{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2024:VOG2CEI7TMNP3UP576F2KFW7EX","short_pith_number":"pith:VOG2CEI7","canonical_record":{"source":{"id":"2401.04314","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AG","submitted_at":"2024-01-09T02:06:06Z","cross_cats_sorted":[],"title_canon_sha256":"4973a4b6c9b7b8f54423230e8f849ea6bcbb07ec0e1fbb2e0681c4c593e9b1d2","abstract_canon_sha256":"cadcc99df576d18c52230d0de4b430d420ee71cea0d4d0e5925e731d1abb60a5"},"schema_version":"1.0"},"canonical_sha256":"ab8da1111f9b1afdd1fdff8ba516df25cb15db1b2c9409c09a2a1f87511fefad","source":{"kind":"arxiv","id":"2401.04314","version":3},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2401.04314","created_at":"2026-07-05T11:04:35Z"},{"alias_kind":"arxiv_version","alias_value":"2401.04314v3","created_at":"2026-07-05T11:04:35Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2401.04314","created_at":"2026-07-05T11:04:35Z"},{"alias_kind":"pith_short_12","alias_value":"VOG2CEI7TMNP","created_at":"2026-07-05T11:04:35Z"},{"alias_kind":"pith_short_16","alias_value":"VOG2CEI7TMNP3UP5","created_at":"2026-07-05T11:04:35Z"},{"alias_kind":"pith_short_8","alias_value":"VOG2CEI7","created_at":"2026-07-05T11:04:35Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2024:VOG2CEI7TMNP3UP576F2KFW7EX","target":"record","payload":{"canonical_record":{"source":{"id":"2401.04314","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AG","submitted_at":"2024-01-09T02:06:06Z","cross_cats_sorted":[],"title_canon_sha256":"4973a4b6c9b7b8f54423230e8f849ea6bcbb07ec0e1fbb2e0681c4c593e9b1d2","abstract_canon_sha256":"cadcc99df576d18c52230d0de4b430d420ee71cea0d4d0e5925e731d1abb60a5"},"schema_version":"1.0"},"canonical_sha256":"ab8da1111f9b1afdd1fdff8ba516df25cb15db1b2c9409c09a2a1f87511fefad","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T11:04:35.426403Z","signature_b64":"trmgVbtRfzDMRE2sGozTYn4oCs7iN4DfhjGwNWqcuJlHcubZV+awiABwJ3IjEHWYECDeN9UEqlvxqIFvCKS+DA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"ab8da1111f9b1afdd1fdff8ba516df25cb15db1b2c9409c09a2a1f87511fefad","last_reissued_at":"2026-07-05T11:04:35.425931Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T11:04:35.425931Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"2401.04314","source_version":3,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-07-05T11:04:35Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"gRsmna59PlnPuhn4FLArWgn5pNNEfxsrC5yldqJnZR5OCzFvrSYPiRTIkoBVyQ+HMKBvbNQQr2eeIccSe+WkCw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-20T06:39:57.910325Z"},"content_sha256":"00d223d1a43d971f9cb6a9d1b8a68a8d6a98158130bf76854b48f71fbabdb633","schema_version":"1.0","event_id":"sha256:00d223d1a43d971f9cb6a9d1b8a68a8d6a98158130bf76854b48f71fbabdb633"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2024:VOG2CEI7TMNP3UP576F2KFW7EX","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"The affine Grassmannian as a presheaf quotient","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AG","authors_text":"Kestutis Cesnavicius","submitted_at":"2024-01-09T02:06:06Z","abstract_excerpt":"For a reductive group $G$ over a ring $A$, its affine Grassmannian $\\mathrm{Gr}_G$ plays important roles in a wide range of subjects and is typically defined as the \\'etale sheafification of the presheaf quotient $LG/L^+G$ of the loop group $LG$ by its positive loop subgroup $L^+G$. We show that the Zariski sheafification gives the same result. Moreover, for totally isotropic $G$ (for instance, for quasi-split $G$), we show that no sheafification is needed at all: $\\mathrm{Gr}_G$ is already the presheaf quotient $LG/L^+G$, which seems new already in the classical case of $G$ over $\\mathbb{C}$."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2401.04314","kind":"arxiv","version":3},"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/2401.04314/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":"2026-07-05T11:04:35Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"PzskLQUnS2Us+d8Sz+yvHO30dulJ9Nhc+L1lrCazq56iJFtsSBz97CQ2d4j7i03Qhl/4ImnImF9DBL2O8uKVCg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-20T06:39:57.910898Z"},"content_sha256":"a5690aab56a257cd918f1c8e81a9d3e04ffd8b7f67d60ddbce2912b5c015838e","schema_version":"1.0","event_id":"sha256:a5690aab56a257cd918f1c8e81a9d3e04ffd8b7f67d60ddbce2912b5c015838e"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/VOG2CEI7TMNP3UP576F2KFW7EX/bundle.json","state_url":"https://pith.science/pith/VOG2CEI7TMNP3UP576F2KFW7EX