{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2011:ZMXKIDS5M3OX725TLTK3QLSMK3","short_pith_number":"pith:ZMXKIDS5","canonical_record":{"source":{"id":"1110.6836","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.KT","submitted_at":"2011-10-31T15:35:38Z","cross_cats_sorted":["math-ph","math.AT","math.MP","math.OA"],"title_canon_sha256":"3f252158a05647b1717965ecd4c8c879baf25cf966a23408bf90822010899d80","abstract_canon_sha256":"0c00fe00d8fc1c3e04b8c1276a31657f98658525d8e44919fe33935734186674"},"schema_version":"1.0"},"canonical_sha256":"cb2ea40e5d66dd7febb35cd5b82e4c56d6a62139c84e72180401276dc9c570a9","source":{"kind":"arxiv","id":"1110.6836","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1110.6836","created_at":"2026-05-18T04:09:54Z"},{"alias_kind":"arxiv_version","alias_value":"1110.6836v1","created_at":"2026-05-18T04:09:54Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1110.6836","created_at":"2026-05-18T04:09:54Z"},{"alias_kind":"pith_short_12","alias_value":"ZMXKIDS5M3OX","created_at":"2026-05-18T12:26:47Z"},{"alias_kind":"pith_short_16","alias_value":"ZMXKIDS5M3OX725T","created_at":"2026-05-18T12:26:47Z"},{"alias_kind":"pith_short_8","alias_value":"ZMXKIDS5","created_at":"2026-05-18T12:26:47Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2011:ZMXKIDS5M3OX725TLTK3QLSMK3","target":"record","payload":{"canonical_record":{"source":{"id":"1110.6836","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.KT","submitted_at":"2011-10-31T15:35:38Z","cross_cats_sorted":["math-ph","math.AT","math.MP","math.OA"],"title_canon_sha256":"3f252158a05647b1717965ecd4c8c879baf25cf966a23408bf90822010899d80","abstract_canon_sha256":"0c00fe00d8fc1c3e04b8c1276a31657f98658525d8e44919fe33935734186674"},"schema_version":"1.0"},"canonical_sha256":"cb2ea40e5d66dd7febb35cd5b82e4c56d6a62139c84e72180401276dc9c570a9","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T04:09:54.096039Z","signature_b64":"BdZtavcKS4C9JZw70TrKAnGdyAzmevsvAXgTaFG1JkGRUg7ymhxJMChk3zjwj/x8pWzxD2/62034dou2Z1JEBQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"cb2ea40e5d66dd7febb35cd5b82e4c56d6a62139c84e72180401276dc9c570a9","last_reissued_at":"2026-05-18T04:09:54.095496Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T04:09:54.095496Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1110.6836","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":"2026-05-18T04:09:54Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"8Qmp24xxGzi9bomfSSMgwNaouRQRc8WZljbhAkoBW+lEwURe9k+FosCh8ITvw85+DCgNs1hIKtXY7zlswrUcCw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-10T15:03:41.825182Z"},"content_sha256":"02ce17c9c81be35e0b1f7f92ee9e2db7251d1d0e471c619d70731959e26253d6","schema_version":"1.0","event_id":"sha256:02ce17c9c81be35e0b1f7f92ee9e2db7251d1d0e471c619d70731959e26253d6"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2011:ZMXKIDS5M3OX725TLTK3QLSMK3","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Twistings of KR for Real groupoids","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math-ph","math.AT","math.MP","math.OA"],"primary_cat":"math.KT","authors_text":"El-ka\\\"ioum M. Moutuou","submitted_at":"2011-10-31T15:35:38Z","abstract_excerpt":"B-fields over a groupoid with involution are defined as Real graded Dixmier-Douady bundles. We use these to introduce the Real graded Brauer group which constitutes the set of twistings for Atiyah's KR-functor in the category of locally compact groupoids with involutions. We interpret this group in terms of groupoid extensions and elements of some equivariant Cech cohomology theory. The construction of the twisted KR-functor is outlined."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1110.6836","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":"2026-05-18T04:09:54Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"6lulitxSmSGU6YOw7XOqYhHlrmVhrEQKf/qWN4HePWlfr/9LW32Ywd/spOPMzX1MuOMvbnf/MsNziAZ4qSTtBg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-10T15:03:41.825939Z"},"content_sha256":"da5bd88e97b8cef3f83245838131cdfa2c0b9e5aa42704522db41b46a87cdf23","schema_version":"1.0","event_id":"sha256:da5bd88e97b8cef3f83245838131cdfa2c0b9e5aa42704522db41b46a87cdf23"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/ZMXKIDS5M3OX725TLTK3QLSMK3/bundle.json","state_url":"https://pith.science/pith/ZMXKIDS5M3OX725TLTK3QLSMK3/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/ZMXKIDS5M3OX725TLTK3QLSMK3/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":"2026-06-10T15:03:41Z","links":{"resolver":"https://pith.science/pith/ZMXKIDS5M3OX725TLTK3QLSMK3","bundle":"https://pith.science/pith/ZMXKIDS5M3OX725TLTK3QLSMK3/bundle.json","state":"https://pith.science/pith/ZMXKIDS5M3OX725TLTK3QLSMK3/state.json","well_known_bundle":"https://pith.science/.well-known/pith/ZMXKIDS5M3OX725TLTK3QLSMK3/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2011:ZMXKIDS5M3OX725TLTK3QLSMK3","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":"0c00fe00d8fc1c3e04b8c1276a31657f98658525d8e44919fe33935734186674","cross_cats_sorted":["math-ph","math.AT","math.MP","math.OA"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.KT","submitted_at":"2011-10-31T15:35:38Z","title_canon_sha256":"3f252158a05647b1717965ecd4c8c879baf25cf966a23408bf90822010899d80"},"schema_version":"1.0","source":{"id":"1110.6836","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1110.6836","created_at":"2026-05-18T04:09:54Z"},{"alias_kind":"arxiv_version","alias_value":"1110.6836v1","created_at":"2026-05-18T04:09:54Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1110.6836","created_at":"2026-05-18T04:09:54Z"},{"alias_kind":"pith_short_12","alias_value":"ZMXKIDS5M3OX","created_at":"2026-05-18T12:26:47Z"},{"alias_kind":"pith_short_16","alias_value":"ZMXKIDS5M3OX725T","created_at":"2026-05-18T12:26:47Z"},{"alias_kind":"pith_short_8","alias_value":"ZMXKIDS5","created_at":"2026-05-18T12:26:47Z"}],"graph_snapshots":[{"event_id":"sha256:da5bd88e97b8cef3f83245838131cdfa2c0b9e5aa42704522db41b46a87cdf23","target":"graph","created_at":"2026-05-18T04:09:54Z","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":"B-fields over a groupoid with involution are defined as Real graded Dixmier-Douady bundles. We use these to introduce the Real graded Brauer group which constitutes the set of twistings for Atiyah's KR-functor in the category of locally compact groupoids with involutions. We interpret this group in terms of groupoid extensions and elements of some equivariant Cech cohomology theory. The construction of the twisted KR-functor is outlined.","authors_text":"El-ka\\\"ioum M. Moutuou","cross_cats":["math-ph","math.AT","math.MP","math.OA"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.KT","submitted_at":"2011-10-31T15:35:38Z","title":"Twistings of KR for Real groupoids"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1110.6836","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":"sha256:02ce17c9c81be35e0b1f7f92ee9e2db7251d1d0e471c619d70731959e26253d6","target":"record","created_at":"2026-05-18T04:09:54Z","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":"0c00fe00d8fc1c3e04b8c1276a31657f98658525d8e44919fe33935734186674","cross_cats_sorted":["math-ph","math.AT","math.MP","math.OA"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.KT","submitted_at":"2011-10-31T15:35:38Z","title_canon_sha256":"3f252158a05647b1717965ecd4c8c879baf25cf966a23408bf90822010899d80"},"schema_version":"1.0","source":{"id":"1110.6836","kind":"arxiv","version":1}},"canonical_sha256":"cb2ea40e5d66dd7febb35cd5b82e4c56d6a62139c84e72180401276dc9c570a9","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"cb2ea40e5d66dd7febb35cd5b82e4c56d6a62139c84e72180401276dc9c570a9","first_computed_at":"2026-05-18T04:09:54.095496Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T04:09:54.095496Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"BdZtavcKS4C9JZw70TrKAnGdyAzmevsvAXgTaFG1JkGRUg7ymhxJMChk3zjwj/x8pWzxD2/62034dou2Z1JEBQ==","signature_status":"signed_v1","signed_at":"2026-05-18T04:09:54.096039Z","signed_message":"canonical_sha256_bytes"},"source_id":"1110.6836","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:02ce17c9c81be35e0b1f7f92ee9e2db7251d1d0e471c619d70731959e26253d6","sha256:da5bd88e97b8cef3f83245838131cdfa2c0b9e5aa42704522db41b46a87cdf23"],"state_sha256":"66f7dd3ff8a27ea4535814be08b5c9070e3ec956e68ab24944ba47ddf09c6606"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"oxXyPLGrtcfCHoj+Z5/OMwhzKbmD1WHh+5jksn8CKK22Oy8yU52KJG63K8caIHid8Y3lePRuTYUMt9NeMoCyDg==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-10T15:03:41.830061Z","bundle_sha256":"dee2f3f2487653f52fe7c7dfc7d0b6e27cec93fabcb4b9aa1caf618658cab6f9"}}