{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2010:KRIBPROBEN4XEZCXXIAVFFXQ3D","short_pith_number":"pith:KRIBPROB","canonical_record":{"source":{"id":"1005.0156","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.RA","submitted_at":"2010-05-02T19:23:52Z","cross_cats_sorted":["math.CT"],"title_canon_sha256":"0a22492e41a01d0f46d79f16401fe178d75dc5efaf142b9260727bcbe80abcb8","abstract_canon_sha256":"f03ba3c0664fd6c289805376f653819295c78d799b45993dafc4bb22b94c6e1f"},"schema_version":"1.0"},"canonical_sha256":"545017c5c12379726457ba015296f0d8d73ba6926f43fc66a9daeb622a34f6e8","source":{"kind":"arxiv","id":"1005.0156","version":3},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1005.0156","created_at":"2026-05-18T02:24:09Z"},{"alias_kind":"arxiv_version","alias_value":"1005.0156v3","created_at":"2026-05-18T02:24:09Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1005.0156","created_at":"2026-05-18T02:24:09Z"},{"alias_kind":"pith_short_12","alias_value":"KRIBPROBEN4X","created_at":"2026-05-18T12:26:09Z"},{"alias_kind":"pith_short_16","alias_value":"KRIBPROBEN4XEZCX","created_at":"2026-05-18T12:26:09Z"},{"alias_kind":"pith_short_8","alias_value":"KRIBPROB","created_at":"2026-05-18T12:26:09Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2010:KRIBPROBEN4XEZCXXIAVFFXQ3D","target":"record","payload":{"canonical_record":{"source":{"id":"1005.0156","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.RA","submitted_at":"2010-05-02T19:23:52Z","cross_cats_sorted":["math.CT"],"title_canon_sha256":"0a22492e41a01d0f46d79f16401fe178d75dc5efaf142b9260727bcbe80abcb8","abstract_canon_sha256":"f03ba3c0664fd6c289805376f653819295c78d799b45993dafc4bb22b94c6e1f"},"schema_version":"1.0"},"canonical_sha256":"545017c5c12379726457ba015296f0d8d73ba6926f43fc66a9daeb622a34f6e8","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T02:24:09.386416Z","signature_b64":"Zuv2lOAd39jATA8Yjqs7+aBRIsaPsc0n5nl1KlD3jQNRTl5JHaHyGLt6s3CIxbE6/D0bfyFvIwk6lxoXydX4Cw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"545017c5c12379726457ba015296f0d8d73ba6926f43fc66a9daeb622a34f6e8","last_reissued_at":"2026-05-18T02:24:09.385759Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T02:24:09.385759Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1005.0156","source_version":3,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T02:24:09Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"ILOj7tSash08eafO8qtTAko3uMfEj5Wdf8+YF/dtXKFX83l/lafPD1ALw+dV7D150/9UoPbWmfprCa/zfhpnDA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-02T11:21:19.597058Z"},"content_sha256":"1b793ab6aa49aecc956025a43969fc1bd0d85e22aa42c07670b6c44848686b71","schema_version":"1.0","event_id":"sha256:1b793ab6aa49aecc956025a43969fc1bd0d85e22aa42c07670b6c44848686b71"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2010:KRIBPROBEN4XEZCXXIAVFFXQ3D","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Four problems regarding representable functors","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.CT"],"primary_cat":"math.RA","authors_text":"Gigel Militaru","submitted_at":"2010-05-02T19:23:52Z","abstract_excerpt":"Let $R$, $S$ be two rings, $C$ an $R$-coring and ${}_{R}^C{\\mathcal M}$ the category of left $C$-comodules. The category ${\\bf Rep}\\, ( {}_{R}^C{\\mathcal M}, {}_{S}{\\mathcal M} )$ of all representable functors ${}_{R}^C{\\mathcal M} \\to {}_{S}{\\mathcal M}$ is shown to be equivalent to the opposite of the category ${}_{R}^C{\\mathcal M}_S$. For $U$ an $(S,R)$-bimodule we give necessary and sufficient conditions for the induction functor $U\\otimes_R - : {}_{R}^C\\mathcal{M} \\to {}_{S}\\mathcal{M}$ to be: a representable functor, an equivalence of categories, a separable or a Frobenius functor. The l"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1005.0156","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"},"verdict_id":null},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T02:24:09Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"PPc+S+Z3uN25Egm9srjugcfTJ6TbNwwsvXnnRquO4GeVglPzope1EkM4VVMxkPk3RWw07uvkXvttyytYOFh9Dw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-02T11:21:19.597428Z"},"content_sha256":"b3f3b087480724840e4af2b2ab4b744c60d05399396749bd160319a9c463d0e5","schema_version":"1.0","event_id":"sha256:b3f3b087480724840e4af2b2ab4b744c60d05399396749bd160319a9c463d0e5"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/KRIBPROBEN4XEZCXXIAVFFXQ3D/bundle.json","state_url":"https://pith.science/pith/KRIBPROBEN4XEZCXXIAVFFXQ3D