{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2016:XED3J2VEHSPI4ANHDVKBCEYUKR","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":"e23ff7d317ce03e2edded85fc482c3e86a46867bfdf1716b26950ba49eb9f2ca","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CT","submitted_at":"2016-06-29T01:13:47Z","title_canon_sha256":"f77eb7e7bf6aeaf9e4371878c0b1ca4239e2f4da16eac694abbb86b6091ed81d"},"schema_version":"1.0","source":{"id":"1606.08930","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1606.08930","created_at":"2026-05-18T00:14:41Z"},{"alias_kind":"arxiv_version","alias_value":"1606.08930v2","created_at":"2026-05-18T00:14:41Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1606.08930","created_at":"2026-05-18T00:14:41Z"},{"alias_kind":"pith_short_12","alias_value":"XED3J2VEHSPI","created_at":"2026-05-18T12:30:51Z"},{"alias_kind":"pith_short_16","alias_value":"XED3J2VEHSPI4ANH","created_at":"2026-05-18T12:30:51Z"},{"alias_kind":"pith_short_8","alias_value":"XED3J2VE","created_at":"2026-05-18T12:30:51Z"}],"graph_snapshots":[{"event_id":"sha256:764fd54ed36e8f250ec35b59a83e5e03fbca94bf92704825f67e002242328a5a","target":"graph","created_at":"2026-05-18T00:14:41Z","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":"It is well known that a relation $\\varphi$ between sets is regular if, and only if, $\\mathcal{K}\\varphi$ is completely distributive (cd), where $\\mathcal{K}\\varphi$ is the complete lattice consisting of fixed points of the Kan adjunction induced by $\\varphi$. For a small quantaloid $\\mathcal{Q}$, we investigate the $\\mathcal{Q}$-enriched version of this classical result, i.e., the regularity of $\\mathcal{Q}$-distributors versus the constructive complete distributivity (ccd) of $\\mathcal{Q}$-categories, and prove that \"the dual of $\\mathcal{K}\\varphi$ is (ccd) $\\implies$ $\\varphi$ is regular $\\","authors_text":"Hongliang Lai, Lili Shen","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CT","submitted_at":"2016-06-29T01:13:47Z","title":"Regularity vs. constructive complete (co)distributivity"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1606.08930","kind":"arxiv","version":2},"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:d2376dd774d0b448bc0526422cea8cc4f4995675ca4378b0de0cf47213228b4e","target":"record","created_at":"2026-05-18T00:14:41Z","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":"e23ff7d317ce03e2edded85fc482c3e86a46867bfdf1716b26950ba49eb9f2ca","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CT","submitted_at":"2016-06-29T01:13:47Z","title_canon_sha256":"f77eb7e7bf6aeaf9e4371878c0b1ca4239e2f4da16eac694abbb86b6091ed81d"},"schema_version":"1.0","source":{"id":"1606.08930","kind":"arxiv","version":2}},"canonical_sha256":"b907b4eaa43c9e8e01a71d5411131454591793bd23ffccba7870550b97f1d226","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"b907b4eaa43c9e8e01a71d5411131454591793bd23ffccba7870550b97f1d226","first_computed_at":"2026-05-18T00:14:41.495430Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T00:14:41.495430Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"N9rZDWWApGOWzLBxgd5G/cIw5WtaNXI/zRh/atokeFzrmaO7DxUHfoPooLboF1nFo920DoucyYEfvmwE74hDAA==","signature_status":"signed_v1","signed_at":"2026-05-18T00:14:41.496275Z","signed_message":"canonical_sha256_bytes"},"source_id":"1606.08930","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:d2376dd774d0b448bc0526422cea8cc4f4995675ca4378b0de0cf47213228b4e","sha256:764fd54ed36e8f250ec35b59a83e5e03fbca94bf92704825f67e002242328a5a"],"state_sha256":"c85c97a68296ab90156150aa91b4f0fe3fd5c63a5533ebb4311351bdb12ba12f"}