{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2018:WMZU2HBWQ7BKPPVKUD3RXZXUPM","short_pith_number":"pith:WMZU2HBW","canonical_record":{"source":{"id":"1805.05156","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CT","submitted_at":"2018-05-14T13:16:34Z","cross_cats_sorted":[],"title_canon_sha256":"060679d1ac3d6024a7555174370f019cacd056ef42199fb19930256587f9c9c0","abstract_canon_sha256":"7fd98039e390d8e81c00be8818f7e48cda8abe3847e73ca6a687046d62febaea"},"schema_version":"1.0"},"canonical_sha256":"b3334d1c3687c2a7beaaa0f71be6f47b144a4d7d93b4985825ee84b93a85f13b","source":{"kind":"arxiv","id":"1805.05156","version":3},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1805.05156","created_at":"2026-05-17T23:51:05Z"},{"alias_kind":"arxiv_version","alias_value":"1805.05156v3","created_at":"2026-05-17T23:51:05Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1805.05156","created_at":"2026-05-17T23:51:05Z"},{"alias_kind":"pith_short_12","alias_value":"WMZU2HBWQ7BK","created_at":"2026-05-18T12:33:01Z"},{"alias_kind":"pith_short_16","alias_value":"WMZU2HBWQ7BKPPVK","created_at":"2026-05-18T12:33:01Z"},{"alias_kind":"pith_short_8","alias_value":"WMZU2HBW","created_at":"2026-05-18T12:33:01Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2018:WMZU2HBWQ7BKPPVKUD3RXZXUPM","target":"record","payload":{"canonical_record":{"source":{"id":"1805.05156","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CT","submitted_at":"2018-05-14T13:16:34Z","cross_cats_sorted":[],"title_canon_sha256":"060679d1ac3d6024a7555174370f019cacd056ef42199fb19930256587f9c9c0","abstract_canon_sha256":"7fd98039e390d8e81c00be8818f7e48cda8abe3847e73ca6a687046d62febaea"},"schema_version":"1.0"},"canonical_sha256":"b3334d1c3687c2a7beaaa0f71be6f47b144a4d7d93b4985825ee84b93a85f13b","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-17T23:51:05.846062Z","signature_b64":"q23ppESsFdRG+Mf4YFwYWiZhZ+563jPry0TdEBR+XNzZ7CmygdjoeklzB2/z2TsPsNr6x4r1lbxoP3+OwjiVCw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"b3334d1c3687c2a7beaaa0f71be6f47b144a4d7d93b4985825ee84b93a85f13b","last_reissued_at":"2026-05-17T23:51:05.845525Z","signature_status":"signed_v1","first_computed_at":"2026-05-17T23:51:05.845525Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1805.05156","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":"2026-05-17T23:51:05Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"RbDX7JNfRooujya2+/yKkszLqX9amYl0O4PmvzmiPk62I8F1NvkIvE3/oOUVUyEm3tMfgCOEmEBCTwzGakwgDw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-02T18:38:10.339941Z"},"content_sha256":"650c124f8010ccb4f35d39f63d1d3b2f0efdf80c7a2fbead3f647f91e691b2aa","schema_version":"1.0","event_id":"sha256:650c124f8010ccb4f35d39f63d1d3b2f0efdf80c7a2fbead3f647f91e691b2aa"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2018:WMZU2HBWQ7BKPPVKUD3RXZXUPM","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Exactness of direct limits for abelian categories with an injective cogenerator","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CT","authors_text":"Jan Stovicek, Leonid Positselski","submitted_at":"2018-05-14T13:16:34Z","abstract_excerpt":"We prove that the exactness of direct limits in an abelian category with products and an injective cogenerator J is equivalent to a condition on J which is well-known to characterize pure-injectivity in module categories, and we describe an application of this result to the tilting theory. We derive our result as a consequence of a more general characterization of when inverse limits in the Eilenberg-Moore category of a monad on the category of sets preserve regular epimorphisms."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1805.05156","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":"2026-05-17T23:51:05Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"pAyjWeg3PoufQix7mgmCM2/uNgj3ajxwn1VP+Lmz12TYijaIeHRG/Q3KQQLhVULlXonGi9Jm/+2YhH9PIpU3Cw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-02T18:38:10.340281Z"},"content_sha256":"cf73ed3da2f9bd2eefba4d734a73b087e129375138cf2f16b3a8944d1e825c7e","schema_version":"1.0","event_id":"sha256:cf73ed3da2f9bd2eefba4d734a73b087e129375138cf2f16b3a8944d1e825c7e"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/WMZU2HBWQ7BKPPVKUD3RXZXUPM/bundle.json","state_url":"https://pith.science/pith/WMZU2HBWQ7BKPPVKUD