{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2003:H5B5SXJWO74XUNQZ3XAAFOZGH4","short_pith_number":"pith:H5B5SXJW","canonical_record":{"source":{"id":"math/0312277","kind":"arxiv","version":1},"metadata":{"license":"","primary_cat":"math.AT","submitted_at":"2003-12-14T09:37:23Z","cross_cats_sorted":["math.QA"],"title_canon_sha256":"1b1cf8f7bb71770cb5b4eb8be63efae8f58589541481261028ee51e223359c9c","abstract_canon_sha256":"2e629ef80bfe0677f597969e930e4eed4a2c6224b742c3ab7b81bc9fe311a6a0"},"schema_version":"1.0"},"canonical_sha256":"3f43d95d3677f97a3619ddc002bb263f22d5509b80a6ac7568798c13ff61acea","source":{"kind":"arxiv","id":"math/0312277","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"math/0312277","created_at":"2026-05-18T01:05:26Z"},{"alias_kind":"arxiv_version","alias_value":"math/0312277v1","created_at":"2026-05-18T01:05:26Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.math/0312277","created_at":"2026-05-18T01:05:26Z"},{"alias_kind":"pith_short_12","alias_value":"H5B5SXJWO74X","created_at":"2026-05-18T12:25:52Z"},{"alias_kind":"pith_short_16","alias_value":"H5B5SXJWO74XUNQZ","created_at":"2026-05-18T12:25:52Z"},{"alias_kind":"pith_short_8","alias_value":"H5B5SXJW","created_at":"2026-05-18T12:25:52Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2003:H5B5SXJWO74XUNQZ3XAAFOZGH4","target":"record","payload":{"canonical_record":{"source":{"id":"math/0312277","kind":"arxiv","version":1},"metadata":{"license":"","primary_cat":"math.AT","submitted_at":"2003-12-14T09:37:23Z","cross_cats_sorted":["math.QA"],"title_canon_sha256":"1b1cf8f7bb71770cb5b4eb8be63efae8f58589541481261028ee51e223359c9c","abstract_canon_sha256":"2e629ef80bfe0677f597969e930e4eed4a2c6224b742c3ab7b81bc9fe311a6a0"},"schema_version":"1.0"},"canonical_sha256":"3f43d95d3677f97a3619ddc002bb263f22d5509b80a6ac7568798c13ff61acea","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:05:26.668714Z","signature_b64":"LNb/yZmVDVk9+ARrDvX7dAV7pusVntZIkXfmhVS5AeWTGuTP90md3CaP57nvKVBM+QoEMCSrF4pvCv0V2Yv7Dg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"3f43d95d3677f97a3619ddc002bb263f22d5509b80a6ac7568798c13ff61acea","last_reissued_at":"2026-05-18T01:05:26.668171Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:05:26.668171Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"math/0312277","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":"2026-05-18T01:05:26Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"l7qHrS/X4SzWWl7s50iiz1i0mkIlwyIiN/V6gzaFHvZPMuDuoumpSO/qxTlLjtaI/EoEo6Fqzka0in9nF+yADA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-29T17:20:12.482102Z"},"content_sha256":"4814fa0c03d5c31bc6c50417072c02d029672ead89b70155710fc27787805361","schema_version":"1.0","event_id":"sha256:4814fa0c03d5c31bc6c50417072c02d029672ead89b70155710fc27787805361"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2003:H5B5SXJWO74XUNQZ3XAAFOZGH4","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Associahedra, cellular W-construction and products of $A_\\infty$-algebras","license":"","headline":"","cross_cats":["math.QA"],"primary_cat":"math.AT","authors_text":"Martin Markl, Steve Shnider","submitted_at":"2003-12-14T09:37:23Z","abstract_excerpt":"Our aim is to construct a functorial tensor product of $A_\\infty$-algebras or, equivalently, an explicit diagonal for the operad of cellular chains, over the integers, of the Stasheff associahedron. These construction were in fact already indicated by R. Umble and S. Saneblidze in [9]; we will try to give a more satisfactory presentation. We also prove that there does not exist an associative tensor product of $A_\\infty$-algebras."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"math/0312277","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":"2026-05-18T01:05:26Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"sk2MAsDjHcsAoGjNuTUbAEYpmZ9Unh6y0d4uW0+z2z3xOePR+4aukeCV7sGf670lppT80XkKHeVRGlSLb7QTAQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-29T17:20:12.482774Z"},"content_sha256":"ab3817569fa46885553f206685dd78465e1a745be5e1e925eab90d804902cb03","schema_version":"1.0","event_id":"sha256:ab3817569fa46885553f206685dd78465e1a745be5e1e925eab90d804902cb03"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/H5B5SXJWO74XUNQZ3XAAFOZGH4/bundle.json","state_url":"https://pith.science