{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2019:B57M7RUOKB7V3MONGQEPCV2B2F","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":"ed258143475a8d0849da5b40f3bad5509ba2a06e536924e33d92ad0c40657b01","cross_cats_sorted":["math.AT","math.CT"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AG","submitted_at":"2019-10-31T16:53:43Z","title_canon_sha256":"099c9d0491a2e267ed377ab15d62a2a8ac2d8cf0295b5fb0368096b9087a2fef"},"schema_version":"1.0","source":{"id":"1910.14602","kind":"arxiv","version":5}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1910.14602","created_at":"2026-07-05T07:11:03Z"},{"alias_kind":"arxiv_version","alias_value":"1910.14602v5","created_at":"2026-07-05T07:11:03Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1910.14602","created_at":"2026-07-05T07:11:03Z"},{"alias_kind":"pith_short_12","alias_value":"B57M7RUOKB7V","created_at":"2026-07-05T07:11:03Z"},{"alias_kind":"pith_short_16","alias_value":"B57M7RUOKB7V3MON","created_at":"2026-07-05T07:11:03Z"},{"alias_kind":"pith_short_8","alias_value":"B57M7RUO","created_at":"2026-07-05T07:11:03Z"}],"graph_snapshots":[{"event_id":"sha256:3f24eddf482cf343df69a16a588ef11d5a7168ab6f06939c58028d8cbfa12ed3","target":"graph","created_at":"2026-07-05T07:11:03Z","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"},"integrity":{"available":true,"clean":true,"detectors_run":[],"endpoint":"/pith/1910.14602/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"We introduce a theory of stratifications of noncommutative stacks (i.e. presentable stable $\\infty$-categories), and we prove a reconstruction theorem that expresses them in terms of their strata and gluing data. This reconstruction theorem is compatible with symmetric monoidal structures, and with more general operadic structures such as $E_n$-monoidal structures. We also provide a suite of fundamental operations for constructing new stratifications from old ones: restriction, pullback, quotient, pushforward, and refinement. Moreover, we establish a dual form of reconstruction; this is closel","authors_text":"Aaron Mazel-Gee, David Ayala, Nick Rozenblyum","cross_cats":["math.AT","math.CT"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AG","submitted_at":"2019-10-31T16:53:43Z","title":"Stratified noncommutative geometry"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1910.14602","kind":"arxiv","version":5},"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":"sha256:2a2d94163e4243c5ec0ab931f87e707c1d530d621e039f1043df1d96d17c3f17","target":"record","created_at":"2026-07-05T07:11:03Z","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":"ed258143475a8d0849da5b40f3bad5509ba2a06e536924e33d92ad0c40657b01","cross_cats_sorted":["math.AT","math.CT"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AG","submitted_at":"2019-10-31T16:53:43Z","title_canon_sha256":"099c9d0491a2e267ed377ab15d62a2a8ac2d8cf0295b5fb0368096b9087a2fef"},"schema_version":"1.0","source":{"id":"1910.14602","kind":"arxiv","version":5}},"canonical_sha256":"0f7ecfc68e507f5db1cd3408f15741d174a99de3b0e403fda36de1daca765c74","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"0f7ecfc68e507f5db1cd3408f15741d174a99de3b0e403fda36de1daca765c74","first_computed_at":"2026-07-05T07:11:03.628606Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-07-05T07:11:03.628606Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"M7FWU1DdO1DXlnZzgUHFjwn5KoN9ZE8JlAZN258FvE0oDglf4sy6piZINl2Yxtb0TQEGcwV3yXODkgsS7Z5xAA==","signature_status":"signed_v1","signed_at":"2026-07-05T07:11:03.629134Z","signed_message":"canonical_sha256_bytes"},"source_id":"1910.14602","source_kind":"arxiv","source_version":5}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:2a2d94163e4243c5ec0ab931f87e707c1d530d621e039f1043df1d96d17c3f17","sha256:3f24eddf482cf343df69a16a588ef11d5a7168ab6f06939c58028d8cbfa12ed3"],"state_sha256":"b754ca72b74d8694f57ebb1d533a5f31eb42c1e7ed011a4e04a7c3784b4ffc25"}