},"content_sha256":"b7a742bca0a4621f16d9896a3952f2c0aa0879f0ec80f18fc8035e8eb3421efd","schema_version":"1.0","event_id":"sha256:b7a742bca0a4621f16d9896a3952f2c0aa0879f0ec80f18fc8035e8eb3421efd"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/RUQN2VZQJ7KUH7SBAUPY2LXEDX/bundle.json","state_url":"https://pith.science/pith/RUQN2VZQJ7KUH7SBAUPY2LXEDX/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/RUQN2VZQJ7KUH7SBAUPY2LXEDX/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":"2026-06-25T07:39:30Z","links":{"resolver":"https://pith.science/pith/RUQN2VZQJ7KUH7SBAUPY2LXEDX","bundle":"https://pith.science/pith/RUQN2VZQJ7KUH7SBAUPY2LXEDX/bundle.json","state":"https://pith.science/pith/RUQN2VZQJ7KUH7SBAUPY2LXEDX/state.json","well_known_bundle":"https://pith.science/.well-known/pith/RUQN2VZQJ7KUH7SBAUPY2LXEDX/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2009:RUQN2VZQJ7KUH7SBAUPY2LXEDX","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":"a405bc6bba937ca565b077b8269c12bc5d20bcdf46c8bb6779236d51abf505d0","cross_cats_sorted":["math.RA"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CT","submitted_at":"2009-02-24T17:17:38Z","title_canon_sha256":"ef34571f08043e671b4b408093b1d8fd1204f446c2aa5117d83863757fbb4d73"},"schema_version":"1.0","source":{"id":"0902.4197","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"0902.4197","created_at":"2026-05-18T04:40:45Z"},{"alias_kind":"arxiv_version","alias_value":"0902.4197v2","created_at":"2026-05-18T04:40:45Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.0902.4197","created_at":"2026-05-18T04:40:45Z"},{"alias_kind":"pith_short_12","alias_value":"RUQN2VZQJ7KU","created_at":"2026-05-18T12:26:01Z"},{"alias_kind":"pith_short_16","alias_value":"RUQN2VZQJ7KUH7SB","created_at":"2026-05-18T12:26:01Z"},{"alias_kind":"pith_short_8","alias_value":"RUQN2VZQ","created_at":"2026-05-18T12:26:01Z"}],"graph_snapshots":[{"event_id":"sha256:b7a742bca0a4621f16d9896a3952f2c0aa0879f0ec80f18fc8035e8eb3421efd","target":"graph","created_at":"2026-05-18T04:40:45Z","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":"A weak entwining structure in a 2-category K consists of a monad t and a comonad c, together with a 2-cell relating both structures in a way that generalizes a mixed distributive law.A weak entwining structure can be characterized as a compatible pair of a monad and a comonad, in 2-categories generalizing the 2-category of comonads and the 2-category of monads in K, respectively. This observation is used to define a 2-category Entw^w(K) of weak entwining structures in K. If the 2-category K admits Eilenberg-Moore constructions for both monads and comonads and idempotent 2-cells in K split, the","authors_text":"Gabriella B\\\"ohm","cross_cats":["math.RA"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CT","submitted_at":"2009-02-24T17:17:38Z","title":"The 2-category of weak entwining structures"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"0902.4197","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":"sha256:2a43d4c5b2c34f85c0e7de2816e19be6b1e098f466d51c3edeb1a314b992e4bc","target":"record","created_at":"2026-05-18T04:40:45Z","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":"a405bc6bba937ca565b077b8269c12bc5d20bcdf46c8bb6779236d51abf505d0","cross_cats_sorted":["math.RA"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CT","submitted_at":"2009-02-24T17:17:38Z","title_canon_sha256":"ef34571f08043e671b4b408093b1d8fd1204f446c2aa5117d83863757fbb4d73"},"schema_version":"1.0","source":{"id":"0902.4197","kind":"arxiv","version":2}},"canonical_sha256":"8d20dd57304fd543fe41051f8d2ee41dcb052b8c96e0d4b4242ed92a5c02308d","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"8d20dd57304fd543fe41051f8d2ee41dcb052b8c96e0d4b4242ed92a5c02308d","first_computed_at":"2026-05-18T04:40:45.396908Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T04:40:45.396908Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"ip5rXITmxSWEnrxb3X1EzjlPHC9jqgWs4533P5UnKShJNvcRSPh/QjQ2ics/32YK4WvxkiemJmcOldBtUZyxDA==","signature_status":"signed_v1","signed_at":"2026-05-18T04:40:45.397449Z","signed_message":"canonical_sha256_bytes"},"source_id":"0902.4197","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:2a43d4c5b2c34f85c0e7de2816e19be6b1e098f466d51c3edeb1a314b992e4bc","sha256:b7a742bca0a4621f16d9896a3952f2c0aa0879f0ec80f18fc8035e8eb3421efd"],"state_sha256":"5b0abb2f00c6fa00dc9308e99c63566a148623e7c03919650b5e8e60ce808097"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"YYpKBAqCfTPXUHWAoLx/9g8EElKYYNrsCUPcLznoEzkpsK0e0QG37nQxlr+305EfzakSlWtjX8uhYEXgB6hlAQ==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-25T07:39:30.192769Z","bundle_sha256":"5d84982f8189b657c64f4f90a3c2b18592dffd52bb5b50118db841ff4e91e35a"}}