/pith/H5B5SXJWO74XUNQZ3XAAFOZGH4/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/H5B5SXJWO74XUNQZ3XAAFOZGH4/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":"2026-05-29T17:20:12Z","links":{"resolver":"https://pith.science/pith/H5B5SXJWO74XUNQZ3XAAFOZGH4","bundle":"https://pith.science/pith/H5B5SXJWO74XUNQZ3XAAFOZGH4/bundle.json","state":"https://pith.science/pith/H5B5SXJWO74XUNQZ3XAAFOZGH4/state.json","well_known_bundle":"https://pith.science/.well-known/pith/H5B5SXJWO74XUNQZ3XAAFOZGH4/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2003:H5B5SXJWO74XUNQZ3XAAFOZGH4","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":"2e629ef80bfe0677f597969e930e4eed4a2c6224b742c3ab7b81bc9fe311a6a0","cross_cats_sorted":["math.QA"],"license":"","primary_cat":"math.AT","submitted_at":"2003-12-14T09:37:23Z","title_canon_sha256":"1b1cf8f7bb71770cb5b4eb8be63efae8f58589541481261028ee51e223359c9c"},"schema_version":"1.0","source":{"id":"math/0312277","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"math/0312277","created_at":"2026-05-18T01:05:26Z"},{"alias_kind":"arxiv_version","alias_value":"math/0312277v1","created_at":"2026-05-18T01:05:26Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.math/0312277","created_at":"2026-05-18T01:05:26Z"},{"alias_kind":"pith_short_12","alias_value":"H5B5SXJWO74X","created_at":"2026-05-18T12:25:52Z"},{"alias_kind":"pith_short_16","alias_value":"H5B5SXJWO74XUNQZ","created_at":"2026-05-18T12:25:52Z"},{"alias_kind":"pith_short_8","alias_value":"H5B5SXJW","created_at":"2026-05-18T12:25:52Z"}],"graph_snapshots":[{"event_id":"sha256:ab3817569fa46885553f206685dd78465e1a745be5e1e925eab90d804902cb03","target":"graph","created_at":"2026-05-18T01:05:26Z","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":"Our aim is to construct a functorial tensor product of $A_\\infty$-algebras or, equivalently, an explicit diagonal for the operad of cellular chains, over the integers, of the Stasheff associahedron. These construction were in fact already indicated by R. Umble and S. Saneblidze in [9]; we will try to give a more satisfactory presentation. We also prove that there does not exist an associative tensor product of $A_\\infty$-algebras.","authors_text":"Martin Markl, Steve Shnider","cross_cats":["math.QA"],"headline":"","license":"","primary_cat":"math.AT","submitted_at":"2003-12-14T09:37:23Z","title":"Associahedra, cellular W-construction and products of $A_\\infty$-algebras"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"math/0312277","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":"sha256:4814fa0c03d5c31bc6c50417072c02d029672ead89b70155710fc27787805361","target":"record","created_at":"2026-05-18T01:05:26Z","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":"2e629ef80bfe0677f597969e930e4eed4a2c6224b742c3ab7b81bc9fe311a6a0","cross_cats_sorted":["math.QA"],"license":"","primary_cat":"math.AT","submitted_at":"2003-12-14T09:37:23Z","title_canon_sha256":"1b1cf8f7bb71770cb5b4eb8be63efae8f58589541481261028ee51e223359c9c"},"schema_version":"1.0","source":{"id":"math/0312277","kind":"arxiv","version":1}},"canonical_sha256":"3f43d95d3677f97a3619ddc002bb263f22d5509b80a6ac7568798c13ff61acea","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"3f43d95d3677f97a3619ddc002bb263f22d5509b80a6ac7568798c13ff61acea","first_computed_at":"2026-05-18T01:05:26.668171Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T01:05:26.668171Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"LNb/yZmVDVk9+ARrDvX7dAV7pusVntZIkXfmhVS5AeWTGuTP90md3CaP57nvKVBM+QoEMCSrF4pvCv0V2Yv7Dg==","signature_status":"signed_v1","signed_at":"2026-05-18T01:05:26.668714Z","signed_message":"canonical_sha256_bytes"},"source_id":"math/0312277","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:4814fa0c03d5c31bc6c50417072c02d029672ead89b70155710fc27787805361","sha256:ab3817569fa46885553f206685dd78465e1a745be5e1e925eab90d804902cb03"],"state_sha256":"0cfa13257f1e1853a8b71fde830a8ce0a119740ec02b79025032d53acc217945"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"ThzxLpGoHJt/RgGURKnhawFAzgEgZtrlHbvG8d8CGw5LkkhqTwJaZ3rZhMujHHojXLF4w9z/h9c12q1BQQcUBw==","signed_message":"bundle_sha256_bytes","signed_at":"2026-05-29T17:20:12.486728Z","bundle_sha256":"caf4b8fd5f58ba1b6851a672d9ccf05fbfe4162d8839302d0427737e1c38ec8f"}}