{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2012:JDAIUNOOD23BNG2PK6YLHK63XL","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":"4513dffd2fb9b7649412ab8ee7f526a5da0ba0a71b4d1f09a64f5bb30e47b35e","cross_cats_sorted":["math.GR"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AT","submitted_at":"2012-08-29T06:22:44Z","title_canon_sha256":"6a56b33a959cf84f5b1ee41e9abd172e66c3415c0dacc01dd3c2f61757d8ef22"},"schema_version":"1.0","source":{"id":"1208.5844","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1208.5844","created_at":"2026-05-18T03:46:48Z"},{"alias_kind":"arxiv_version","alias_value":"1208.5844v1","created_at":"2026-05-18T03:46:48Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1208.5844","created_at":"2026-05-18T03:46:48Z"},{"alias_kind":"pith_short_12","alias_value":"JDAIUNOOD23B","created_at":"2026-05-18T12:27:11Z"},{"alias_kind":"pith_short_16","alias_value":"JDAIUNOOD23BNG2P","created_at":"2026-05-18T12:27:11Z"},{"alias_kind":"pith_short_8","alias_value":"JDAIUNOO","created_at":"2026-05-18T12:27:11Z"}],"graph_snapshots":[{"event_id":"sha256:acf601ad2a2f428725b964a0c38b18340ae953afd11af64c6259f4f39e37c48d","target":"graph","created_at":"2026-05-18T03:46:48Z","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 define what is meant by a strict total order in a category having subobjects, products and fibre products. This allows us to define the notions of an ordered bundle X and an ordered G-set; when G=\\pi_1(X) we relate these structures to orderings of \\pi_1(X). We apply this to prove a theorem of Farrell relating right-orderings of \\pi_1(X) to embeddings of the universal cover into line bundles over X, and generalize it by relating bi-orderings of \\pi_1(X) to embeddings of the path space into line bundles over X \\times X.","authors_text":"Adam Clay, Mathieu Anel","cross_cats":["math.GR"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AT","submitted_at":"2012-08-29T06:22:44Z","title":"Orderable groups and bundles"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1208.5844","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:e0f427ae5f0c55a287ecac2518bf755c45953cd890a5b39e4a574a9db6861f57","target":"record","created_at":"2026-05-18T03:46:48Z","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":"4513dffd2fb9b7649412ab8ee7f526a5da0ba0a71b4d1f09a64f5bb30e47b35e","cross_cats_sorted":["math.GR"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AT","submitted_at":"2012-08-29T06:22:44Z","title_canon_sha256":"6a56b33a959cf84f5b1ee41e9abd172e66c3415c0dacc01dd3c2f61757d8ef22"},"schema_version":"1.0","source":{"id":"1208.5844","kind":"arxiv","version":1}},"canonical_sha256":"48c08a35ce1eb6169b4f57b0b3abdbbaf3250729d49acd8fbf8c49f258fd1d9c","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"48c08a35ce1eb6169b4f57b0b3abdbbaf3250729d49acd8fbf8c49f258fd1d9c","first_computed_at":"2026-05-18T03:46:48.540816Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T03:46:48.540816Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"6/gb0sLv0RQKe8yaxtc0vJ67a7oa7BBYU0SsSfAN/zK8DKDGQ0QIOu+DF2fh7jrHkwFIe1EZk2wcvhWDR5rhCg==","signature_status":"signed_v1","signed_at":"2026-05-18T03:46:48.541636Z","signed_message":"canonical_sha256_bytes"},"source_id":"1208.5844","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:e0f427ae5f0c55a287ecac2518bf755c45953cd890a5b39e4a574a9db6861f57","sha256:acf601ad2a2f428725b964a0c38b18340ae953afd11af64c6259f4f39e37c48d"],"state_sha256":"5f3f04d40e6261fc7205fdc5adf799abeae01cf487b8511772fcb45d2d465da0"}