{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2015:KO65CVCKNXTT544SOYV4VJFX7V","short_pith_number":"pith:KO65CVCK","canonical_record":{"source":{"id":"1511.07366","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DG","submitted_at":"2015-11-23T19:06:35Z","cross_cats_sorted":[],"title_canon_sha256":"3291e87e94b26b2e5087bd5d6aca37c76de86b69243eaff2b6a230a98eec117c","abstract_canon_sha256":"85752c670b4b5cdf1cf71e3bc0917b1fda774b3d41a900ad2d369c0dca028134"},"schema_version":"1.0"},"canonical_sha256":"53bdd1544a6de73ef392762bcaa4b7fd6e3f51fdf95e0749d74d93e3c139da2b","source":{"kind":"arxiv","id":"1511.07366","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1511.07366","created_at":"2026-05-18T01:26:11Z"},{"alias_kind":"arxiv_version","alias_value":"1511.07366v1","created_at":"2026-05-18T01:26:11Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1511.07366","created_at":"2026-05-18T01:26:11Z"},{"alias_kind":"pith_short_12","alias_value":"KO65CVCKNXTT","created_at":"2026-05-18T12:29:29Z"},{"alias_kind":"pith_short_16","alias_value":"KO65CVCKNXTT544S","created_at":"2026-05-18T12:29:29Z"},{"alias_kind":"pith_short_8","alias_value":"KO65CVCK","created_at":"2026-05-18T12:29:29Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2015:KO65CVCKNXTT544SOYV4VJFX7V","target":"record","payload":{"canonical_record":{"source":{"id":"1511.07366","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DG","submitted_at":"2015-11-23T19:06:35Z","cross_cats_sorted":[],"title_canon_sha256":"3291e87e94b26b2e5087bd5d6aca37c76de86b69243eaff2b6a230a98eec117c","abstract_canon_sha256":"85752c670b4b5cdf1cf71e3bc0917b1fda774b3d41a900ad2d369c0dca028134"},"schema_version":"1.0"},"canonical_sha256":"53bdd1544a6de73ef392762bcaa4b7fd6e3f51fdf95e0749d74d93e3c139da2b","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:26:11.543035Z","signature_b64":"jy+oyxvxeteNGOxxEf5lSILy45v0eEUvjtuuBTdLPh8KFgBn+8HM8PI7UGLbCdbm/v6+Lqf3mRH0qIkGPYbmDw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"53bdd1544a6de73ef392762bcaa4b7fd6e3f51fdf95e0749d74d93e3c139da2b","last_reissued_at":"2026-05-18T01:26:11.542547Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:26:11.542547Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1511.07366","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":"2026-05-18T01:26:11Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"Vdv/4GKHkH5nH2Ckd9ouMhBADv5Ux6DgRDsKxhoGihwV1ISV8EU+XxXFLTODd7NiuFC59yUZ9YDYMQuvV7E+DA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-10T22:57:40.259121Z"},"content_sha256":"01abb40255c0ee07842d5cb5c5de0f27a875d7ad74db517e251a190118d1d0ce","schema_version":"1.0","event_id":"sha256:01abb40255c0ee07842d5cb5c5de0f27a875d7ad74db517e251a190118d1d0ce"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2015:KO65CVCKNXTT544SOYV4VJFX7V","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Lie Algebroids over Differentiable Stacks","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.DG","authors_text":"James Waldron","submitted_at":"2015-11-23T19:06:35Z","abstract_excerpt":"We develop a theory of Lie algebroids over differentiable stacks that extends the standard theory of Lie algebroids over manifolds. In particular we show that Lie algebroids satisfy descent for submersions, define the category of Lie algebroids over a differentiable stack, construct a cohomology theory for these objects, and explain the relation to the theory of $\\mathcal{LA}$-groupoids. We construct a number of examples."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1511.07366","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":""},"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":"2026-05-18T01:26:11Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"skE/XcRyCJZBCVHFnmc6q0NxKVWiIogqKktyAqpllc51vj7vtGKXFxzwddCSzxbQ32s30JxE5NtXKNq5OK4QDQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-10T22:57:40.259833Z"},"content_sha256":"4c9b132673cdb778c45fd66cc254f525d44aad25bbfc6132fc5e701439a14035","schema_version":"1.0","event_id":"sha256:4c9b132673cdb778c45fd66cc254f525d44aad25bbfc6132fc5e701439a14035"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/KO65CVCKNXTT544SOYV4VJFX7V/bundle.json","state_url