{"paper":{"title":"Effectivity of Generalized Double $\\infty$-Categories","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CT","authors_text":"F\\'elix Loubaton","submitted_at":"2025-03-25T01:01:30Z","abstract_excerpt":"We construct an adjunction between $m$-categories internal to $(\\infty,n)$-categories, called $(n,m)$-double $\\infty$-categories, and filtrations $A_0\\to \\dots\\to A_m$ where for all $i<m$, $A_i$ is a $(n+i)$-category. We show that this adjunction induces an equivalence between $(n,m)$-double $\\infty$-categories admitting enough companions and filtrations such that each morphism $A_i\\to A_{i+1}$ is essentially surjective on cells of dimension lower than or equal to $i$.\n  This result can be seen as a $(\\infty,n)$-categorical generalization of the equivalence between internal groupoids and effec"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2503.19242","kind":"arxiv","version":1},"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/2503.19242/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"}