{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2011:QVJEW54IX4FHKYOGHDPY3PXUOU","short_pith_number":"pith:QVJEW54I","canonical_record":{"source":{"id":"1101.3219","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.FA","submitted_at":"2011-01-17T14:08:17Z","cross_cats_sorted":["math.CT"],"title_canon_sha256":"63104e8ec918e993f493e732d6a768fdd49a849cad275405d9896b107311278b","abstract_canon_sha256":"4a22c7aba594d1df165d0d7161d8e6154d68ae76732d756d74f037d5f2d466a9"},"schema_version":"1.0"},"canonical_sha256":"85524b7788bf0a7561c638df8dbef4753f7fb03a56b7537e5c20b5eee83cd0de","source":{"kind":"arxiv","id":"1101.3219","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1101.3219","created_at":"2026-05-18T04:31:30Z"},{"alias_kind":"arxiv_version","alias_value":"1101.3219v1","created_at":"2026-05-18T04:31:30Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1101.3219","created_at":"2026-05-18T04:31:30Z"},{"alias_kind":"pith_short_12","alias_value":"QVJEW54IX4FH","created_at":"2026-05-18T12:26:39Z"},{"alias_kind":"pith_short_16","alias_value":"QVJEW54IX4FHKYOG","created_at":"2026-05-18T12:26:39Z"},{"alias_kind":"pith_short_8","alias_value":"QVJEW54I","created_at":"2026-05-18T12:26:39Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2011:QVJEW54IX4FHKYOGHDPY3PXUOU","target":"record","payload":{"canonical_record":{"source":{"id":"1101.3219","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.FA","submitted_at":"2011-01-17T14:08:17Z","cross_cats_sorted":["math.CT"],"title_canon_sha256":"63104e8ec918e993f493e732d6a768fdd49a849cad275405d9896b107311278b","abstract_canon_sha256":"4a22c7aba594d1df165d0d7161d8e6154d68ae76732d756d74f037d5f2d466a9"},"schema_version":"1.0"},"canonical_sha256":"85524b7788bf0a7561c638df8dbef4753f7fb03a56b7537e5c20b5eee83cd0de","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T04:31:30.315614Z","signature_b64":"qNksIPma99e7HCyHKv0Q9+v+zvAbv88XSn/AIxXQR8L4tn9DI5rRugGsKJO9Y6sbxsRl8FyQjQdn4pdvR8hvAg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"85524b7788bf0a7561c638df8dbef4753f7fb03a56b7537e5c20b5eee83cd0de","last_reissued_at":"2026-05-18T04:31:30.314960Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T04:31:30.314960Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1101.3219","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":"2026-05-18T04:31:30Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"PE/BfBC7RGRQAspAObxcUrBz7NoxnwWWhnYzfbBFn4qf2fSrHslGOKdhQF3NBSkGPzN0kauzMov7R+R6bN6mBg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-04T22:20:10.654243Z"},"content_sha256":"ba215e8ab14079107dfa677c986f0ac6bf6fc3daca47e2c9227c17786cac2f14","schema_version":"1.0","event_id":"sha256:ba215e8ab14079107dfa677c986f0ac6bf6fc3daca47e2c9227c17786cac2f14"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2011:QVJEW54IX4FHKYOGHDPY3PXUOU","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Weak Pullbacks of Topological Groupoids","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.CT"],"primary_cat":"math.FA","authors_text":"Aviv Censor, Daniele Grandini","submitted_at":"2011-01-17T14:08:17Z","abstract_excerpt":"We introduce the category HG, whose objects are topological groupoids endowed with compatible measure theoretic data: a Haar system and a measure on the unit space. We then define and study the notion of weak-pullback in the category of topological groupoids, and subsequently in HG. The category HG is the setting for topological groupoidification, which we present in separate papers, and in which the weak pullback is a key ingredient."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1101.3219","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":"2026-05-18T04:31:30Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"2o0KLeUBz6O8IytSFgpLb7vpa5hoXVpW32Uhy4055odXLgeqPcYkXpji/bD3i1ROGPGaNB2XCVMdbvHMwMCoDw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-04T22:20:10.654946Z"},"content_sha256":"6d7e6a27d0990cb74495d575268b0be8fafb019b7d01fb7d184ec008ede11ce6","schema_version":"1.0","event_id":"sha256:6d7e6a27d0990cb74495d575268b0be8fafb019b7d01fb7d184ec008ede11ce6"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/QVJEW54IX4FHKYOGHDPY3PXUOU/bundle.json","state_url":"https://pith.science/pith/QVJEW54IX4FHKYOGHDPY3PXUOU/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/QVJEW54IX4FHKYOGHDPY3PXUOU/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":"2026-06-04T22:20:10Z","links":{"resolver":"https://pith.science/pith/QVJEW54IX4FHKYOGHDPY3PXUOU","bundle":"https://pith.science/pith/QVJEW54IX4FHKYOGHDPY3PXUOU/bundle.json","state":"https://pith.science/pith/QVJEW54IX4FHKYOGHDPY3PXUOU/state.json","well_known_bundle":"https://pith.science/.well-known/pith/QVJEW54IX4FHKYOGHDPY3PXUOU/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2011:QVJEW54IX4FHKYOGHDPY3PXUOU","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":"4a22c7aba594d1df165d0d7161d8e6154d68ae76732d756d74f037d5f2d466a9","cross_cats_sorted":["math.CT"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.FA","submitted_at":"2011-01-17T14:08:17Z","title_canon_sha256":"63104e8ec918e993f493e732d6a768fdd49a849cad275405d9896b107311278b"},"schema_version":"1.0","source":{"id":"1101.3219","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1101.3219","created_at":"2026-05-18T04:31:30Z"},{"alias_kind":"arxiv_version","alias_value":"1101.3219v1","created_at":"2026-05-18T04:31:30Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1101.3219","created_at":"2026-05-18T04:31:30Z"},{"alias_kind":"pith_short_12","alias_value":"QVJEW54IX4FH","created_at":"2026-05-18T12:26:39Z"},{"alias_kind":"pith_short_16","alias_value":"QVJEW54IX4FHKYOG","created_at":"2026-05-18T12:26:39Z"},{"alias_kind":"pith_short_8","alias_value":"QVJEW54I","created_at":"2026-05-18T12:26:39Z"}],"graph_snapshots":[{"event_id":"sha256:6d7e6a27d0990cb74495d575268b0be8fafb019b7d01fb7d184ec008ede11ce6","target":"graph","created_at":"2026-05-18T04:31:30Z","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 introduce the category HG, whose objects are topological groupoids endowed with compatible measure theoretic data: a Haar system and a measure on the unit space. We then define and study the notion of weak-pullback in the category of topological groupoids, and subsequently in HG. The category HG is the setting for topological groupoidification, which we present in separate papers, and in which the weak pullback is a key ingredient.","authors_text":"Aviv Censor, Daniele Grandini","cross_cats":["math.CT"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.FA","submitted_at":"2011-01-17T14:08:17Z","title":"Weak Pullbacks of Topological Groupoids"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1101.3219","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:ba215e8ab14079107dfa677c986f0ac6bf6fc3daca47e2c9227c17786cac2f14","target":"record","created_at":"2026-05-18T04:31:30Z","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":"4a22c7aba594d1df165d0d7161d8e6154d68ae76732d756d74f037d5f2d466a9","cross_cats_sorted":["math.CT"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.FA","submitted_at":"2011-01-17T14:08:17Z","title_canon_sha256":"63104e8ec918e993f493e732d6a768fdd49a849cad275405d9896b107311278b"},"schema_version":"1.0","source":{"id":"1101.3219","kind":"arxiv","version":1}},"canonical_sha256":"85524b7788bf0a7561c638df8dbef4753f7fb03a56b7537e5c20b5eee83cd0de","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"85524b7788bf0a7561c638df8dbef4753f7fb03a56b7537e5c20b5eee83cd0de","first_computed_at":"2026-05-18T04:31:30.314960Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T04:31:30.314960Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"qNksIPma99e7HCyHKv0Q9+v+zvAbv88XSn/AIxXQR8L4tn9DI5rRugGsKJO9Y6sbxsRl8FyQjQdn4pdvR8hvAg==","signature_status":"signed_v1","signed_at":"2026-05-18T04:31:30.315614Z","signed_message":"canonical_sha256_bytes"},"source_id":"1101.3219","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:ba215e8ab14079107dfa677c986f0ac6bf6fc3daca47e2c9227c17786cac2f14","sha256:6d7e6a27d0990cb74495d575268b0be8fafb019b7d01fb7d184ec008ede11ce6"],"state_sha256":"9c11beaf2dfd2f63bf80a163049366f6b2345f816fe9882587ea23cd2fbff4a3"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"xkv4PoYKquAUqRncJbrDEx/p1TsG8pRX/92kHRE2OJ80K7kJppTgyixNAQdQQlYPS2bApJiyVJwJLajOJe3cCw==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-04T22:20:10.658615Z","bundle_sha256":"5757069ee6e53016a7d8c74dcc84de50a6c4997b1c293587599d254576baa2d3"}}