{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2009:WXUYYYLY6NKLU5BW45O2WRB3Z7","short_pith_number":"pith:WXUYYYLY","schema_version":"1.0","canonical_sha256":"b5e98c6178f354ba7436e75dab443bcfc78d517443c504f81f55a630cc7e52ec","source":{"kind":"arxiv","id":"0908.2959","version":3},"attestation_state":"computed","paper":{"title":"Monomorphisms of Coalgebras","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.RA","authors_text":"A.L. Agore","submitted_at":"2009-08-20T16:49:40Z","abstract_excerpt":"We prove new necessary and sufficient conditions for a morphism of coalgebras to be a monomorphism, different from the ones already available in the literature. More precisely, $\\phi: C \\to D$ is a monomorphism of coalgebras if and only if the first cohomology groups of the coalgebras $C$ and $D$ coincide if and only if $\\sum_{i \\in I}\\epsilon(a^{i})b^{i} = \\sum_{i \\in I} a^{i} \\epsilon(b^{i})$, for all $\\sum_{i \\in I}a^{i} \\otimes b^{i} \\in C \\square_{D} C$. In particular, necessary and sufficient conditions for a Hopf algebra map to be a monomorphism are given."},"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":"0908.2959","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.RA","submitted_at":"2009-08-20T16:49:40Z","cross_cats_sorted":[],"title_canon_sha256":"d9199cdbf9966c9c1144b04844776505b7f3fdcca03bdaa2bc21f38d1b6cca1e","abstract_canon_sha256":"7124f57ba678a48cbef5389db36ccf222e63aabf0e9705dae3d50cc44ea8e635"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T02:58:31.057747Z","signature_b64":"dLMw0Got6rs02MCIc+cl9Asl3y3lFCfVySxMwEkECAdimlcjNilkULW83Ex+sVbLflsqYXEVySSRY5xao+aaBA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"b5e98c6178f354ba7436e75dab443bcfc78d517443c504f81f55a630cc7e52ec","last_reissued_at":"2026-05-18T02:58:31.056758Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T02:58:31.056758Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Monomorphisms of Coalgebras","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.RA","authors_text":"A.L. Agore","submitted_at":"2009-08-20T16:49:40Z","abstract_excerpt":"We prove new necessary and sufficient conditions for a morphism of coalgebras to be a monomorphism, different from the ones already available in the literature. More precisely, $\\phi: C \\to D$ is a monomorphism of coalgebras if and only if the first cohomology groups of the coalgebras $C$ and $D$ coincide if and only if $\\sum_{i \\in I}\\epsilon(a^{i})b^{i} = \\sum_{i \\in I} a^{i} \\epsilon(b^{i})$, for all $\\sum_{i \\in I}a^{i} \\otimes b^{i} \\in C \\square_{D} C$. In particular, necessary and sufficient conditions for a Hopf algebra map to be a monomorphism are given."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"0908.2959","kind":"arxiv","version":3},"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"},"aliases":[{"alias_kind":"arxiv","alias_value":"0908.2959","created_at":"2026-05-18T02:58:31.056908+00:00"},{"alias_kind":"arxiv_version","alias_value":"0908.2959v3","created_at":"2026-05-18T02:58:31.056908+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.0908.2959","created_at":"2026-05-18T02:58:31.056908+00:00"},{"alias_kind":"pith_short_12","alias_value":"WXUYYYLY6NKL","created_at":"2026-05-18T12:26:02.257875+00:00"},{"alias_kind":"pith_short_16","alias_value":"WXUYYYLY6NKLU5BW","created_at":"2026-05-18T12:26:02.257875+00:00"},{"alias_kind":"pith_short_8","alias_value":"WXUYYYLY","created_at":"2026-05-18T12:26:02.257875+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":0,"internal_anchor_count":0,"sample":[]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/WXUYYYLY6NKLU5BW45O2WRB3Z7","json":"https://pith.science/pith/WXUYYYLY6NKLU5BW45O2WRB3Z7.json","graph_json":"https://pith.science/api/pith-number/WXUYYYLY6NKLU5BW45O2WRB3Z7/graph.json","events_json":"https://pith.science/api/pith-number/WXUYYYLY6NKLU5BW45O2WRB3Z7/events.json","paper":"https://pith.science/paper/WXUYYYLY"},"agent_actions":{"view_html":"https://pith.science/pith/WXUYYYLY6NKLU5BW45O2WRB3Z7","download_json":"https://pith.science/pith/WXUYYYLY6NKLU5BW45O2WRB3Z7.json","view_paper":"https://pith.science/paper/WXUYYYLY","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=0908.2959&json=true","fetch_graph":"https://pith.science/api/pith-number/WXUYYYLY6NKLU5BW45O2WRB3Z7/graph.json","fetch_events":"https://pith.science/api/pith-number/WXUYYYLY6NKLU5BW45O2WRB3Z7/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/WXUYYYLY6NKLU5BW45O2WRB3Z7/action/timestamp_anchor","attest_storage":"https://pith.science/pith/WXUYYYLY6NKLU5BW45O2WRB3Z7/action/storage_attestation","attest_author":"https://pith.science/pith/WXUYYYLY6NKLU5BW45O2WRB3Z7/action/author_attestation","sign_citation":"https://pith.science/pith/WXUYYYLY6NKLU5BW45O2WRB3Z7/action/citation_signature","submit_replication":"https://pith.science/pith/WXUYYYLY6NKLU5BW45O2WRB3Z7/action/replication_record"}},"created_at":"2026-05-18T02:58:31.056908+00:00","updated_at":"2026-05-18T02:58:31.056908+00:00"}