{"paper":{"title":"Regularity vs. constructive complete (co)distributivity","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CT","authors_text":"Hongliang Lai, Lili Shen","submitted_at":"2016-06-29T01:13:47Z","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 $\\"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1606.08930","kind":"arxiv","version":2},"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"}