{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2018:O3CQVMJVZZXLC66KKKDJHCK5UL","short_pith_number":"pith:O3CQVMJV","canonical_record":{"source":{"id":"1806.01939","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DG","submitted_at":"2018-06-05T21:37:25Z","cross_cats_sorted":[],"title_canon_sha256":"e63adff64546ea0455928c4a091b1fe385e0a3b75f7d4205c7c598e16ea5acbc","abstract_canon_sha256":"b98a2d2be72f5402e31550d632681a32217bbe2fb213acfb080c50c371f68cf2"},"schema_version":"1.0"},"canonical_sha256":"76c50ab135ce6eb17bca528693895da2e7f4837d0d7f3fac8cf54fe04474fb09","source":{"kind":"arxiv","id":"1806.01939","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1806.01939","created_at":"2026-05-18T00:14:03Z"},{"alias_kind":"arxiv_version","alias_value":"1806.01939v1","created_at":"2026-05-18T00:14:03Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1806.01939","created_at":"2026-05-18T00:14:03Z"},{"alias_kind":"pith_short_12","alias_value":"O3CQVMJVZZXL","created_at":"2026-05-18T12:32:40Z"},{"alias_kind":"pith_short_16","alias_value":"O3CQVMJVZZXLC66K","created_at":"2026-05-18T12:32:40Z"},{"alias_kind":"pith_short_8","alias_value":"O3CQVMJV","created_at":"2026-05-18T12:32:40Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2018:O3CQVMJVZZXLC66KKKDJHCK5UL","target":"record","payload":{"canonical_record":{"source":{"id":"1806.01939","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DG","submitted_at":"2018-06-05T21:37:25Z","cross_cats_sorted":[],"title_canon_sha256":"e63adff64546ea0455928c4a091b1fe385e0a3b75f7d4205c7c598e16ea5acbc","abstract_canon_sha256":"b98a2d2be72f5402e31550d632681a32217bbe2fb213acfb080c50c371f68cf2"},"schema_version":"1.0"},"canonical_sha256":"76c50ab135ce6eb17bca528693895da2e7f4837d0d7f3fac8cf54fe04474fb09","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:14:03.358243Z","signature_b64":"x8SN94MhZkRnQGOHRhzn9FjuS4jC9/4Xf9N8U2J6cInalR+sj8moJxt9C9M+i1N18eVB3Qa1iEdhejsbfsOHAA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"76c50ab135ce6eb17bca528693895da2e7f4837d0d7f3fac8cf54fe04474fb09","last_reissued_at":"2026-05-18T00:14:03.357603Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:14:03.357603Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1806.01939","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":"2026-05-18T00:14:03Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"NEovkY8fH0rCUpCPL8bzoFUFkPI1jxLc2wxxYb8/r9YzZYz96dp1+ayz1y5AKGi6/DRP+l9bkDirXgAclu18Ag==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-21T12:52:44.162248Z"},"content_sha256":"d403c5c985a50e5141e4d96d0597d674421659ace3a5c6c3a5c7b03c5d4d5441","schema_version":"1.0","event_id":"sha256:d403c5c985a50e5141e4d96d0597d674421659ace3a5c6c3a5c7b03c5d4d5441"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2018:O3CQVMJVZZXLC66KKKDJHCK5UL","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Stacks in Poisson Geometry","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.DG","authors_text":"Joel Villatoro","submitted_at":"2018-06-05T21:37:25Z","abstract_excerpt":"This thesis is divided into four chapters. The first chapter discusses the relationship between stacks on a site and groupoids internal to the site. It includes a rigorous proof of the folklore result that there is an equivalence between the bicategory of internal groupoids and the bicategory of geometric stacks. The second chapter discusses standard concepts in the theory of geometric stacks, including Morita equivalence, stack symmetries, and some Morita invariants. The third chapter introduces a new site of Dirac structures and provides a rigorous answer to the question: What is the stack a"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1806.01939","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":"2026-05-18T00:14:03Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"XAXMUpIh7AoZPreLU1o0hrh+r5vy6igx+tig+iIAisKyVBtdAZE4V10ZVvbvx6YC4HTjeWtYCA7IaPMvOpUYDA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-21T12:52:44.162755Z"},"content_sha256":"23c9ec3c9462340f3e5effce36fef85c787c7e5d6334b9ebe5d8af928b31d30c","schema_version":"1.0","event_id":"sha256:23c9ec3c9462340f3e5effce36fef85c787c7e5d6334b9ebe5d8af928b31d30c"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/O3CQVMJVZZXLC66KKKDJHCK5UL/bundle.json","state_url":"https://pith.science/pith/O