{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2015:PM3KSBAJOUNZIJH2X6UJX26U6B","short_pith_number":"pith:PM3KSBAJ","canonical_record":{"source":{"id":"1510.07133","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CT","submitted_at":"2015-10-24T11:45:20Z","cross_cats_sorted":["math.AT"],"title_canon_sha256":"df2f94f308b34f34ccee76b8ea3b51b90789e6a738e1ed1f0712c54cf715ea85","abstract_canon_sha256":"2a2dd0c1872130e1bef7e39951b40e79500b0e830e8872676da0e8f6635aeac8"},"schema_version":"1.0"},"canonical_sha256":"7b36a90409751b9424fabfa89bebd4f0688838f77f290aae1e945ecbeb17a912","source":{"kind":"arxiv","id":"1510.07133","version":2},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1510.07133","created_at":"2026-05-17T23:51:34Z"},{"alias_kind":"arxiv_version","alias_value":"1510.07133v2","created_at":"2026-05-17T23:51:34Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1510.07133","created_at":"2026-05-17T23:51:34Z"},{"alias_kind":"pith_short_12","alias_value":"PM3KSBAJOUNZ","created_at":"2026-05-18T12:29:37Z"},{"alias_kind":"pith_short_16","alias_value":"PM3KSBAJOUNZIJH2","created_at":"2026-05-18T12:29:37Z"},{"alias_kind":"pith_short_8","alias_value":"PM3KSBAJ","created_at":"2026-05-18T12:29:37Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2015:PM3KSBAJOUNZIJH2X6UJX26U6B","target":"record","payload":{"canonical_record":{"source":{"id":"1510.07133","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CT","submitted_at":"2015-10-24T11:45:20Z","cross_cats_sorted":["math.AT"],"title_canon_sha256":"df2f94f308b34f34ccee76b8ea3b51b90789e6a738e1ed1f0712c54cf715ea85","abstract_canon_sha256":"2a2dd0c1872130e1bef7e39951b40e79500b0e830e8872676da0e8f6635aeac8"},"schema_version":"1.0"},"canonical_sha256":"7b36a90409751b9424fabfa89bebd4f0688838f77f290aae1e945ecbeb17a912","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-17T23:51:34.990519Z","signature_b64":"u+QcCc3LI5oUC96/UyCoCSrTfQo11pq5WmpZ/4VgyFz0dzX+ahXjSU0aq49tcWHDmOlm+xxSyVtV/FnGznroBA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"7b36a90409751b9424fabfa89bebd4f0688838f77f290aae1e945ecbeb17a912","last_reissued_at":"2026-05-17T23:51:34.989576Z","signature_status":"signed_v1","first_computed_at":"2026-05-17T23:51:34.989576Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1510.07133","source_version":2,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-17T23:51:34Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"riwBU75yQLVj+fFdz3QpcKM3IptMCS7UvtAOZOoF6Df4ti0zkRU3vJwESejEdU5DNFUDUOXDGhggt/0VnG5xCA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-30T20:29:17.034824Z"},"content_sha256":"0f80d4487b251e1ce1929a71445905e583d76551868d308e1a48f5de2440e99c","schema_version":"1.0","event_id":"sha256:0f80d4487b251e1ce1929a71445905e583d76551868d308e1a48f5de2440e99c"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2015:PM3KSBAJOUNZIJH2X6UJX26U6B","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Two-Fold Homotopy of 2-Crossed Module Maps of Commutative Algebras","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.AT"],"primary_cat":"math.CT","authors_text":"\\.I.\\.Ilker Ak\\c{c}a, Jo\\~ao Faria Martins, Kadir Emir","submitted_at":"2015-10-24T11:45:20Z","abstract_excerpt":"We address the homotopy theory of 2-crossed modules of commutative algebras. In particular, we define the concept of a 2-fold homotopy between a pair of 1-fold homotopies connecting 2-crossed module maps $\\A \\to \\A'$. We also prove that if the domain 2-crossed module $\\A$ is free up to order one (i.e. if the bottom algebra is a polynomial algebra) then we have a 2-groupoid of 2-crossed module maps $\\A \\to \\A'$ and their homotopies and 2-fold homotopies."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1510.07133","kind":"arxiv","version":2},"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":"2026-05-17T23:51:34Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"3DHwZPhCi/pDuhIuGqI3SmcNNk1xx0HfTnelSJvPYUi05XoVy9iok+tNXdV3FlDdHJvceb6WQmtmtrVa80wxCA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-30T20:29:17.035527Z"},"content_sha256":"97f68ed29c219672304f9fe90c00a65123b158e1963a8d6c53373ed357e17a4a","schema_version":"1.0","event_id":"sha256:97f68ed29c219672304f9fe90c00a65123b158e1963a8d6c53373ed357e17a4a"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/PM3KSBAJOUNZIJH2X6UJX26U6B/bundle.json","state_url":"https://pith.science/pith/PM3