/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/KRIBPROBEN4XEZCXXIAVFFXQ3D/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":"2026-06-02T11:21:19Z","links":{"resolver":"https://pith.science/pith/KRIBPROBEN4XEZCXXIAVFFXQ3D","bundle":"https://pith.science/pith/KRIBPROBEN4XEZCXXIAVFFXQ3D/bundle.json","state":"https://pith.science/pith/KRIBPROBEN4XEZCXXIAVFFXQ3D/state.json","well_known_bundle":"https://pith.science/.well-known/pith/KRIBPROBEN4XEZCXXIAVFFXQ3D/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2010:KRIBPROBEN4XEZCXXIAVFFXQ3D","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":"f03ba3c0664fd6c289805376f653819295c78d799b45993dafc4bb22b94c6e1f","cross_cats_sorted":["math.CT"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.RA","submitted_at":"2010-05-02T19:23:52Z","title_canon_sha256":"0a22492e41a01d0f46d79f16401fe178d75dc5efaf142b9260727bcbe80abcb8"},"schema_version":"1.0","source":{"id":"1005.0156","kind":"arxiv","version":3}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1005.0156","created_at":"2026-05-18T02:24:09Z"},{"alias_kind":"arxiv_version","alias_value":"1005.0156v3","created_at":"2026-05-18T02:24:09Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1005.0156","created_at":"2026-05-18T02:24:09Z"},{"alias_kind":"pith_short_12","alias_value":"KRIBPROBEN4X","created_at":"2026-05-18T12:26:09Z"},{"alias_kind":"pith_short_16","alias_value":"KRIBPROBEN4XEZCX","created_at":"2026-05-18T12:26:09Z"},{"alias_kind":"pith_short_8","alias_value":"KRIBPROB","created_at":"2026-05-18T12:26:09Z"}],"graph_snapshots":[{"event_id":"sha256:b3f3b087480724840e4af2b2ab4b744c60d05399396749bd160319a9c463d0e5","target":"graph","created_at":"2026-05-18T02:24:09Z","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":"Let $R$, $S$ be two rings, $C$ an $R$-coring and ${}_{R}^C{\\mathcal M}$ the category of left $C$-comodules. The category ${\\bf Rep}\\, ( {}_{R}^C{\\mathcal M}, {}_{S}{\\mathcal M} )$ of all representable functors ${}_{R}^C{\\mathcal M} \\to {}_{S}{\\mathcal M}$ is shown to be equivalent to the opposite of the category ${}_{R}^C{\\mathcal M}_S$. For $U$ an $(S,R)$-bimodule we give necessary and sufficient conditions for the induction functor $U\\otimes_R - : {}_{R}^C\\mathcal{M} \\to {}_{S}\\mathcal{M}$ to be: a representable functor, an equivalence of categories, a separable or a Frobenius functor. The l","authors_text":"Gigel Militaru","cross_cats":["math.CT"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.RA","submitted_at":"2010-05-02T19:23:52Z","title":"Four problems regarding representable functors"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1005.0156","kind":"arxiv","version":3},"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":"sha256:1b793ab6aa49aecc956025a43969fc1bd0d85e22aa42c07670b6c44848686b71","target":"record","created_at":"2026-05-18T02:24:09Z","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":"f03ba3c0664fd6c289805376f653819295c78d799b45993dafc4bb22b94c6e1f","cross_cats_sorted":["math.CT"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.RA","submitted_at":"2010-05-02T19:23:52Z","title_canon_sha256":"0a22492e41a01d0f46d79f16401fe178d75dc5efaf142b9260727bcbe80abcb8"},"schema_version":"1.0","source":{"id":"1005.0156","kind":"arxiv","version":3}},"canonical_sha256":"545017c5c12379726457ba015296f0d8d73ba6926f43fc66a9daeb622a34f6e8","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"545017c5c12379726457ba015296f0d8d73ba6926f43fc66a9daeb622a34f6e8","first_computed_at":"2026-05-18T02:24:09.385759Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T02:24:09.385759Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"Zuv2lOAd39jATA8Yjqs7+aBRIsaPsc0n5nl1KlD3jQNRTl5JHaHyGLt6s3CIxbE6/D0bfyFvIwk6lxoXydX4Cw==","signature_status":"signed_v1","signed_at":"2026-05-18T02:24:09.386416Z","signed_message":"canonical_sha256_bytes"},"source_id":"1005.0156","source_kind":"arxiv","source_version":3}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:1b793ab6aa49aecc956025a43969fc1bd0d85e22aa42c07670b6c44848686b71","sha256:b3f3b087480724840e4af2b2ab4b744c60d05399396749bd160319a9c463d0e5"],"state_sha256":"27f0c32fc33738fd363a2b6d3f41aca1221a695384e315aa62e94333ba2e78aa"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"/WxK/34YdFNO7939SaPqWRlXEpBxOIlBDX0ClZY8ILz5uiY0TK3P4AH6lUrxa9kPOHUOG5HRox1zSqYAUfO9AA==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-02T11:21:19.599436Z","bundle_sha256":"f102dbdb2df821008d52b6cbd75688b83340ee06d06e40f1ee1f6e35ef412e2f"}}