{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2025:5Z3MIZLKZVF4C6ZTWO2BXEODEZ","short_pith_number":"pith:5Z3MIZLK","canonical_record":{"source":{"id":"2502.20699","kind":"arxiv","version":1},"metadata":{"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.CT","submitted_at":"2025-02-28T04:17:31Z","cross_cats_sorted":["math.DG"],"title_canon_sha256":"c3269e78f53ea12baf2ff27b82195c7efd68cbcfd14b0eef329d9375b8754ba5","abstract_canon_sha256":"ab05a84e28b748f17980a9083f87b1356997b094e6fff3eb2cbdb061860c8a17"},"schema_version":"1.0"},"canonical_sha256":"ee76c4656acd4bc17b33b3b41b91c326453a54ee772d975d9dac3e6a3acd59a1","source":{"kind":"arxiv","id":"2502.20699","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2502.20699","created_at":"2026-07-05T10:21:38Z"},{"alias_kind":"arxiv_version","alias_value":"2502.20699v1","created_at":"2026-07-05T10:21:38Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2502.20699","created_at":"2026-07-05T10:21:38Z"},{"alias_kind":"pith_short_12","alias_value":"5Z3MIZLKZVF4","created_at":"2026-07-05T10:21:38Z"},{"alias_kind":"pith_short_16","alias_value":"5Z3MIZLKZVF4C6ZT","created_at":"2026-07-05T10:21:38Z"},{"alias_kind":"pith_short_8","alias_value":"5Z3MIZLK","created_at":"2026-07-05T10:21:38Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2025:5Z3MIZLKZVF4C6ZTWO2BXEODEZ","target":"record","payload":{"canonical_record":{"source":{"id":"2502.20699","kind":"arxiv","version":1},"metadata":{"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.CT","submitted_at":"2025-02-28T04:17:31Z","cross_cats_sorted":["math.DG"],"title_canon_sha256":"c3269e78f53ea12baf2ff27b82195c7efd68cbcfd14b0eef329d9375b8754ba5","abstract_canon_sha256":"ab05a84e28b748f17980a9083f87b1356997b094e6fff3eb2cbdb061860c8a17"},"schema_version":"1.0"},"canonical_sha256":"ee76c4656acd4bc17b33b3b41b91c326453a54ee772d975d9dac3e6a3acd59a1","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T10:21:38.161068Z","signature_b64":"N7TBOGbwU4xGGIFUXJDnaVaV49GfURuzfVQzZioFRHOUYHqrdVAdWm9wdBzowFlgAVIaD7qQB5T/V5AevuT0DA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"ee76c4656acd4bc17b33b3b41b91c326453a54ee772d975d9dac3e6a3acd59a1","last_reissued_at":"2026-07-05T10:21:38.160489Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T10:21:38.160489Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"2502.20699","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":"2026-07-05T10:21:38Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"Hn9LCxlzfE15zKtoC7BEpXmeyj55QWm3leBRIG72xBwCnlm7pq45+qoAfS9iKBA5RsECQFOyrk6iE2CrFIrbCA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-07-09T23:06:37.365565Z"},"content_sha256":"113f15311f27c8abedc72cd1cf3d12fac94edbc8d57b50699226f23666eac7ff","schema_version":"1.0","event_id":"sha256:113f15311f27c8abedc72cd1cf3d12fac94edbc8d57b50699226f23666eac7ff"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2025:5Z3MIZLKZVF4C6ZTWO2BXEODEZ","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Pullbacks in tangent categories and tangent display maps","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":["math.DG"],"primary_cat":"math.CT","authors_text":"Geoffrey Cruttwell, Marcello Lanfranchi","submitted_at":"2025-02-28T04:17:31Z","abstract_excerpt":"In differential geometry, the existence of pullbacks is a delicate matter, since the category of smooth manifolds does not admit all of them. When pullbacks are required, often submersions are employed as an ideal class of maps which behaves well under this operation and the tangent bundle functor. This issue is reflected in tangent category theory, which aims to axiomatize the tangent bundle functor of differential geometry categorically. Key constructions such as connections, tangent fibrations, or reverse tangent categories require one to work with pullbacks preserved by the tangent bundle "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2502.20699","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":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2502.20699/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"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":"2026-07-05T10:21:38Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"N681usnoPnFCyErboE3JcNb2FbvhV4XS29PFSJ4F9JM17L2Ih0T7snD2PFY/v44RNTwNOLgSxJ4BQH0lZejiCg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-07-09T23:06:37.365942Z"},"content_sha256":"9e20fe1afdde190b8422c25170b63758b1d8f92c8f150576f88c4197210d92e8","schema_version":"1.0","event_id":"sha256:9e20fe1afdde190b8422c25170b63758b1d8f92c8f150576f88c4197210d92e8"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/5Z3MIZLKZVF4C6ZTWO2BXEODEZ/bundle.json","state_url":"https