":"https://pith.science/pith/KO65CVCKNXTT544SOYV4VJFX7V/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/KO65CVCKNXTT544SOYV4VJFX7V/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":"2026-06-10T22:57:40Z","links":{"resolver":"https://pith.science/pith/KO65CVCKNXTT544SOYV4VJFX7V","bundle":"https://pith.science/pith/KO65CVCKNXTT544SOYV4VJFX7V/bundle.json","state":"https://pith.science/pith/KO65CVCKNXTT544SOYV4VJFX7V/state.json","well_known_bundle":"https://pith.science/.well-known/pith/KO65CVCKNXTT544SOYV4VJFX7V/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2015:KO65CVCKNXTT544SOYV4VJFX7V","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":"85752c670b4b5cdf1cf71e3bc0917b1fda774b3d41a900ad2d369c0dca028134","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DG","submitted_at":"2015-11-23T19:06:35Z","title_canon_sha256":"3291e87e94b26b2e5087bd5d6aca37c76de86b69243eaff2b6a230a98eec117c"},"schema_version":"1.0","source":{"id":"1511.07366","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1511.07366","created_at":"2026-05-18T01:26:11Z"},{"alias_kind":"arxiv_version","alias_value":"1511.07366v1","created_at":"2026-05-18T01:26:11Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1511.07366","created_at":"2026-05-18T01:26:11Z"},{"alias_kind":"pith_short_12","alias_value":"KO65CVCKNXTT","created_at":"2026-05-18T12:29:29Z"},{"alias_kind":"pith_short_16","alias_value":"KO65CVCKNXTT544S","created_at":"2026-05-18T12:29:29Z"},{"alias_kind":"pith_short_8","alias_value":"KO65CVCK","created_at":"2026-05-18T12:29:29Z"}],"graph_snapshots":[{"event_id":"sha256:4c9b132673cdb778c45fd66cc254f525d44aad25bbfc6132fc5e701439a14035","target":"graph","created_at":"2026-05-18T01:26:11Z","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 develop a theory of Lie algebroids over differentiable stacks that extends the standard theory of Lie algebroids over manifolds. In particular we show that Lie algebroids satisfy descent for submersions, define the category of Lie algebroids over a differentiable stack, construct a cohomology theory for these objects, and explain the relation to the theory of $\\mathcal{LA}$-groupoids. We construct a number of examples.","authors_text":"James Waldron","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DG","submitted_at":"2015-11-23T19:06:35Z","title":"Lie Algebroids over Differentiable Stacks"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1511.07366","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":"sha256:01abb40255c0ee07842d5cb5c5de0f27a875d7ad74db517e251a190118d1d0ce","target":"record","created_at":"2026-05-18T01:26:11Z","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":"85752c670b4b5cdf1cf71e3bc0917b1fda774b3d41a900ad2d369c0dca028134","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DG","submitted_at":"2015-11-23T19:06:35Z","title_canon_sha256":"3291e87e94b26b2e5087bd5d6aca37c76de86b69243eaff2b6a230a98eec117c"},"schema_version":"1.0","source":{"id":"1511.07366","kind":"arxiv","version":1}},"canonical_sha256":"53bdd1544a6de73ef392762bcaa4b7fd6e3f51fdf95e0749d74d93e3c139da2b","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"53bdd1544a6de73ef392762bcaa4b7fd6e3f51fdf95e0749d74d93e3c139da2b","first_computed_at":"2026-05-18T01:26:11.542547Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T01:26:11.542547Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"jy+oyxvxeteNGOxxEf5lSILy45v0eEUvjtuuBTdLPh8KFgBn+8HM8PI7UGLbCdbm/v6+Lqf3mRH0qIkGPYbmDw==","signature_status":"signed_v1","signed_at":"2026-05-18T01:26:11.543035Z","signed_message":"canonical_sha256_bytes"},"source_id":"1511.07366","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:01abb40255c0ee07842d5cb5c5de0f27a875d7ad74db517e251a190118d1d0ce","sha256:4c9b132673cdb778c45fd66cc254f525d44aad25bbfc6132fc5e701439a14035"],"state_sha256":"5b6632d965ffb444c3135aab92669324e867f66a51a0e4fc8eb795f3e5a817a3"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"5e8QBVYoi3cFgPmCPQhKg+q3K63nSSuk1TOldZagU3Ra93KsSeCcodqi5SxW6h86tAOD+BLzbmi3MYOpvJngBg==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-10T22:57:40.266030Z","bundle_sha256":"5d5bde16e8def3867489749e53f3e501983ae4e1c71846743e3f673a33a837d8"}}