{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2012:HHFMLXZ4MTFZBHP533CZJBQVEB","short_pith_number":"pith:HHFMLXZ4","canonical_record":{"source":{"id":"1212.1545","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CT","submitted_at":"2012-12-07T07:08:18Z","cross_cats_sorted":["math.QA"],"title_canon_sha256":"3f2fd6d9e6a006015f92012c3fd265423e1ae46bd7cff245ef51d3b4cc496da3","abstract_canon_sha256":"30515b070e8b47ee6a622caeb4405ed80a39b335de53edbbf30d91baccfaee96"},"schema_version":"1.0"},"canonical_sha256":"39cac5df3c64cb909dfddec594861520740ce9ee29765bfbbe5e9f95113cb584","source":{"kind":"arxiv","id":"1212.1545","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1212.1545","created_at":"2026-05-18T03:38:56Z"},{"alias_kind":"arxiv_version","alias_value":"1212.1545v1","created_at":"2026-05-18T03:38:56Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1212.1545","created_at":"2026-05-18T03:38:56Z"},{"alias_kind":"pith_short_12","alias_value":"HHFMLXZ4MTFZ","created_at":"2026-05-18T12:27:09Z"},{"alias_kind":"pith_short_16","alias_value":"HHFMLXZ4MTFZBHP5","created_at":"2026-05-18T12:27:09Z"},{"alias_kind":"pith_short_8","alias_value":"HHFMLXZ4","created_at":"2026-05-18T12:27:09Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2012:HHFMLXZ4MTFZBHP533CZJBQVEB","target":"record","payload":{"canonical_record":{"source":{"id":"1212.1545","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CT","submitted_at":"2012-12-07T07:08:18Z","cross_cats_sorted":["math.QA"],"title_canon_sha256":"3f2fd6d9e6a006015f92012c3fd265423e1ae46bd7cff245ef51d3b4cc496da3","abstract_canon_sha256":"30515b070e8b47ee6a622caeb4405ed80a39b335de53edbbf30d91baccfaee96"},"schema_version":"1.0"},"canonical_sha256":"39cac5df3c64cb909dfddec594861520740ce9ee29765bfbbe5e9f95113cb584","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T03:38:56.966067Z","signature_b64":"BMcpn3Zv7ZQiOQLoVnp0dYkA8XOKTJfas2OaDT+Wqz0NU7ljvvN+WvrpIthDjZIQ2Ty+hHcO7NS57/TS59peDw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"39cac5df3c64cb909dfddec594861520740ce9ee29765bfbbe5e9f95113cb584","last_reissued_at":"2026-05-18T03:38:56.965572Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T03:38:56.965572Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1212.1545","source_version":1,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T03:38:56Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"iiqcbD4ff79RkyjtM4FdqJDOX1uxordXeDi/A2854r98BF87lIRg5nIXDG8uGpOTruEuUcids41IxzOPDTKEBQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-25T18:02:17.516287Z"},"content_sha256":"dde136e97c954f103e8719e9ef630e92078d63c47f4a77c38895234215382e34","schema_version":"1.0","event_id":"sha256:dde136e97c954f103e8719e9ef630e92078d63c47f4a77c38895234215382e34"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2012:HHFMLXZ4MTFZBHP533CZJBQVEB","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Tensor products of finitely cocomplete and abelian categories","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.QA"],"primary_cat":"math.CT","authors_text":"Ignacio Lopez Franco","submitted_at":"2012-12-07T07:08:18Z","abstract_excerpt":"The purpose of this article is to study the existence of Deligne's tensor product of abelian categories by comparing it with the well-known ten- sor product of finitely cocomplete categories. The main result states that the former exists precisely when the latter is an abelian category, and moreover in this case both tensor products coincide. An example of two abelian categories whose Deligne tensor product does not exist is given."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1212.1545","kind":"arxiv","version":1},"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":"2026-05-18T03:38:56Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"9hy7YQNyCFcPtlzBu8q7Lc+yKYpiLLVa+u9wYPgY35j5PdEmDh8nP+FYy5Z3WU6eZkcIDdKgR/JmuVUI0k/IDw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-25T18:02:17.516663Z"},"content_sha256":"d3512e83796176283ad734237d93be4127ed94924f0a74ca046878c3c536c053","schema_version":"1.0","event_id":"sha256:d3512e83796176283ad734237d93be4127ed94924f0a74ca046878c3c536c053"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/HHFMLXZ4MTFZBHP533CZJBQVEB/bundle.json","state_url":"https://pith.science/pith/HHFMLXZ4MTFZBHP533CZJBQVEB/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/HHFMLXZ4MTFZBHP533CZJBQVEB/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":"2026-06-25T18:02:17Z","links":{"resolver":"https://pith.science/pith/HHFMLXZ4MTFZBHP533CZJBQVEB","bundle":"https://pith.science/pith/HHFMLXZ4MTFZBHP533CZJBQVEB/bundle.json","state":"https://pith.science/pith/HHFMLXZ4MTFZBHP533CZJBQVEB/state.json","well_known_bundle":"https://pith.science/.well-known/pith/HHFMLXZ4MTFZBHP533CZJBQVEB/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2012:HHFMLXZ4MTFZBHP533CZJBQVEB","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":"30515b070e8b47ee6a622caeb4405ed80a39b335de53edbbf30d91baccfaee96","cross_cats_sorted":["math.QA"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CT","submitted_at":"2012-12-07T07:08:18Z","title_canon_sha256":"3f2fd6d9e6a006015f92012c3fd265423e1ae46bd7cff245ef51d3b4cc496da3"},"schema_version":"1.0","source":{"id":"1212.1545","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1212.1545","created_at":"2026-05-18T03:38:56Z"},{"alias_kind":"arxiv_version","alias_value":"1212.1545v1","created_at":"2026-05-18T03:38:56Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1212.1545","created_at":"2026-05-18T03:38:56Z"},{"alias_kind":"pith_short_12","alias_value":"HHFMLXZ4MTFZ","created_at":"2026-05-18T12:27:09Z"},{"alias_kind":"pith_short_16","alias_value":"HHFMLXZ4MTFZBHP5","created_at":"2026-05-18T12:27:09Z"},{"alias_kind":"pith_short_8","alias_value":"HHFMLXZ4","created_at":"2026-05-18T12:27:09Z"}],"graph_snapshots":[{"event_id":"sha256:d3512e83796176283ad734237d93be4127ed94924f0a74ca046878c3c536c053","target":"graph","created_at":"2026-05-18T03:38:56Z","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":"The purpose of this article is to study the existence of Deligne's tensor product of abelian categories by comparing it with the well-known ten- sor product of finitely cocomplete categories. The main result states that the former exists precisely when the latter is an abelian category, and moreover in this case both tensor products coincide. An example of two abelian categories whose Deligne tensor product does not exist is given.","authors_text":"Ignacio Lopez Franco","cross_cats":["math.QA"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CT","submitted_at":"2012-12-07T07:08:18Z","title":"Tensor products of finitely cocomplete and abelian categories"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1212.1545","kind":"arxiv","version":1},"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:dde136e97c954f103e8719e9ef630e92078d63c47f4a77c38895234215382e34","target":"record","created_at":"2026-05-18T03:38:56Z","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":"30515b070e8b47ee6a622caeb4405ed80a39b335de53edbbf30d91baccfaee96","cross_cats_sorted":["math.QA"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CT","submitted_at":"2012-12-07T07:08:18Z","title_canon_sha256":"3f2fd6d9e6a006015f92012c3fd265423e1ae46bd7cff245ef51d3b4cc496da3"},"schema_version":"1.0","source":{"id":"1212.1545","kind":"arxiv","version":1}},"canonical_sha256":"39cac5df3c64cb909dfddec594861520740ce9ee29765bfbbe5e9f95113cb584","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"39cac5df3c64cb909dfddec594861520740ce9ee29765bfbbe5e9f95113cb584","first_computed_at":"2026-05-18T03:38:56.965572Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T03:38:56.965572Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"BMcpn3Zv7ZQiOQLoVnp0dYkA8XOKTJfas2OaDT+Wqz0NU7ljvvN+WvrpIthDjZIQ2Ty+hHcO7NS57/TS59peDw==","signature_status":"signed_v1","signed_at":"2026-05-18T03:38:56.966067Z","signed_message":"canonical_sha256_bytes"},"source_id":"1212.1545","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:dde136e97c954f103e8719e9ef630e92078d63c47f4a77c38895234215382e34","sha256:d3512e83796176283ad734237d93be4127ed94924f0a74ca046878c3c536c053"],"state_sha256":"4dfd580714a231c59721d66ac7a189b69a681c3605dbe45b3e8544813a3a6175"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"cdtL3QJF1014FVlaCQFQC/cARLQiNwMVFyY4iCoYn4upTCoaeXKydS7fV5vJ1SXwSgrcMzbBOKD1qciheeVHAQ==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-25T18:02:17.518579Z","bundle_sha256":"7d2ffaa8021a0c36136fd6f2c0f8873a2df95d1973faa2a4b50d587e746ae1d9"}}