3RXZXUPM/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/WMZU2HBWQ7BKPPVKUD3RXZXUPM/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":"2026-06-02T18:38:10Z","links":{"resolver":"https://pith.science/pith/WMZU2HBWQ7BKPPVKUD3RXZXUPM","bundle":"https://pith.science/pith/WMZU2HBWQ7BKPPVKUD3RXZXUPM/bundle.json","state":"https://pith.science/pith/WMZU2HBWQ7BKPPVKUD3RXZXUPM/state.json","well_known_bundle":"https://pith.science/.well-known/pith/WMZU2HBWQ7BKPPVKUD3RXZXUPM/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2018:WMZU2HBWQ7BKPPVKUD3RXZXUPM","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":"7fd98039e390d8e81c00be8818f7e48cda8abe3847e73ca6a687046d62febaea","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CT","submitted_at":"2018-05-14T13:16:34Z","title_canon_sha256":"060679d1ac3d6024a7555174370f019cacd056ef42199fb19930256587f9c9c0"},"schema_version":"1.0","source":{"id":"1805.05156","kind":"arxiv","version":3}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1805.05156","created_at":"2026-05-17T23:51:05Z"},{"alias_kind":"arxiv_version","alias_value":"1805.05156v3","created_at":"2026-05-17T23:51:05Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1805.05156","created_at":"2026-05-17T23:51:05Z"},{"alias_kind":"pith_short_12","alias_value":"WMZU2HBWQ7BK","created_at":"2026-05-18T12:33:01Z"},{"alias_kind":"pith_short_16","alias_value":"WMZU2HBWQ7BKPPVK","created_at":"2026-05-18T12:33:01Z"},{"alias_kind":"pith_short_8","alias_value":"WMZU2HBW","created_at":"2026-05-18T12:33:01Z"}],"graph_snapshots":[{"event_id":"sha256:cf73ed3da2f9bd2eefba4d734a73b087e129375138cf2f16b3a8944d1e825c7e","target":"graph","created_at":"2026-05-17T23:51:05Z","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 prove that the exactness of direct limits in an abelian category with products and an injective cogenerator J is equivalent to a condition on J which is well-known to characterize pure-injectivity in module categories, and we describe an application of this result to the tilting theory. We derive our result as a consequence of a more general characterization of when inverse limits in the Eilenberg-Moore category of a monad on the category of sets preserve regular epimorphisms.","authors_text":"Jan Stovicek, Leonid Positselski","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CT","submitted_at":"2018-05-14T13:16:34Z","title":"Exactness of direct limits for abelian categories with an injective cogenerator"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1805.05156","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":"sha256:650c124f8010ccb4f35d39f63d1d3b2f0efdf80c7a2fbead3f647f91e691b2aa","target":"record","created_at":"2026-05-17T23:51:05Z","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":"7fd98039e390d8e81c00be8818f7e48cda8abe3847e73ca6a687046d62febaea","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CT","submitted_at":"2018-05-14T13:16:34Z","title_canon_sha256":"060679d1ac3d6024a7555174370f019cacd056ef42199fb19930256587f9c9c0"},"schema_version":"1.0","source":{"id":"1805.05156","kind":"arxiv","version":3}},"canonical_sha256":"b3334d1c3687c2a7beaaa0f71be6f47b144a4d7d93b4985825ee84b93a85f13b","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"b3334d1c3687c2a7beaaa0f71be6f47b144a4d7d93b4985825ee84b93a85f13b","first_computed_at":"2026-05-17T23:51:05.845525Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-17T23:51:05.845525Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"q23ppESsFdRG+Mf4YFwYWiZhZ+563jPry0TdEBR+XNzZ7CmygdjoeklzB2/z2TsPsNr6x4r1lbxoP3+OwjiVCw==","signature_status":"signed_v1","signed_at":"2026-05-17T23:51:05.846062Z","signed_message":"canonical_sha256_bytes"},"source_id":"1805.05156","source_kind":"arxiv","source_version":3}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:650c124f8010ccb4f35d39f63d1d3b2f0efdf80c7a2fbead3f647f91e691b2aa","sha256:cf73ed3da2f9bd2eefba4d734a73b087e129375138cf2f16b3a8944d1e825c7e"],"state_sha256":"ad652c25a2a3545deaabb2e03e9a62ed336888b6eb9a9c0d8d370d6fde2d7f62"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"87g6JWdeNwB5kaP5JkqJlPzaeSTvGOmFS4KqJS4aoyJfMKMTxo5dyNcJabO6bJHpECjfebwvv5aLPPS90GmGAw==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-02T18:38:10.342418Z","bundle_sha256":"38bb9f3dc7580ab980e45a584fa93bd82cc30f8701c048ccf363a5dbf56f2e85"}}