://pith.science/pith/5Z3MIZLKZVF4C6ZTWO2BXEODEZ/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/5Z3MIZLKZVF4C6ZTWO2BXEODEZ/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":"2026-07-09T23:06:37Z","links":{"resolver":"https://pith.science/pith/5Z3MIZLKZVF4C6ZTWO2BXEODEZ","bundle":"https://pith.science/pith/5Z3MIZLKZVF4C6ZTWO2BXEODEZ/bundle.json","state":"https://pith.science/pith/5Z3MIZLKZVF4C6ZTWO2BXEODEZ/state.json","well_known_bundle":"https://pith.science/.well-known/pith/5Z3MIZLKZVF4C6ZTWO2BXEODEZ/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2025:5Z3MIZLKZVF4C6ZTWO2BXEODEZ","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":"ab05a84e28b748f17980a9083f87b1356997b094e6fff3eb2cbdb061860c8a17","cross_cats_sorted":["math.DG"],"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.CT","submitted_at":"2025-02-28T04:17:31Z","title_canon_sha256":"c3269e78f53ea12baf2ff27b82195c7efd68cbcfd14b0eef329d9375b8754ba5"},"schema_version":"1.0","source":{"id":"2502.20699","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2502.20699","created_at":"2026-07-05T10:21:38Z"},{"alias_kind":"arxiv_version","alias_value":"2502.20699v1","created_at":"2026-07-05T10:21:38Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2502.20699","created_at":"2026-07-05T10:21:38Z"},{"alias_kind":"pith_short_12","alias_value":"5Z3MIZLKZVF4","created_at":"2026-07-05T10:21:38Z"},{"alias_kind":"pith_short_16","alias_value":"5Z3MIZLKZVF4C6ZT","created_at":"2026-07-05T10:21:38Z"},{"alias_kind":"pith_short_8","alias_value":"5Z3MIZLK","created_at":"2026-07-05T10:21:38Z"}],"graph_snapshots":[{"event_id":"sha256:9e20fe1afdde190b8422c25170b63758b1d8f92c8f150576f88c4197210d92e8","target":"graph","created_at":"2026-07-05T10:21:38Z","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"},"integrity":{"available":true,"clean":true,"detectors_run":[],"endpoint":"/pith/2502.20699/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"In differential geometry, the existence of pullbacks is a delicate matter, since the category of smooth manifolds does not admit all of them. When pullbacks are required, often submersions are employed as an ideal class of maps which behaves well under this operation and the tangent bundle functor. This issue is reflected in tangent category theory, which aims to axiomatize the tangent bundle functor of differential geometry categorically. Key constructions such as connections, tangent fibrations, or reverse tangent categories require one to work with pullbacks preserved by the tangent bundle ","authors_text":"Geoffrey Cruttwell, Marcello Lanfranchi","cross_cats":["math.DG"],"headline":"","license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.CT","submitted_at":"2025-02-28T04:17:31Z","title":"Pullbacks in tangent categories and tangent display maps"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2502.20699","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":"sha256:113f15311f27c8abedc72cd1cf3d12fac94edbc8d57b50699226f23666eac7ff","target":"record","created_at":"2026-07-05T10:21:38Z","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":"ab05a84e28b748f17980a9083f87b1356997b094e6fff3eb2cbdb061860c8a17","cross_cats_sorted":["math.DG"],"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.CT","submitted_at":"2025-02-28T04:17:31Z","title_canon_sha256":"c3269e78f53ea12baf2ff27b82195c7efd68cbcfd14b0eef329d9375b8754ba5"},"schema_version":"1.0","source":{"id":"2502.20699","kind":"arxiv","version":1}},"canonical_sha256":"ee76c4656acd4bc17b33b3b41b91c326453a54ee772d975d9dac3e6a3acd59a1","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"ee76c4656acd4bc17b33b3b41b91c326453a54ee772d975d9dac3e6a3acd59a1","first_computed_at":"2026-07-05T10:21:38.160489Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-07-05T10:21:38.160489Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"N7TBOGbwU4xGGIFUXJDnaVaV49GfURuzfVQzZioFRHOUYHqrdVAdWm9wdBzowFlgAVIaD7qQB5T/V5AevuT0DA==","signature_status":"signed_v1","signed_at":"2026-07-05T10:21:38.161068Z","signed_message":"canonical_sha256_bytes"},"source_id":"2502.20699","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:113f15311f27c8abedc72cd1cf3d12fac94edbc8d57b50699226f23666eac7ff","sha256:9e20fe1afdde190b8422c25170b63758b1d8f92c8f150576f88c4197210d92e8"],"state_sha256":"299714e8d06de322b2a1d39145d8329f5eeaab1ddbe22cbe390f600ea31550cd"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"pBWvRZ5K0xOFBU0WApoRh1JjJo9hmbFZblfdAg943+zDdXS1zbu7F2ybMfH+Z06bPmbZhpX/MVe9mXZmTREUBQ==","signed_message":"bundle_sha256_bytes","signed_at":"2026-07-09T23:06:37.368029Z","bundle_sha256":"77a6d096f23993356bc56871fe9fda2a4bc4130fa7b21b077330a0a620ee57d0"}}