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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"},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"2503.19242","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CT","submitted_at":"2025-03-25T01:01:30Z","cross_cats_sorted":[],"title_canon_sha256":"35b1b5a123e340b60420909a133725503ed57b5029d1fc4162cd3ff01cf89b8e","abstract_canon_sha256":"b00c09afc3167a39be0855f5b355244d4f38082d6d76aa42506dc313ba501c03"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T10:38:51.661828Z","signature_b64":"ce+6nLWV4DgniqpCJMw5xxn7qHhzOFsckxqN7hbuXWsILuFZGdlbxWzZLUjahSh5Vqel5J/UBsDipnfckVSKBA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"2cb97f3581562cb7b2d736da8b5de4455109b563987cd9a89cecd8802bab944c","last_reissued_at":"2026-07-05T10:38:51.661263Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T10:38:51.661263Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"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. 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