/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/VOG2CEI7TMNP3UP576F2KFW7EX/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":"2026-08-20T06:39:57Z","links":{"resolver":"https://pith.science/pith/VOG2CEI7TMNP3UP576F2KFW7EX","bundle":"https://pith.science/pith/VOG2CEI7TMNP3UP576F2KFW7EX/bundle.json","state":"https://pith.science/pith/VOG2CEI7TMNP3UP576F2KFW7EX/state.json","well_known_bundle":"https://pith.science/.well-known/pith/VOG2CEI7TMNP3UP576F2KFW7EX/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2024:VOG2CEI7TMNP3UP576F2KFW7EX","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":"cadcc99df576d18c52230d0de4b430d420ee71cea0d4d0e5925e731d1abb60a5","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AG","submitted_at":"2024-01-09T02:06:06Z","title_canon_sha256":"4973a4b6c9b7b8f54423230e8f849ea6bcbb07ec0e1fbb2e0681c4c593e9b1d2"},"schema_version":"1.0","source":{"id":"2401.04314","kind":"arxiv","version":3}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2401.04314","created_at":"2026-07-05T11:04:35Z"},{"alias_kind":"arxiv_version","alias_value":"2401.04314v3","created_at":"2026-07-05T11:04:35Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2401.04314","created_at":"2026-07-05T11:04:35Z"},{"alias_kind":"pith_short_12","alias_value":"VOG2CEI7TMNP","created_at":"2026-07-05T11:04:35Z"},{"alias_kind":"pith_short_16","alias_value":"VOG2CEI7TMNP3UP5","created_at":"2026-07-05T11:04:35Z"},{"alias_kind":"pith_short_8","alias_value":"VOG2CEI7","created_at":"2026-07-05T11:04:35Z"}],"graph_snapshots":[{"event_id":"sha256:a5690aab56a257cd918f1c8e81a9d3e04ffd8b7f67d60ddbce2912b5c015838e","target":"graph","created_at":"2026-07-05T11:04:35Z","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/2401.04314/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"For a reductive group $G$ over a ring $A$, its affine Grassmannian $\\mathrm{Gr}_G$ plays important roles in a wide range of subjects and is typically defined as the \\'etale sheafification of the presheaf quotient $LG/L^+G$ of the loop group $LG$ by its positive loop subgroup $L^+G$. We show that the Zariski sheafification gives the same result. Moreover, for totally isotropic $G$ (for instance, for quasi-split $G$), we show that no sheafification is needed at all: $\\mathrm{Gr}_G$ is already the presheaf quotient $LG/L^+G$, which seems new already in the classical case of $G$ over $\\mathbb{C}$.","authors_text":"Kestutis Cesnavicius","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AG","submitted_at":"2024-01-09T02:06:06Z","title":"The affine Grassmannian as a presheaf quotient"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2401.04314","kind":"arxiv","version":3},"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":"sha256:00d223d1a43d971f9cb6a9d1b8a68a8d6a98158130bf76854b48f71fbabdb633","target":"record","created_at":"2026-07-05T11:04:35Z","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":"cadcc99df576d18c52230d0de4b430d420ee71cea0d4d0e5925e731d1abb60a5","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AG","submitted_at":"2024-01-09T02:06:06Z","title_canon_sha256":"4973a4b6c9b7b8f54423230e8f849ea6bcbb07ec0e1fbb2e0681c4c593e9b1d2"},"schema_version":"1.0","source":{"id":"2401.04314","kind":"arxiv","version":3}},"canonical_sha256":"ab8da1111f9b1afdd1fdff8ba516df25cb15db1b2c9409c09a2a1f87511fefad","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"ab8da1111f9b1afdd1fdff8ba516df25cb15db1b2c9409c09a2a1f87511fefad","first_computed_at":"2026-07-05T11:04:35.425931Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-07-05T11:04:35.425931Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"trmgVbtRfzDMRE2sGozTYn4oCs7iN4DfhjGwNWqcuJlHcubZV+awiABwJ3IjEHWYECDeN9UEqlvxqIFvCKS+DA==","signature_status":"signed_v1","signed_at":"2026-07-05T11:04:35.426403Z","signed_message":"canonical_sha256_bytes"},"source_id":"2401.04314","source_kind":"arxiv","source_version":3}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:00d223d1a43d971f9cb6a9d1b8a68a8d6a98158130bf76854b48f71fbabdb633","sha256:a5690aab56a257cd918f1c8e81a9d3e04ffd8b7f67d60ddbce2912b5c015838e"],"state_sha256":"ff592b2b5748518ee4ff3ee8c18e458f09c5aa4ee0b381e48fd81a69b6375288"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"63ilkhZx+enUdgKeLa2tThFuk3wAAiEnNc1kaTrnl1z16hFzQbODDooRZpl7HUjrdVhzjYpXGiYUEzF6wiWJCA==","signed_message":"bundle_sha256_bytes","signed_at":"2026-08-20T06:39:57.915952Z","bundle_sha256":"fe43aa062b2b9fb79d573719ee560db86bb5b83a23cfe7911513d599a8a1a433"}}