3CQVMJVZZXLC66KKKDJHCK5UL/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/O3CQVMJVZZXLC66KKKDJHCK5UL/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":"2026-08-21T12:52:44Z","links":{"resolver":"https://pith.science/pith/O3CQVMJVZZXLC66KKKDJHCK5UL","bundle":"https://pith.science/pith/O3CQVMJVZZXLC66KKKDJHCK5UL/bundle.json","state":"https://pith.science/pith/O3CQVMJVZZXLC66KKKDJHCK5UL/state.json","well_known_bundle":"https://pith.science/.well-known/pith/O3CQVMJVZZXLC66KKKDJHCK5UL/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2018:O3CQVMJVZZXLC66KKKDJHCK5UL","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":"b98a2d2be72f5402e31550d632681a32217bbe2fb213acfb080c50c371f68cf2","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DG","submitted_at":"2018-06-05T21:37:25Z","title_canon_sha256":"e63adff64546ea0455928c4a091b1fe385e0a3b75f7d4205c7c598e16ea5acbc"},"schema_version":"1.0","source":{"id":"1806.01939","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1806.01939","created_at":"2026-05-18T00:14:03Z"},{"alias_kind":"arxiv_version","alias_value":"1806.01939v1","created_at":"2026-05-18T00:14:03Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1806.01939","created_at":"2026-05-18T00:14:03Z"},{"alias_kind":"pith_short_12","alias_value":"O3CQVMJVZZXL","created_at":"2026-05-18T12:32:40Z"},{"alias_kind":"pith_short_16","alias_value":"O3CQVMJVZZXLC66K","created_at":"2026-05-18T12:32:40Z"},{"alias_kind":"pith_short_8","alias_value":"O3CQVMJV","created_at":"2026-05-18T12:32:40Z"}],"graph_snapshots":[{"event_id":"sha256:23c9ec3c9462340f3e5effce36fef85c787c7e5d6334b9ebe5d8af928b31d30c","target":"graph","created_at":"2026-05-18T00:14:03Z","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":"This thesis is divided into four chapters. The first chapter discusses the relationship between stacks on a site and groupoids internal to the site. It includes a rigorous proof of the folklore result that there is an equivalence between the bicategory of internal groupoids and the bicategory of geometric stacks. The second chapter discusses standard concepts in the theory of geometric stacks, including Morita equivalence, stack symmetries, and some Morita invariants. The third chapter introduces a new site of Dirac structures and provides a rigorous answer to the question: What is the stack a","authors_text":"Joel Villatoro","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DG","submitted_at":"2018-06-05T21:37:25Z","title":"Stacks in Poisson Geometry"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1806.01939","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:d403c5c985a50e5141e4d96d0597d674421659ace3a5c6c3a5c7b03c5d4d5441","target":"record","created_at":"2026-05-18T00:14:03Z","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":"b98a2d2be72f5402e31550d632681a32217bbe2fb213acfb080c50c371f68cf2","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DG","submitted_at":"2018-06-05T21:37:25Z","title_canon_sha256":"e63adff64546ea0455928c4a091b1fe385e0a3b75f7d4205c7c598e16ea5acbc"},"schema_version":"1.0","source":{"id":"1806.01939","kind":"arxiv","version":1}},"canonical_sha256":"76c50ab135ce6eb17bca528693895da2e7f4837d0d7f3fac8cf54fe04474fb09","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"76c50ab135ce6eb17bca528693895da2e7f4837d0d7f3fac8cf54fe04474fb09","first_computed_at":"2026-05-18T00:14:03.357603Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T00:14:03.357603Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"x8SN94MhZkRnQGOHRhzn9FjuS4jC9/4Xf9N8U2J6cInalR+sj8moJxt9C9M+i1N18eVB3Qa1iEdhejsbfsOHAA==","signature_status":"signed_v1","signed_at":"2026-05-18T00:14:03.358243Z","signed_message":"canonical_sha256_bytes"},"source_id":"1806.01939","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:d403c5c985a50e5141e4d96d0597d674421659ace3a5c6c3a5c7b03c5d4d5441","sha256:23c9ec3c9462340f3e5effce36fef85c787c7e5d6334b9ebe5d8af928b31d30c"],"state_sha256":"42963bd9cdcec4adfa1621e081c3e126d37dccbddaa474c9c2c01121016f0f11"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"Lo9iRY4EJjIwFta0hOY6M1tXwmwKloQe6IVc/p8XcG4DKKqn9FSu3ZKqI6PCaThhNKYP0MfG9iJdxI+BAh6lDQ==","signed_message":"bundle_sha256_bytes","signed_at":"2026-08-21T12:52:44.166706Z","bundle_sha256":"a468f225e4adc9fd3ecfe1779f8f17ca4d35d8886890f714d9d69a898c06a20d"}}