{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2026:2XF56672SVMPP4JE4THVYTB6TK","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":"d47a03f368e9dcfbb564ba42757c05667c34d7515b27e614df41ecefb9716729","cross_cats_sorted":[],"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.CT","submitted_at":"2026-04-24T09:02:20Z","title_canon_sha256":"4f91e134b763df988af0121cddea33a272ae16ebda6f7e3e58f7fa2373d0008e"},"schema_version":"1.0","source":{"id":"2604.22370","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2604.22370","created_at":"2026-07-28T02:23:30Z"},{"alias_kind":"arxiv_version","alias_value":"2604.22370v2","created_at":"2026-07-28T02:23:30Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2604.22370","created_at":"2026-07-28T02:23:30Z"},{"alias_kind":"pith_short_12","alias_value":"2XF56672SVMP","created_at":"2026-07-28T02:23:30Z"},{"alias_kind":"pith_short_16","alias_value":"2XF56672SVMPP4JE","created_at":"2026-07-28T02:23:30Z"},{"alias_kind":"pith_short_8","alias_value":"2XF56672","created_at":"2026-07-28T02:23:30Z"}],"graph_snapshots":[{"event_id":"sha256:4d52de231b0e0aef05243b0c91153f4af54d5664d46c3ab75075793314c908ad","target":"graph","created_at":"2026-07-28T02:23:30Z","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":4,"items":[{"attestation":"unclaimed","claim_id":"C1","kind":"strongest_claim","source":"verdict.strongest_claim","status":"machine_extracted","text":"We show that, in a virtual equipment satisfying mild assumptions, free cocompletions under classes of weights are exhibited by presheaf constructions."},{"attestation":"unclaimed","claim_id":"C2","kind":"weakest_assumption","source":"verdict.weakest_assumption","status":"machine_extracted","text":"The virtual equipment satisfies mild assumptions (not specified in detail in the abstract) that allow presheaf constructions to exhibit free cocompletions and the extension of weighted colimits."},{"attestation":"unclaimed","claim_id":"C3","kind":"one_line_summary","source":"verdict.one_line_summary","status":"machine_extracted","text":"In virtual equipments with mild assumptions, presheaf constructions provide free cocompletions under classes of weights, extending weighted colimits and enabling constructions for enriched categories in monoidal categories and bicategories."},{"attestation":"unclaimed","claim_id":"C4","kind":"headline","source":"verdict.pith_extraction.headline","status":"machine_extracted","text":"In virtual equipments satisfying mild assumptions, presheaf constructions exhibit free cocompletions under classes of weights."}],"snapshot_sha256":"c0ff4c0fa4560daaa7ca5e6a4d3f85ca09c93cef186a593314d846284aee314b"},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"integrity":{"available":true,"clean":true,"detectors_run":[{"findings_count":0,"name":"ai_meta_artifact","ran_at":"2026-05-21T10:41:04.698505Z","status":"completed","version":"1.0.0"},{"findings_count":0,"name":"doi_compliance","ran_at":"2026-05-20T00:04:32.800068Z","status":"completed","version":"1.0.0"}],"endpoint":"/pith/2604.22370/integrity.json","findings":[],"snapshot_sha256":"f056a0241febd2598886fb3e51efe73ef1d8f3e20acb9ac3d8e2f0ef77428b3b","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"We study the relationship between presheaf constructions and free cocompletions in the context of formal category theory, elucidating the coincidence between the two concepts in familiar settings. We show that, in a virtual equipment satisfying mild assumptions, free cocompletions under classes of weights are exhibited by presheaf constructions. We furthermore extend the theory of weighted colimits from enriched category theory to this setting, developing the concepts of atomicity and rank, and providing recognition theorems for presheaf objects, free cocompletions, and cocomplete objects. As ","authors_text":"Dylan McDermott, Nathanael Arkor","cross_cats":[],"headline":"In virtual equipments satisfying mild assumptions, presheaf constructions exhibit free cocompletions under classes of weights.","license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.CT","submitted_at":"2026-04-24T09:02:20Z","title":"Presheaves and cocompletions in formal category theory"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2604.22370","kind":"arxiv","version":2},"verdict":{"created_at":"2026-05-08T08:50:08.367805Z","id":"87ec0805-b762-45da-813d-0f1c0546e9aa","model_set":{"reader":"grok-4.3"},"one_line_summary":"In virtual equipments with mild assumptions, presheaf constructions provide free cocompletions under classes of weights, extending weighted colimits and enabling constructions for enriched categories in monoidal categories and bicategories.","pipeline_version":"pith-pipeline@v0.9.0","pith_extraction_headline":"In virtual equipments satisfying mild assumptions, presheaf constructions exhibit free cocompletions under classes of weights.","strongest_claim":"We show that, in a virtual equipment satisfying mild assumptions, free cocompletions under classes of weights are exhibited by presheaf constructions.","weakest_assumption":"The virtual equipment satisfies mild assumptions (not specified in detail in the abstract) that allow presheaf constructions to exhibit free cocompletions and the extension of weighted colimits."}},"verdict_id":"87ec0805-b762-45da-813d-0f1c0546e9aa"}}],"author_attestations":[],"timestamp_anchors":[],"storage_attestations":[],"citation_signatures":[],"replication_records":[],"corrections":[],"mirror_hints":[],"record_created":{"event_id":"sha256:a9b494b14a5c86f306630328c1825788f0d927fb77db3cdde343a1267e2e8034","target":"record","created_at":"2026-07-28T02:23:30Z","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":"d47a03f368e9dcfbb564ba42757c05667c34d7515b27e614df41ecefb9716729","cross_cats_sorted":[],"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.CT","submitted_at":"2026-04-24T09:02:20Z","title_canon_sha256":"4f91e134b763df988af0121cddea33a272ae16ebda6f7e3e58f7fa2373d0008e"},"schema_version":"1.0","source":{"id":"2604.22370","kind":"arxiv","version":2}},"canonical_sha256":"d5cbdf7bfa9558f7f124e4cf5c4c3e9a8c0a13d6d852b6594981286655f64825","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"d5cbdf7bfa9558f7f124e4cf5c4c3e9a8c0a13d6d852b6594981286655f64825","first_computed_at":"2026-07-28T02:23:30.954513Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-07-28T02:23:30.954513Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"6eRdK4sLE2toXs5Lgmmcb8N865xlW8ncNsh4LHbvjfsp0UtQ6//PSbgc3Tm5/87NYL4mgOD+Eut4NoyfH61FDg==","signature_status":"signed_v1","signed_at":"2026-07-28T02:23:30.955553Z","signed_message":"canonical_sha256_bytes"},"source_id":"2604.22370","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:a9b494b14a5c86f306630328c1825788f0d927fb77db3cdde343a1267e2e8034","sha256:4d52de231b0e0aef05243b0c91153f4af54d5664d46c3ab75075793314c908ad"],"state_sha256":"f7742d5adc25de6719c8bacaef9fd605275b3498ac0d266a800a4206e6f407ba"}