{"paper":{"title":"Borel functors, interpretations, and strong conceptual completeness for $\\mathcal L_{\\omega_1\\omega}$","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.CT"],"primary_cat":"math.LO","authors_text":"Ruiyuan Chen","submitted_at":"2017-10-06T00:42:54Z","abstract_excerpt":"We prove a strong conceptual completeness theorem (in the sense of Makkai) for the infinitary logic $\\mathcal L_{\\omega_1\\omega}$: every countable $\\mathcal L_{\\omega_1\\omega}$-theory can be canonically recovered from its standard Borel groupoid of countable models, up to a suitable syntactical notion of equivalence. This implies that given two theories $(\\mathcal L, \\mathcal T)$ and $(\\mathcal L', \\mathcal T')$ (in possibly different languages $\\mathcal L, \\mathcal L'$), every Borel functor $\\mathsf{Mod}(\\mathcal L', \\mathcal T') \\to \\mathsf{Mod}(\\mathcal L, \\mathcal T)$ between the respectiv"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1710.02246","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":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/1710.02246/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"}