{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2001:CYE2CB6642EXNLQ5C4WDK27H7P","short_pith_number":"pith:CYE2CB66","canonical_record":{"source":{"id":"math/0109170","kind":"arxiv","version":3},"metadata":{"license":"","primary_cat":"math.AT","submitted_at":"2001-09-23T00:41:53Z","cross_cats_sorted":["math.CT"],"title_canon_sha256":"803a2dda8df6485adcd20cc733bebadc7a96bd2bd7bce593356ad267b0ce4ec8","abstract_canon_sha256":"02a04b0edb968b4aab85dd8bdad67429e224f6eb1e6e13519c44fb651d13295c"},"schema_version":"1.0"},"canonical_sha256":"1609a107dee68976ae1d172c356be7fbcfdac27c1d0f08393c9974b493a1fd46","source":{"kind":"arxiv","id":"math/0109170","version":3},"source_aliases":[{"alias_kind":"arxiv","alias_value":"math/0109170","created_at":"2026-05-18T00:40:35Z"},{"alias_kind":"arxiv_version","alias_value":"math/0109170v3","created_at":"2026-05-18T00:40:35Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.math/0109170","created_at":"2026-05-18T00:40:35Z"},{"alias_kind":"pith_short_12","alias_value":"CYE2CB6642EX","created_at":"2026-05-18T12:25:50Z"},{"alias_kind":"pith_short_16","alias_value":"CYE2CB6642EXNLQ5","created_at":"2026-05-18T12:25:50Z"},{"alias_kind":"pith_short_8","alias_value":"CYE2CB66","created_at":"2026-05-18T12:25:50Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2001:CYE2CB6642EXNLQ5C4WDK27H7P","target":"record","payload":{"canonical_record":{"source":{"id":"math/0109170","kind":"arxiv","version":3},"metadata":{"license":"","primary_cat":"math.AT","submitted_at":"2001-09-23T00:41:53Z","cross_cats_sorted":["math.CT"],"title_canon_sha256":"803a2dda8df6485adcd20cc733bebadc7a96bd2bd7bce593356ad267b0ce4ec8","abstract_canon_sha256":"02a04b0edb968b4aab85dd8bdad67429e224f6eb1e6e13519c44fb651d13295c"},"schema_version":"1.0"},"canonical_sha256":"1609a107dee68976ae1d172c356be7fbcfdac27c1d0f08393c9974b493a1fd46","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:40:35.300749Z","signature_b64":"JTlm0KmKOaNkYzrQEuyv/O6mRf5t9LESKUWRW5LQ67ITLiiRzJcZrqUH4ZTBGgIn7Yd15JLNQmIB7xgfjxZMAQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"1609a107dee68976ae1d172c356be7fbcfdac27c1d0f08393c9974b493a1fd46","last_reissued_at":"2026-05-18T00:40:35.300000Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:40:35.300000Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"math/0109170","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":"2026-05-18T00:40:35Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"Hudt2FlYxjMSAJq3RSQjL4/pIjzMWb/3M7TVqWpT4ET2au4lkrZPIPi5p5ep0Enr8IzYbKhdT+Wr2no2pk+VAw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-07T15:32:16.139259Z"},"content_sha256":"20d42a657f20d02011cbe6bb4efdd6f65857bc1fbb6d3f676a1b1c0c2b695c5d","schema_version":"1.0","event_id":"sha256:20d42a657f20d02011cbe6bb4efdd6f65857bc1fbb6d3f676a1b1c0c2b695c5d"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2001:CYE2CB6642EXNLQ5C4WDK27H7P","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Obstruction Theory in Model Categories","license":"","headline":"","cross_cats":["math.CT"],"primary_cat":"math.AT","authors_text":"Daniel C. Isaksen, J. Daniel Christensen, William G. Dwyer","submitted_at":"2001-09-23T00:41:53Z","abstract_excerpt":"Many examples of obstruction theory can be formulated as the study of when a lift exists in a commutative square. Typically, one of the maps is a cofibration of some sort and the opposite map is a fibration, and there is a functorial obstruction class that determines whether a lift exists. Working in an arbitrary pointed proper model category, we classify the cofibrations that have such an obstruction theory with respect to all fibrations. Up to weak equivalence, retract, and cobase change, they are the cofibrations with weakly contractible target. Equivalently, they are the retracts of princi"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"math/0109170","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":"2026-05-18T00:40:35Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"24B5ESsH/h9X5TBnWaJNgBSuHep+IkG/oHDsq24lEl6bt8sRP4duq1AI9ovmoKISDclbxLcAqM9vieHjEJ+lDw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-07T15:32:16.139958Z"},"content_sha256":"af154dba908d97f7022557e1eb6a457cfc9ecaca410d09570cd42b4034c9dca3","schema_version":"1.0","event_id":"sha256:af154dba908d97f7022557e1eb6a457cfc9ecaca410d09570cd42b4034c9dca3"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/CYE2CB6642EXNLQ5C4WDK27H7P/bundle.json","state_url":"https://pith.science/pith/CYE2CB6642EXNLQ5C4WDK27H7P/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/CYE2CB6642EXNLQ5C4WDK27H7P/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":"2026-06-07T15:32:16Z","links":{"resolver":"https://pith.science/pith/CYE2CB6642EXNLQ5C4WDK27H7P","bundle":"https://pith.science/pith/CYE2CB6642EXNLQ5C4WDK27H7P/bundle.json","state":"https://pith.science/pith/CYE2CB6642EXNLQ5C4WDK27H7P/state.json","well_known_bundle":"https://pith.science/.well-known/pith/CYE2CB6642EXNLQ5C4WDK27H7P/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2001:CYE2CB6642EXNLQ5C4WDK27H7P","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":"02a04b0edb968b4aab85dd8bdad67429e224f6eb1e6e13519c44fb651d13295c","cross_cats_sorted":["math.CT"],"license":"","primary_cat":"math.AT","submitted_at":"2001-09-23T00:41:53Z","title_canon_sha256":"803a2dda8df6485adcd20cc733bebadc7a96bd2bd7bce593356ad267b0ce4ec8"},"schema_version":"1.0","source":{"id":"math/0109170","kind":"arxiv","version":3}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"math/0109170","created_at":"2026-05-18T00:40:35Z"},{"alias_kind":"arxiv_version","alias_value":"math/0109170v3","created_at":"2026-05-18T00:40:35Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.math/0109170","created_at":"2026-05-18T00:40:35Z"},{"alias_kind":"pith_short_12","alias_value":"CYE2CB6642EX","created_at":"2026-05-18T12:25:50Z"},{"alias_kind":"pith_short_16","alias_value":"CYE2CB6642EXNLQ5","created_at":"2026-05-18T12:25:50Z"},{"alias_kind":"pith_short_8","alias_value":"CYE2CB66","created_at":"2026-05-18T12:25:50Z"}],"graph_snapshots":[{"event_id":"sha256:af154dba908d97f7022557e1eb6a457cfc9ecaca410d09570cd42b4034c9dca3","target":"graph","created_at":"2026-05-18T00:40:35Z","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":"Many examples of obstruction theory can be formulated as the study of when a lift exists in a commutative square. Typically, one of the maps is a cofibration of some sort and the opposite map is a fibration, and there is a functorial obstruction class that determines whether a lift exists. Working in an arbitrary pointed proper model category, we classify the cofibrations that have such an obstruction theory with respect to all fibrations. Up to weak equivalence, retract, and cobase change, they are the cofibrations with weakly contractible target. Equivalently, they are the retracts of princi","authors_text":"Daniel C. Isaksen, J. Daniel Christensen, William G. Dwyer","cross_cats":["math.CT"],"headline":"","license":"","primary_cat":"math.AT","submitted_at":"2001-09-23T00:41:53Z","title":"Obstruction Theory in Model Categories"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"math/0109170","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":"sha256:20d42a657f20d02011cbe6bb4efdd6f65857bc1fbb6d3f676a1b1c0c2b695c5d","target":"record","created_at":"2026-05-18T00:40:35Z","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":"02a04b0edb968b4aab85dd8bdad67429e224f6eb1e6e13519c44fb651d13295c","cross_cats_sorted":["math.CT"],"license":"","primary_cat":"math.AT","submitted_at":"2001-09-23T00:41:53Z","title_canon_sha256":"803a2dda8df6485adcd20cc733bebadc7a96bd2bd7bce593356ad267b0ce4ec8"},"schema_version":"1.0","source":{"id":"math/0109170","kind":"arxiv","version":3}},"canonical_sha256":"1609a107dee68976ae1d172c356be7fbcfdac27c1d0f08393c9974b493a1fd46","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"1609a107dee68976ae1d172c356be7fbcfdac27c1d0f08393c9974b493a1fd46","first_computed_at":"2026-05-18T00:40:35.300000Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T00:40:35.300000Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"JTlm0KmKOaNkYzrQEuyv/O6mRf5t9LESKUWRW5LQ67ITLiiRzJcZrqUH4ZTBGgIn7Yd15JLNQmIB7xgfjxZMAQ==","signature_status":"signed_v1","signed_at":"2026-05-18T00:40:35.300749Z","signed_message":"canonical_sha256_bytes"},"source_id":"math/0109170","source_kind":"arxiv","source_version":3}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:20d42a657f20d02011cbe6bb4efdd6f65857bc1fbb6d3f676a1b1c0c2b695c5d","sha256:af154dba908d97f7022557e1eb6a457cfc9ecaca410d09570cd42b4034c9dca3"],"state_sha256":"2cdff1444a429439c909115928a20e725301bf09a77d6d45a5130f4c1b190583"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"Al3bgEJ6c40Q4qkfS8+l0r2tyhPkMhdmV+0Co4dD8sXWycbKlTGpgLfD/2bemS4DDOJRSNyeLrY1L9V8diIYDw==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-07T15:32:16.143879Z","bundle_sha256":"ded3b6e343fa571345687969552efd9ad0560ed7730f9636617f06118498891c"}}