KSBAJOUNZIJH2X6UJX26U6B/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/PM3KSBAJOUNZIJH2X6UJX26U6B/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":"2026-05-30T20:29:17Z","links":{"resolver":"https://pith.science/pith/PM3KSBAJOUNZIJH2X6UJX26U6B","bundle":"https://pith.science/pith/PM3KSBAJOUNZIJH2X6UJX26U6B/bundle.json","state":"https://pith.science/pith/PM3KSBAJOUNZIJH2X6UJX26U6B/state.json","well_known_bundle":"https://pith.science/.well-known/pith/PM3KSBAJOUNZIJH2X6UJX26U6B/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2015:PM3KSBAJOUNZIJH2X6UJX26U6B","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":"2a2dd0c1872130e1bef7e39951b40e79500b0e830e8872676da0e8f6635aeac8","cross_cats_sorted":["math.AT"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CT","submitted_at":"2015-10-24T11:45:20Z","title_canon_sha256":"df2f94f308b34f34ccee76b8ea3b51b90789e6a738e1ed1f0712c54cf715ea85"},"schema_version":"1.0","source":{"id":"1510.07133","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1510.07133","created_at":"2026-05-17T23:51:34Z"},{"alias_kind":"arxiv_version","alias_value":"1510.07133v2","created_at":"2026-05-17T23:51:34Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1510.07133","created_at":"2026-05-17T23:51:34Z"},{"alias_kind":"pith_short_12","alias_value":"PM3KSBAJOUNZ","created_at":"2026-05-18T12:29:37Z"},{"alias_kind":"pith_short_16","alias_value":"PM3KSBAJOUNZIJH2","created_at":"2026-05-18T12:29:37Z"},{"alias_kind":"pith_short_8","alias_value":"PM3KSBAJ","created_at":"2026-05-18T12:29:37Z"}],"graph_snapshots":[{"event_id":"sha256:97f68ed29c219672304f9fe90c00a65123b158e1963a8d6c53373ed357e17a4a","target":"graph","created_at":"2026-05-17T23:51:34Z","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 address the homotopy theory of 2-crossed modules of commutative algebras. In particular, we define the concept of a 2-fold homotopy between a pair of 1-fold homotopies connecting 2-crossed module maps $\\A \\to \\A'$. We also prove that if the domain 2-crossed module $\\A$ is free up to order one (i.e. if the bottom algebra is a polynomial algebra) then we have a 2-groupoid of 2-crossed module maps $\\A \\to \\A'$ and their homotopies and 2-fold homotopies.","authors_text":"\\.I.\\.Ilker Ak\\c{c}a, Jo\\~ao Faria Martins, Kadir Emir","cross_cats":["math.AT"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CT","submitted_at":"2015-10-24T11:45:20Z","title":"Two-Fold Homotopy of 2-Crossed Module Maps of Commutative Algebras"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1510.07133","kind":"arxiv","version":2},"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":"sha256:0f80d4487b251e1ce1929a71445905e583d76551868d308e1a48f5de2440e99c","target":"record","created_at":"2026-05-17T23:51:34Z","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":"2a2dd0c1872130e1bef7e39951b40e79500b0e830e8872676da0e8f6635aeac8","cross_cats_sorted":["math.AT"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CT","submitted_at":"2015-10-24T11:45:20Z","title_canon_sha256":"df2f94f308b34f34ccee76b8ea3b51b90789e6a738e1ed1f0712c54cf715ea85"},"schema_version":"1.0","source":{"id":"1510.07133","kind":"arxiv","version":2}},"canonical_sha256":"7b36a90409751b9424fabfa89bebd4f0688838f77f290aae1e945ecbeb17a912","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"7b36a90409751b9424fabfa89bebd4f0688838f77f290aae1e945ecbeb17a912","first_computed_at":"2026-05-17T23:51:34.989576Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-17T23:51:34.989576Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"u+QcCc3LI5oUC96/UyCoCSrTfQo11pq5WmpZ/4VgyFz0dzX+ahXjSU0aq49tcWHDmOlm+xxSyVtV/FnGznroBA==","signature_status":"signed_v1","signed_at":"2026-05-17T23:51:34.990519Z","signed_message":"canonical_sha256_bytes"},"source_id":"1510.07133","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:0f80d4487b251e1ce1929a71445905e583d76551868d308e1a48f5de2440e99c","sha256:97f68ed29c219672304f9fe90c00a65123b158e1963a8d6c53373ed357e17a4a"],"state_sha256":"d4bae909032ac3bfd45670dd8e88131370d0b799329c7ca27ee89871aefbee8f"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"1TbC+ZrG4hXOqp7tyvBlpSc515QEheo8/l+QXQtvvz+6iFv7JrbxFVCcwMm9DaGLq2pDzf5SjvuFqa6gHmHiDA==","signed_message":"bundle_sha256_bytes","signed_at":"2026-05-30T20:29:17.038527Z","bundle_sha256":"23e894658ac62c0db32e910a256034966815d6c9cf227b383d9b494a9ced6c07"}}