{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2014:OUI6OVXYHLGFHUOFR5DI2DSZWP","short_pith_number":"pith:OUI6OVXY","canonical_record":{"source":{"id":"1412.6560","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CT","submitted_at":"2014-12-19T23:39:47Z","cross_cats_sorted":["math.AT"],"title_canon_sha256":"46aeec2ede6187154dd2795678d137f8becd1cf82c79e9c29ad862938972f1ff","abstract_canon_sha256":"dbda7f9396cbea249809a5b1972cb5e265f077e619496e794967457996ef31c0"},"schema_version":"1.0"},"canonical_sha256":"7511e756f83acc53d1c58f468d0e59b3c3c27ecd5e7f2646abe16654064f2c6a","source":{"kind":"arxiv","id":"1412.6560","version":3},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1412.6560","created_at":"2026-05-18T01:33:16Z"},{"alias_kind":"arxiv_version","alias_value":"1412.6560v3","created_at":"2026-05-18T01:33:16Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1412.6560","created_at":"2026-05-18T01:33:16Z"},{"alias_kind":"pith_short_12","alias_value":"OUI6OVXYHLGF","created_at":"2026-05-18T12:28:43Z"},{"alias_kind":"pith_short_16","alias_value":"OUI6OVXYHLGFHUOF","created_at":"2026-05-18T12:28:43Z"},{"alias_kind":"pith_short_8","alias_value":"OUI6OVXY","created_at":"2026-05-18T12:28:43Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2014:OUI6OVXYHLGFHUOFR5DI2DSZWP","target":"record","payload":{"canonical_record":{"source":{"id":"1412.6560","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CT","submitted_at":"2014-12-19T23:39:47Z","cross_cats_sorted":["math.AT"],"title_canon_sha256":"46aeec2ede6187154dd2795678d137f8becd1cf82c79e9c29ad862938972f1ff","abstract_canon_sha256":"dbda7f9396cbea249809a5b1972cb5e265f077e619496e794967457996ef31c0"},"schema_version":"1.0"},"canonical_sha256":"7511e756f83acc53d1c58f468d0e59b3c3c27ecd5e7f2646abe16654064f2c6a","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:33:16.267008Z","signature_b64":"55p2O/xRFZUPh4VriXJ2VNrzCn/gFU2gu7fRIGl7Pij2rpx9v8DAj25Y2fAoJ+fXGdGL3IfJq1SlXz+cb+uODA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"7511e756f83acc53d1c58f468d0e59b3c3c27ecd5e7f2646abe16654064f2c6a","last_reissued_at":"2026-05-18T01:33:16.266450Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:33:16.266450Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1412.6560","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":"2026-05-18T01:33:16Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"CpN4vgrHjjTGbon7d4xfqhcz1YUGA6UzKr0+EXywXTAzdtkJRvPq7kzCXkXNpsvyvplaFRh86ZE9JZjRI0DWBg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-03T07:28:09.676941Z"},"content_sha256":"a9f2c15d4ea63f86a41c9a973912d536982a824352c921a25de6a47e2b6aad17","schema_version":"1.0","event_id":"sha256:a9f2c15d4ea63f86a41c9a973912d536982a824352c921a25de6a47e2b6aad17"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2014:OUI6OVXYHLGFHUOFR5DI2DSZWP","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Algebraic weak factorisation systems II: categories of weak maps","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.AT"],"primary_cat":"math.CT","authors_text":"John Bourke, Richard Garner","submitted_at":"2014-12-19T23:39:47Z","abstract_excerpt":"We investigate the categories of weak maps associated to an algebraic weak factorisation system (AWFS) in the sense of Grandis-Tholen. For any AWFS on a category with an initial object, cofibrant replacement forms a comonad, and the category of (left) weak maps associated to the AWFS is by definition the Kleisli category of this comonad. We exhibit categories of weak maps as a kind of \"homotopy category\", that freely adjoins a section for every \"acyclic fibration\" (=right map) of the AWFS; and using this characterisation, we give an alternate description of categories of weak maps in terms of "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1412.6560","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":"2026-05-18T01:33:16Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"8PnKa/cpUzhtjK7s6rulDcccupmzg1MPr9yM0vp+DR43kjYKGK/5BxsFUdYbC6TDKy9dWExj9PftmLf2Gx9SCA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-03T07:28:09.677297Z"},"content_sha256":"f4bfc97e61fc552701397fe85f26428012cef5e3ad07fe0c20809540518763a2","schema_version":"1.0","event_id":"sha256:f4bfc97e61fc552701397fe85f26428012cef5e3ad07fe0c20809540518763a2"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/OUI6OVXYHLGFHUOFR5DI2DSZWP/bundle.json","state_url":"https://pith.science/pith/OUI6OVXYHLGFHUOFR5DI2DSZWP/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/OUI6OVXYHLGFHUOFR5DI2DSZWP/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":"2026-06-03T07:28:09Z","links":{"resolver":"https://pith.science/pith/OUI6OVXYHLGFHUOFR5DI2DSZWP","bundle":"https://pith.science/pith/OUI6OVXYHLGFHUOFR5DI2DSZWP/bundle.json","state":"https://pith.science/pith/OUI6OVXYHLGFHUOFR5DI2DSZWP/state.json","well_known_bundle":"https://pith.science/.well-known/pith/OUI6OVXYHLGFHUOFR5DI2DSZWP/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2014:OUI6OVXYHLGFHUOFR5DI2DSZWP","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":"dbda7f9396cbea249809a5b1972cb5e265f077e619496e794967457996ef31c0","cross_cats_sorted":["math.AT"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CT","submitted_at":"2014-12-19T23:39:47Z","title_canon_sha256":"46aeec2ede6187154dd2795678d137f8becd1cf82c79e9c29ad862938972f1ff"},"schema_version":"1.0","source":{"id":"1412.6560","kind":"arxiv","version":3}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1412.6560","created_at":"2026-05-18T01:33:16Z"},{"alias_kind":"arxiv_version","alias_value":"1412.6560v3","created_at":"2026-05-18T01:33:16Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1412.6560","created_at":"2026-05-18T01:33:16Z"},{"alias_kind":"pith_short_12","alias_value":"OUI6OVXYHLGF","created_at":"2026-05-18T12:28:43Z"},{"alias_kind":"pith_short_16","alias_value":"OUI6OVXYHLGFHUOF","created_at":"2026-05-18T12:28:43Z"},{"alias_kind":"pith_short_8","alias_value":"OUI6OVXY","created_at":"2026-05-18T12:28:43Z"}],"graph_snapshots":[{"event_id":"sha256:f4bfc97e61fc552701397fe85f26428012cef5e3ad07fe0c20809540518763a2","target":"graph","created_at":"2026-05-18T01:33:16Z","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 investigate the categories of weak maps associated to an algebraic weak factorisation system (AWFS) in the sense of Grandis-Tholen. For any AWFS on a category with an initial object, cofibrant replacement forms a comonad, and the category of (left) weak maps associated to the AWFS is by definition the Kleisli category of this comonad. We exhibit categories of weak maps as a kind of \"homotopy category\", that freely adjoins a section for every \"acyclic fibration\" (=right map) of the AWFS; and using this characterisation, we give an alternate description of categories of weak maps in terms of ","authors_text":"John Bourke, Richard Garner","cross_cats":["math.AT"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CT","submitted_at":"2014-12-19T23:39:47Z","title":"Algebraic weak factorisation systems II: categories of weak maps"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1412.6560","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:a9f2c15d4ea63f86a41c9a973912d536982a824352c921a25de6a47e2b6aad17","target":"record","created_at":"2026-05-18T01:33:16Z","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":"dbda7f9396cbea249809a5b1972cb5e265f077e619496e794967457996ef31c0","cross_cats_sorted":["math.AT"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CT","submitted_at":"2014-12-19T23:39:47Z","title_canon_sha256":"46aeec2ede6187154dd2795678d137f8becd1cf82c79e9c29ad862938972f1ff"},"schema_version":"1.0","source":{"id":"1412.6560","kind":"arxiv","version":3}},"canonical_sha256":"7511e756f83acc53d1c58f468d0e59b3c3c27ecd5e7f2646abe16654064f2c6a","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"7511e756f83acc53d1c58f468d0e59b3c3c27ecd5e7f2646abe16654064f2c6a","first_computed_at":"2026-05-18T01:33:16.266450Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T01:33:16.266450Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"55p2O/xRFZUPh4VriXJ2VNrzCn/gFU2gu7fRIGl7Pij2rpx9v8DAj25Y2fAoJ+fXGdGL3IfJq1SlXz+cb+uODA==","signature_status":"signed_v1","signed_at":"2026-05-18T01:33:16.267008Z","signed_message":"canonical_sha256_bytes"},"source_id":"1412.6560","source_kind":"arxiv","source_version":3}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:a9f2c15d4ea63f86a41c9a973912d536982a824352c921a25de6a47e2b6aad17","sha256:f4bfc97e61fc552701397fe85f26428012cef5e3ad07fe0c20809540518763a2"],"state_sha256":"e158607f82064420c1572a94ad33d947f48c346a6f1d534444a27454fd322cef"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"s8IqcChtCXUHzJhhCfgeaS/2CqWCSkS6x1zfJls5kZ0TCvnYiNU0nn4wzoeHE6JqGqRpOI12dAdvg3bjN7crAQ==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-03T07:28:09.679388Z","bundle_sha256":"f14331547ad9f10eef1db0cb992d474481a0259403ca9392e782c47559eabee3"}}